Running vacuum cosmology with bulk viscous matter

Abstract: We study the late acceleration of the universe by incorporating bulk viscous matter with the running vacuum. The vacuum energy density varies as the squares of the Hubble parameter ($\rho_{\Lambda}\propto H^2$), and the coefficient of bulk viscosity of matter is proportional to the velocity of expansion ($\xi \propto H$). We obtained an analytical solution to the Friedmann equations and estimated the model parameters using the combined data set SN1a+CMB+BAO+OHD. We have evaluated the universe's age as 14 Gyr, which is slightly higher than the age-predicted by the $\Lambda$CDM model. However, it is an improved result compared to the age-predicted by a class of bulk viscous matter-dominated models. Interestingly, we have obtained the coefficient of bulk viscosity of the matter component as $1.316\times 10^5$ kg $\textnormal m^{-1}$ $\textnormal s^{-1}$ which is one to two orders of magnitude less than the value predicted by most of the bulk viscous matter-dominated models and it falls in the range of highly viscous materials found on the earth. The Hubble parameter is a decreasing function of the scale factor, and it attains a constant value in the far future that corresponds to an end deSitter phase of evolution. The deceleration parameter shows a transition from matter-dominated decelerated phase to vacuum energy-dominated accelerating phase, and the transition redshift is obtained as $z_T = 0.73$. The statefinder analysis distinguishes our model from the $\Lambda$CDM model at present, and the $r-s$ trajectory reveals the quintessence behaviour of the vacuum energy. The phase space analysis shows that the universe is evolving towards a mechanically stable state in the far future. The entropy evolution satisfies the generalised second law of thermodynamics, and the entropy is maximised in the far future evolution.