Over the past two decades, observational cosmology has made remarkable progress, significantly advancing our understanding of the universe. This progress has been driven by major surveys and telescope observations, including the Baryon Oscillation Spectroscopic Survey [1], the Planck Collaboration [2], and the Atacama Cosmology Telescope Polarimeter [3]. One of the most important milestones in modern cosmology was the discovery of the universe’s accelerated expansion, achieved through Type I a supernova observations conducted by the Supernova Cosmology Project and the High-Redshift Supernova Search Team [4]-[6], which provided the first direct evidence for cosmic acceleration. Further support came from measurements of the cosmic microwave background [7][8], studies of large-scale cosmic structure [9][10], and additional CMB analyses [11][12]. Observations strongly indicate that the universe transitioned from an early decelerating phase to a late-time accelerating phase; although some recent studies have questioned the robustness of this interpretation [13]. According to the Planck results [2], the energy content of the universe consists of approximately 5% baryonic matter, 26% dark matter, and 69% dark energy. Despite extensive research, the physical nature of dark energy remains one of the biggest unresolved problems in cosmology. Two main approaches have been adopted to address this issue: introducing exotic matter components such as quintessence [14], Chaplygin gas [15][16], polytropic fluids [17], phantom fields [18], tachyons [19], as well as the cosmological constant Λ, which suffers from fine-tuning and coincidence problems [20]. The second approach involves modifications of General Relativity, notably f ( R ) gravity, where the Ricci scalar in the Einstein-Hilbert action is replaced by a general function [21][22]. Further developments included coupling the matter Lagrangian with arbitrary f ( R ) models [23], and Lorentz-violating extensions [24]. A major breakthrough came with f ( R , T ) gravity, where the action depends on both the Ricci scalar and the trace of the energy-momentum tensor [25]. This theory enables matter-geometry coupling and provides a promising geometric explanation for late-time cosmic acceleration. Various cosmological investigations have supported the potential of f ( R , T ) theory, including anisotropic and bulk viscous models [26]-[29], observational reconstruction methods [30], and recent studies focusing on dark-energy evolution and cosmic transit behaviour [7][31][32]. Section 1 introduces the basic ideas of gravity and reviews earlier research. In Section 2, the concept of f ( R , T ) gravity has been introduced. In Section 3, the metric is defined, and the corresponding field equations are obtained. In Section 4, a time-varying deceleration parameter is considered and the field equation are solved. Important quantities such as energy density, viscous pressure, viscosity coefficient, matter trace, Ricci scalar, and energy conditions are derived and shown through graphs. In Section 5, the dynamical behaviour of the model is analyzed and physical parameters. While Section 6 a detailed discussion of the figures is presented. Finally, Section 7 presents the main conclusions of the study.

Assuming that the matter Lagrangian density L_m depends explicitly on the metric tensor, the corresponding field equations in f ( R , T ) gravity can be obtained through the application of the Hilbert-Einstein variational principle.

The standard matter Lagrangian density, denoted as L_m, corresponds to the matter source, while f ( R , T ) represents a general function of the Ricci scalar R and the trace T of the energy-momentum tensor T_ij associated with the matter source. Additionally, g stands for the determinant of the metric tensor g_ij. The energy-momentum tensor derived from the matter Lagrangian is defined as follows:

and its trace T = g i j T i j It is assumed that the matter Lagrangian L_m depends solely on the metric components g_ij and not on their derivatives. Consequently, one obtains

Varying the action S in Equation (1) with respect to g i j the f ( R , T ) metric tensor yields the field equations of gravity

where

Here, f R ( R , T ) = ∂ f R ( R , T ) ∂ R , f T = ∂ f R ( R , T ) ∂ T , □   ≡ ∇ i ∇ j where ∇ i is the covariant Derivative.

Using Equations (4) and (6), the field equation of f ( R , T ) gravity take the form:

It is important to note that the physical properties of the matter field, through the pressure components, influence the structure of the field equations in f ( R , T ) gravity. Based on the nature of the matter source, proposed three specific formulatis of f ( R , T ) gravity, which are outlined below

The respective

field equation corresponding to each frame in f ( R , T ) gravity are expressed as follow

In recent years, numerous researchers have developed cosmological models incorporating perfect fluid matter to investigate the universe’s accelerated expansion. Observational evidence strongly confirms this acceleration, attributed to an unknown energy component known as dark energy, characterized by negative pressure. However, increasing effort has been devoted to constructing cosmological models that explain the acceleration without invoking dark energy or dark matter, instead relying on realistic matter components. One promising approach is cosmic bulk viscosity, which can mimic the dynamical behaviour of dark energy by generating negative effective pressure [33][34], thereby driving late-time acceleration. Consequently, dissipative fluid matter is now considered a more realistic framework compared to perfect fluid or dust models for describing homogeneous and isotropic cosmology. It is widely believed that during the early universe, particularly near the neutrino decoupling epoch in the radiation-dominated era, the cosmic medium exhibited viscous behaviour [35]-[37]. Viscous cosmological models significantly modify singular be behaviour and provide explanations for high entropy production, phase transitions, and baryogenesis. It has further been shown that a scalar field interacting with viscous fluid can account for cosmic acceleration more effectively than when coupled with a perfect fluid alone [38]. As a result, viscous fluid cosmologies have attracted considerable attention in early-universe research [19][39][40]. Observations also investigate the homogeneity and isotropy of the universe. Although the post-inflationary universe is modeled using the FLRW metric, anomalies in CMB data from the Planck mission suggest an early anisotropic stage [41]. This motivates the use of Bianchi-type models, particularly Bianchi Type-I, the simplest anisotropic generalization of FLRW. Studies indicate that viscosity influences singularity behaviour but does not completely remove the initial singularity [42]. More recent research has focused on bulk viscous matter in Bianchi Type-I geometry within modified gravity frameworks such as, and theories, demonstrating strong compatibility with observational constraints on late-time acceleration [16][43]-[46]. The present work extends recent investigations on bulk viscous cosmology in modified gravity by introducing a variable deceleration parameter and studying the combined influence of anisotropy, viscosity, and matter-geometry coupling on the late-time accelerated expansion of the universe. Motivated by recent developments in modified gravity and viscous cosmology, in the present work we investigate an LRS Bianchi type-I cosmological model in the framework of f ( R , T ) gravity with bulk viscous matter. Unlike several earlier studies based on perfect fluid cosmology or constant deceleration parameters, the present model employs a time-dependent deceleration parameter capable of describing the transition from an early decelerating phase to the present accelerated phase of the universe. The combined effects of anisotropy, bulk viscosity, and matter-geometry coupling are analyzed through various cosmological and dynamical parameters, including the jerk and statefinder diagnostics.

We examine the spatially homogeneous LRS Bianchi type-I metric as follows

Here, A(t) and B(t) are assumed to be functions dependent only on the cosmic time t.

The energy-momentum tensor corresponding to a bulk viscous fluid is considered in the following form:

where u i = ( 0 , 0 , 0 , 1 ) is the four velocity vector in co-moving coordinate system satisfyin

The bulk viscous pressure satisfies the linear equation of state p = γ ρ , where 0 ≤ γ ≤ 1 . Here, ξ is the bulk viscosity coefficient, H is the Hubble parameter, p ¯ denotes the effective bulk viscous pressure, p denotes the pressure, and ρ represents the energy density.

The trace of the energy-momentum tensor is given by:

Considering the function f 1 ( R ) = μ T and f 2 ( T ) = μ T (where μ is an arbitrary constant), the field Equation (9) for the metric (10) becomes:

Here, a ˙ denotes differentiation with respect to cosmic time t.

The mean scale factor a and the spatial volume V of LRS Bianchi type -I metric, are defined as

and the mean Hubble parameter H may be given by

from Equations (15)-(17) we have

and

where K is a constant of integration.

The deceleration parameter (DP) is Defined as

The deceleration parameter is a crucial tool for understanding the universe’s expansion [47]-[55]. Previous research indicates that this parameter may either remain constant or change over time. When the expansion rate is constant, the scale factor increases linearly with time, resulting in a deceleration parameter of zero. Conversely, if the Hubble parameter H remains constant, the deceleration parameter takes a constant value of −1. Based on Equation (21), we get

On solving

Here, t₀ is the constant of integration.

This solution suggests that the universe could initially experience a decelerating phase and later transition to an accelerating phase. Such behaviour is consistent with observations from Type Ia supernovae (SNeIa), which support the concept of an accelerating universe.

For a flat FRW universe, we assume:

where c, d, n are positive constants. In the early universe (small a), the function f ( a ) is non-zero. At late times a → ∞ , f ( a ) → 0 .

The chosen functional form of the deceleration parameter is motivated by the observational requirement that the universe should evolve from an early decelerating phase to the present accelerated phase. The adopted parametrization provides a smooth cosmic transit behaviour and yields exact analytical solutions of the highly nonlinear modified field equations. Moreover, the model asymptotically approaches a de-Sitter type accelerating universe at late times.

From Equations (23) and (24), we get:

The time varying deceleration parameter is defined from above Equation (21), we obtain

The Hubble parameter is defined as H = a ˙ a and from the above equation we get

Figure 1. (i) Scale factor a(t) versus Cosmic time t; (ii) Hubble parameter H(t) versus Cosmic time t.

Figure 1(i) and Figure 1(ii) show the combined behavior of the scale factor and the Hubble parameter. The scale factor increases continuously with time, indicating the expansion of the universe, while the Hubble parameter decreases gradually and approaches a constant value at late times. This demonstrates a transition from early rapid expansion to a stable accelerated phase.

Solving the field Equations (14)-(16) the value of ρ and p ¯ obtained as

The expressions for the coefficient of bulk viscosity ξ and the pressure are obtained as follows:

Figure 2. Energy density ρ versus cosmic time t.

Figure 3. Effective bulk viscous pressure p ¯ versus cosmic time t.

Figure 4. Bulk viscosity coefficient ξ versus cosmic time t.

In our model, Figure 2 & Figure 3 shows that the energy density remains positive throughout the evolution of the universe. It gradually decreases over time, starting from a high value in the early universe and approaching zero as t → ∞. Conversely, the bulk viscous pressure p ¯ shows an opposite behavior: it begins with a significantly negative value and steadily increases, moving toward zero in the current epoch. This negative pressure is consistent with the influence of dark energy in driving the universe’s accelerated expansion. Consequently, the behavior of the bulk viscous pressure in our model aligns well with current observations.

Figure 4 demonstrates that the bulk viscous coefficient ξ stays positive during cosmic evolution and reaches a finite limit as t → ∞. To better understand the universe’s matter content, we also examine the energy conditions. These conditions play a vital role in general relativity, helping to establish important results about singularities and black holes, [56] and serve as tools to test the physical soundness of cosmological models. Our study focuses on three widely used energy conditions the Weak Energy Condition, the Dominant Energy Condition, and the Strong Energy Condition and evaluates their applicability to our model.

Figure 5. Behaviour of WEC versus Cosmic time t and n.

Figure 6. Behaviour of DEC versus Cosmic time t and n.

Figure 7. Behaviour of SEC versus Cosmic time t.

Energy conditions are essential because each provides unique physical insights. For instance, the Dominant Energy Condition (DEC) helps assess the stability of the matter source and restricts the equation of state parameter to ω ≥ − 1 Violating this bound can lead to a future singularity known as the Big Rip [57]. The Strong Energy Condition (SEC) is often violated in the presence of a positive cosmological constant Λ [58]. Meanwhile, the Weak Energy Condition (WEC) ensures that the matter-energy density remains non-negative in all situations. In gravity theories, energy conditions have been extensively used to investigate the matter supporting wormhole geometries. Similarly, for the Friedmann-Robertson-Walker (FRW) model with perfect fluid matter, energy conditions have been studied to understand the behaviour of solutions [59][60]. Accordingly, prior research demonstrates that energy conditions are valuable tools for analyzing the evolution of cosmological solutions throughout the universe. In this study, we consider three key energy conditions—the WEC, DEC, and SEC to evaluate our solutions under various scenarios. The behavior of these conditions is illustrated in Figures 5-7 using appropriate parameter choices. Figures 5-7 demonstrate that all the energy conditions are satisfied in the proposed model. The expressions for the Ricci scalar R and the trace T of the energy-momentum tensor (arising from the matter source) are provided below:

By utilizing the above equation, the functional form of f ( R , T ) is derived as follows

Figure 8. Ricci scalar R versus Cosmic time t.

Figure 9.Trace T versus Cosmic time t.

Figure 10. Shows the behaviour of the function f ( R , T ) for this model.

The variation of the Ricci scalar R with cosmic time is depicted in Figure 8. It is observed that the Ricci scalar remains finite throughout the cosmic evolution and gradually approaches a constant value at late times. This behaviour indicates that the space-time geometry evolves smoothly and supports the late-time accelerated expansion of the universe.

The evolution of the trace of the energy-momentum tensor T is illustrated in Figure 9. The quantity T decreases with cosmic time and reflects the dynamical influence of bulk viscous matter. The behaviour confirms that matter contributions become less significant during the late stages of cosmic evolution.

The functional behaviour of f(R,T) is shown in Figure 10. The figure demonstrates the combined influence of curvature and matter-geometry coupling on the evolution of the model. The obtained behaviour remains regular throughout the cosmic history and is consistent with the accelerated expansion scenario.

The deceleration parameter can also be expressed as a function of redshift z. To achieve this, we begin with the relation

and, utilizing Equation (26) with the normalized scale factor a 0 = 1 , we derive the corresponding t-z relation. T 0 is the present cosmic time , defined so that redshift z = 0.

Substituting this relation into Equation (27) yields the expression for the deceleration parameter in terms of redshift.

The deceleration parameter as a function of redshift is presented in Figure 11. The figure clearly exhibits the transition from an early decelerating phase q > 0 to a late-time accelerating phase q < 0. Such a cosmic transit behaviour is in agreement with current observational evidence.

Figure 11. Deceleration Parameter q versus z redshift with n.

The Hubble’s parameter H, expansion scalar θ, shear scalar σ 2 , anisotropy parameter Δ, Jerk Parameter j, and Statefinder Parameters (r-s) become

The statefinder pair (r,s)→(1,0) at late times, which corresponds to the standard ΛCDM cosmology. This confirms that the present model behaves consistently with the observed late-time universe. The physical behaviour of the universe can

be understood by studying the kinematical parameters of the model. For the chosen mean scale factor a = ( e c n T − d c ) 1 n the Hubble parameter is obtained as H = c e c n T e c n T − d while the, corresponding expansion scalar is θ = 3H. These quantities are positive and monotonically decreasing functions of cosmic time, tending

to constant values in the late universe, which indicates that the model asymptotically approaches a de-Sitter phase of accelerated expansion. The deceleration parameter evolves according to q = − 1 + n d e − c n T . At early times, the second term dominates and q > 0 ( e − c n T ≈ 1 , n d > 1 ), so the universe experiences deceleration, whereas at late times q → −1, corresponding to accelerated expansion. Thus, the model naturally provides a transition from a decelerated phase to the current acceleration. The anisotropic nature of the space-time is described through the shear scalar ( σ 2 ) the mean anisotropy parameter (Δ). Both of these quantities decrease as time increases, which shows that anisotropy is gradually suppressed and the universe evolves towards isotropy at late times. The jerk parameter, which is a higher-order diagnostic of cosmic acceleration, is obtained as towards isotropy at late times. The jerk parameter, which is a higher-order diagnostic of cosmic acceleration, is obtained as j(T) For large values of T, the exponential terms vanish and j → 1, which is exactly the value predicted by the concordance ΛCDM model. This behaviour is also confirmed through the statefinder analysis: The pair (r,s) tends to the fixed point (1,0) as T→ ∞. From the above analysis it is evident that the model describes a realistic cosmic evolution. The universe begins in a decelerating regime, undergoes a smooth transition to acceleration, and asymptotically approaches a ΛCDM-like stage. The anisotropy decays with time, the expansion rate stabilizes, and the higher-order kinematical parameters converge to their standard values, confirming the physical viability of the model.

Today, scientists have many different cosmological models to explain dark energy. Current observations cannot rule out most of these models. A few years ago, researchers introduced two new cosmological parameters [61][62] to help compare and separate these models. These parameters are called statefinder parameters. They are written using the scale factor and its time derivatives, up to the third

order. The statefinder parameters are defined for late cosmic times ( T → ∞ ), the statefinder pair becomes (r,s) = (1,0). The model show a singularity at T = 1 n a log b , which represents the beginning of the universe with a Big Bang. We observe that the expansion θ of the universe decrease with time and slowly approaches zero. The scale factor a increases with time. The expansion rate become finite for large value of T. At T = 1 n a log b , the deceleration parameter is q = n − 1 > 0 , which shows decelerated expansion. For large time ( T → ∞ ), we get q = −1, showing accelerated expansion.

The variation of the jerk parameter for different values of n is displayed in Figure 12. It is observed that the jerk parameter approaches unity at late times, which is a characteristic feature of the standard ΛCDM cosmological model. This indicates the observational consistency of the proposed model.

Figure 12. Behaviour of Jerk parameter versus cosmic time t with different n.

The statefinder trajectory (r,s) is illustrated in Figure 13. The trajectory converges towards the fixed point (1,0) corresponding to the ΛCDM model. Therefore, the statefinder diagnostic confirms that the present model successfully reproduces the observed late-time behaviour of the universe.

Figure 13. Statefinder Pair r versus s with different n.

In this model we studied a Bianchi type-I universe in f ( R , T ) gravity with bulk viscous matter. The results indicate that the energy density remains positive throughout cosmic evolution (Figure 2) and decreases with time, starting from a large value in the early universe and approaching zero in the distant future. The effective bulk viscous pressure is negative (Figure 3), and although its magnitude decreases gradually, it is this negative pressure that drives the accelerated expansion of the universe. The bulk viscosity coefficient (Figure 4) stays positive during the whole evolution and gradually approaches a constant value at late times, which indicates a stabilizing role of viscosity in cosmic dynamics. The physical acceptability of the model is supported by the fact that the weak, dominant, and strong energy conditions (Figures 5-7) are satisfied. The deceleration parameter (Figure 11) clearly shows a transition from a positive value in the early phase (deceleration) to a negative value in the later phase (acceleration), in qualitatively agrees with observational cosmology. The higher-order diagnostics also confirm this behaviour the jerk parameter (Figure 12) approaches unity and the statefinder pair (Figure 13) tends to the fixed point (1,0), which is exactly the prediction of the standard ΛCDM model. Thus, the present model provides a smooth description of cosmic evolution, beginning with a decelerating expansion, undergoing a transition, and finally approaching an accelerated de Sitter-like phase driven by bulk viscosity in f ( R , T ) gravity.

In this study, we investigated an LRS Bianchi type-I cosmological model with bulk viscosity in the framework of gravity. Exact analytical solutions of the modified

field equations were obtained by assuming e ∫ q a d a d a is a suitable functional

relationship between the expansion scalar and the scale factor. Furthermore, by considering an alternative choice of the function and employing a specific form of the deceleration parameter, we derived an additional exact solution. The resulting model represents an accelerating universe undergoing exponential expansion. The energy density and the bulk viscosity coefficient are positive and decrease monotonically with cosmic time, approaching as ρ → 0 as t → ∞ .The effective bulk viscous pressure is negative, thereby supporting accelerated cosmic expansion. In addition, all major energy conditions strong (SEC), weak (WEC), and dominant (DEC) are satisfied, ensuring the physical viability of the model. A detailed observational constraint analysis using recent cosmological datasets such as SNe Ia, BAO, and CMB remains an important direction for future investigation. Therefore, the obtained model successfully describes a realistic transition from an early decelerating anisotropic universe to the present accelerating epoch and asymptotically approaches the standard ΛCDM behaviour at late times.