Modified gravity versus shear viscosity: imprints on the scalar matter perturbations

Cosmological scalar perturbation theory studied in the Newtonian gauge depends on two potentials $\Phi$ and $\Psi$. In General Relativity (GR) they must coincide ($\Phi=\Psi$) in the absence of anisotropic stresses sourced by the energy momentum tensor. On the other hand, it is widely accepted in the literature that potential deviations from GR can be parameterized by $\Phi\neq \Psi$. The latter feature is therefore present in both GR cosmologies equipped with shear viscous fluids or modified gravity. We study the evolution of scalar matter density perturbations using the redshift-space-distortion based $f(z)\sigma_8(z)$ data as a tool to differentiate and characterize the imprints of both scenarios. We show that in the $f(z)\sigma_8(z)$ evolution both scenarios yields to completely different imprints in comparison to the standard cosmology. While the current available data is not sensitive to distinguish modified gravity from viscous shear cosmologies, future precise data can be used to break this indistinguishability.