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