The Integrated Sachs-Wolfe Effect in the Bulk Viscous Dark Energy Model

We examine linear perturbation theory to evaluate the contribution of viscosity coefficient in the growing of dark matter perturbations in the context of the bulk viscous dark energy model inspired by thermodynamical dissipative phenomena proposed by [B. Mostaghel et al., 2017, The European Physical Journal C, 77, 541]. As the cosmological implementations, we investigate the {\it Integrated Sachs-Wolfe} (ISW) auto-power spectrum, the ISW-galaxy cross-power spectrum and derive limits on $f\sigma_8$. The dimensionless bulk viscosity coefficient ($\gamma$) in the absence of interaction between dark sectors, modifies the Hubble parameter and the growth function, while the Poisson equation remains unchanged. Increasing $\gamma$ reduces the dark matter growing mode at the early epoch while a considerable enhancement will be achieved at the late time. This behavior imposes non-monotonic variation in the time evolution of gravitational potential generating a unique signature on the CMB photons. The bulk viscous dark energy model leads to almost a decreasing in ISW source function at the late time. Implementation of the Redshift Space Distortion (RSD) observations based on "Gold-2017" catalogue, shows $\Omega^0_{\rm m} =0.303^{+0.044}_{-0.038}$, $\gamma=0.033^{+0.098}_{-0.033}$ and $\sigma_8= 0.769^{+0.080}_{-0.089}$ at $1\sigma$ level of confidence. Finally, tension in the $\sigma_8$ is alleviated in our viscous dark energy model.