The effective fluid approach for modified gravity

Current and coming surveys will require sub-percent agreement in theoretical accuracy to test the different cosmological and gravity scenarios. This can be performed with Boltzmann solvers, i.e. codes that solve the linear evolution of cosmological perturbations. Given the plethora of gravity models, it is crucial to have a standardized unified way to describe all of them and take them into account in a Boltzmann code. Dark Energy (DE) and Modified Gravity (MG) models, although at a first glance quite dissimilar, are possible to unify them within the same framework. In this paper we present a scenario, based on the effective fluid approach that allows to map any modified gravity model as an effective dark energy fluid and then we show how to implement it into existing Boltzmann codes in a simple and straightforward way. This approach has also the advantage that only a handful of variables are needed to compute, i.e the equation of state $w(a)$ at the background level and the sound speed $c_s^2(a,k)$ and the anisotropic stress $\pi(a,k)$ at the linear perturbation. In particular we show that with simple modifications to the latter Cosmic Linear Anisotropy Solving System (CLASS) code, which we called EFCLASS, we provide competitive results in a much simpler and less error-prone approach in including the effects of modified gravity models. To test our modifications, we particularize the effective fluid approach to f(R) theories and a surviving class of Horndeski models, the designer Horndeski (HDES) which have a background exactly to the standard cosmological model $\Lambda$CDM.