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Friedmann model with viscous cosmology in modified $f(R,T)$ gravity theory

In this paper, we introduce bulk viscosity in the formalism of modified gravity theory in which the gravitational action contains a general function $f(R,T)$, where $R$ and $T$ denote the curvature scalar and the trace of the energy-momentum tensor, respectively within the framework of a flat Friedmann-Robertson-Walker model. As an equation of state for prefect fluid, we take $p=(γ-1)ρ$, where $0 \leq γ\leq 2$ and viscous term as a bulk viscosity due to isotropic model, of the form $ζ=ζ_{0}+ζ_{1}H$, where $ζ_{0}$ and $ζ_{1}$ are constants, and $H$ is the Hubble parameter. The exact non-singular solutions to the corresponding field equations are obtained with non- viscous and viscous fluids, respectively by assuming a simplest particular model of the form of $f(R,T) = R+2f(T)$, where $f(T)=αT$ ( $α$ is a constant). A big-rip singularity is also observed for $γ<0$ at a finite value of cosmic time under certain constraints. We study all possible scenarios with the possible positive and negative ranges of $α$ to analyze the expansion history of the universe. It is observed that the universe accelerates or exhibits transition from decelerated phase to accelerated phase under certain constraints of $ζ_0$ and $ζ_1$. We compare the viscous models with the non-viscous one through the graph plotted between scale factor and cosmic time and find that bulk viscosity plays the major role in the expansion of the universe. A similar graph is plotted for deceleration parameter with non-viscous and viscous fluids and find a transition from decelerated to accelerated phase with some form of bulk viscosity.