Unraveling the effective fluid approach for $f(R)$ models in the sub-horizon approximation

We provide explicit formulas for the effective fluid approach of $f(R)$ theories, such as the Hu & Sawicki and the designer models. Using the latter and simple modifications to the CLASS code, which we call EFCLASS, in conjunction with very accurate analytic approximations for the background evolution, we obtain competitive results in a much simpler and less error-prone approach. We also derive the initial conditions in matter domination and we find they differ from those already found in the literature for a constant $w$ model. A clear example is the designer model that behaves as $\Lambda$CDM in the background, but has nonetheless dark energy perturbations. We then use the aforementioned models to derive constraints from the latest cosmological data, including supernovae, BAO, CMB, $H(z)$ and growth-rate data, and find they are statistically consistent to the $\Lambda$CDM model. Finally, we show that the viscosity parameter $c_{vis}^2$ in realistic models is not constant as commonly assumed, but rather evolves significantly over several orders of magnitude, something which could affect forecasts of upcoming surveys.