Extensions of Einstein’s General Relativity (GR) can formally be given a GR structure in which additional geometric degrees of freedom are mapped on an effective energy-momentum tensor. The corresponding effective cosmic medium can then be modeled as an imperfect fluid within GR. The imperfect fluid structure allows us to include, on a phenomenological basis, anisotropic stresses and energy fluxes which are considered as potential signatures for deviations from the cosmological standard Λ -cold-dark-matter ( Λ CDM) model. As an example, we consider the dynamics of a scalar-tensor extension of the standard model, the e Φ Λ CDM model. We constrain the magnitudes of anisotropic pressure and energy flux with the help of redshift-space distortion (RSD) data for the matter growth function f σ 8 .

On the basis of a previously established scalar–tensor extension of the [Formula: see text]CDM model, we develop an effective fluid approach for the matter growth function. This extended [Formula: see text]CDM (henceforth [Formula: see text]CDM) cosmology takes into account deviations from the Standard Model both via a modified background expansion and by the inclusion of geometric anisotropic stresses as well as of perturbations of the geometric dark-energy equivalent. The background dynamics is governed by an explicit analytic expression for the Hubble rate in which modifications of the Standard Model are given in terms of a single constant parameter [W. C. Algoner, H. E. S. Velten and W. Zimdahl, J. Cosmol. Astropart. Phys. 1611 (2016) 034]. To close the system of fluid-dynamical perturbation equations, we introduce two phenomenological parameters through which the anisotropic stress is related both to the total energy density perturbation of the cosmic substratum and to relative perturbations in the effective two-component system. We quantify the impact of deviations from the standard background, of anisotropic stresses and of nonvanishing perturbations of the effective dark-energy component on the matter growth rate function [Formula: see text] and confront the results with recent redshift-space distortion (RSD) measurements.

We construct a cosmological scalar-tensor-theory model in which the Brans-Dicke type scalar Φ enters the effective (Jordan-frame) Hubble rate as a simple modification of the Hubble rate of the ΛCDM model. This allows us to quantify differences between the background dynamics of scalar-tensor theories and general relativity (GR) in a transparent and observationally testable manner in terms of one single parameter. Problems of the mapping of the scalar-field degrees of freedom on an effective fluid description in a GR context are discused. Data from supernovae, the differential age of old galaxies and baryon acoustic oscillations are shown to strongly limit potential deviations from the standard model.