Exploring interacting bulk viscous model with decaying vacuum density

In the present work, we study a cosmological model composed of a viscous dark matter interacting with decaying vacuum energy in a spatially flat Universe. In the first part, we find the analytical solution of different cosmological parameters by assuming the physically viable forms of bulk viscosity and decaying vacuum density with the interaction term. The second part is dedicated to constrain the free parameters of the interacting viscous model with decaying vacuum energy by employing latest observational data of $Pantheon+$, Cosmic Chronometer and $f(z)\sigma_{8}(z)$. We find that the interacting model just deviate very slightly from well-known concordance $\Lambda$CDM model and can alleviate effectively the current $H_0$ tension between local measurement by R21 and global measurement by Planck 2018, and the excess in the mass fluctuation amplitude $\sigma_{8}$ essentially vanish in this context. We report the Hubble constants as $H_0=72.150^{+0.989}_{-0.779}$, and $72.202^{+0.796}_{-0.937}$ \;$km s^{-1} Mpc^{-1}$, deceleration parameters as $q_0=-0.533 \pm 0.024$, and $-0.531 \pm 0.024$, and equation of state parameters as $w_0=-0.689 \pm 0.016$, and $-0.687 \pm 0.016$ for $\Lambda$CDM and interacting models, respectively. It is found that the interacting model is in good agreement with $\Lambda$CDM. Further, we discuss the amplitude of matter power spectrum $\sigma_8$ and its associated parameter $S_8$ using $f(z)\sigma_8(z)$ data. Finally, the information selection criterion and Bayesian inference are discussed to distinguish the interacting model with $\Lambda$CDM model.