In this article, the bulk viscosity is introduced in a modified gravity model. The gravitational action has a general $f(R,T)$ form, where $R$ and $ T $ are the curvature scalar and the trace of energy momentum tensor respectively. An effective equation of state (EoS) has been investigated in the cosmological evolution with bulk viscosity. In the present scenario, the Hubble parameter which has a scaling relation with the redshift can be obtained generically. The role of deceleration parameter $q$ and equation of state parameter $\omega $ is discussed to explain the late-time accelerating expansion of the universe. The statefinder parameters and Om diagnostic analysis are discussed for our obtained model to distinguish from other dark energy models together with the analysis of energy conditions and velocity of sound for the model. We have also numerically investigated the model by detailed maximum likelihood analysis of $580$ Type Ia supernovae from Union $ 2.1$ compilation datasets and updated $57$ Hubble datasets ($31$ data points from differential age method and $26$ points from BAO and other methods). It is with efforts found that the present model is in good agreement with observations.

中文翻译：

---

修正重力和观测约束条件下的有效状态方程

在本文中，体积粘度是在修改后的重力模型中引入的。引力作用具有一般的$f(R,T)$形式，其中$R$和$T$分别是曲率标量和能量动量张量的迹。已经在具有体积粘度的宇宙学演化中研究了有效状态方程 (EoS)。在本场景中，与红移具有比例关系的哈勃参数可以通用获得。讨论了减速参数 $q$ 和状态方程参数 $\omega $ 的作用，以解释宇宙后期加速膨胀。讨论了我们获得的模型的状态查找器参数和 Om 诊断分析，以区别于其他暗能量模型，并分析了该模型的能量条件和声速。我们还通过对来自 Union $2.1$ 编译数据集的 $580$ Ia 型超新星的详细最大似然分析和更新的 $57$ 哈勃数据集（来自差异年龄方法的 $31$ 数据点和来自 BAO 和其他方法的 $26$ 点）对模型进行了数值研究。 . 经过努力发现，目前的模型与观察结果非常吻合。