Designing Horndeski and the effective fluid approach

We present a family of designer Horndeski models, i.e. models that have a background exactly equal to that of the $\Lambda$CDM model but perturbations given by the Horndeski theory. Then, we extend the effective fluid approach to Horndeski theories, providing simple analytic formulae for the equivalent dark energy effective fluid pressure, density and velocity. We implement the dark energy effective fluid formulae in our code EFCLASS, a modified version of the widely used Boltzmann solver CLASS, and compare the solution of the perturbation equations with those of the code hi_CLASS which already includes Horndeski models. We find that our simple modifications to the vanilla code are accurate to the level of $\sim 0.1\%$ with respect to the more complicated hi_CLASS code. Furthermore, we study the kinetic braiding model both on and off the attractor and we find that even though the full case has a proper $\Lambda$CDM model limit for large $n$, it is not appropriately smooth, thus causing the quasistatic approximation to break down. Finally, we focus on our designer model (HDES), which has both a smooth $\Lambda$CDM limit and well-behaved perturbations, and we use it to perform Markov Chain Monte Carlo analyses to constrain its parameters with the latest cosmological data. We find that our HDES model can also alleviate the soft $2\sigma$ tension between the growth data and Planck 18 due to a degeneracy between $\sigma_8$ and one of its model parameters that indicates the deviation from the $\Lambda$CDM model.