Comparison of different approaches to the quasi-static approximation in Horndeski models.

A quasi-static approximation (QSA) for modified gravity can be applied in a number of ways. We consider three different analytical formulations based on applying this approximation to: (1) the field equations; (2) the equations for the two metric potentials; (3) the use of the attractor solution derived within the Equation of State (EoS) approach. We assess the veracity of these implementations on the effective gravitational constant ($\mu$) and the slip parameter ($\eta$), within the framework of Horndeski models. In particular, for a set of models we compare cosmological observables, i.e., the matter power spectrum and the CMB temperature and lensing angular power spectra, computed using the QSA, with exact numerical solutions. To do that, we use a newly developed branch of the CLASS code: QSA_class. All three approaches agree exactly on very small scales. Typically, we find that, except for $f(R)$ models where all the three approaches lead to the same result, the quasi-static approximations differ from the numerical calculations on large scales ($k \lesssim 3 - 4 \times 10^{-3}\,h\,{\rm Mpc}^{-1}$). Cosmological observables are reproduced to within 1% up to scales ${\rm K} = k/H_0$ of the order of a few and $\ell>5$ for the approaches based on the field equations and on the EoS, and we also do not find any appreciable difference if we use the scale-dependent expressions for $\mu$ and $\eta$ with respect to the value on small scales, showing that the formalism and the conclusions are reliable and robust, fixing the range of applicability of the formalism. We discuss why the expressions derived from the equations for the potentials have limited applicability. Our results are in agreement with previous analytical estimates and show that the QSA is a reliable tool and can be used for comparison with current and future observations to constrain models beyond $\Lambda$CDM.