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Constraints on deviations from $Λ$CDM within Horndeski gravity

Recent anomalies found in cosmological datasets such as the low multipoles of the Cosmic Microwave Background or the low redshift amplitude and growth of clustering measured by e.g., abundance of galaxy clusters and redshift space distortions in galaxy surveys, have motivated explorations of models beyond standard $Λ$CDM. Of particular interest are models where general relativity (GR) is modified on large cosmological scales. Here we consider deviations from $Λ$CDM+GR within the context of Horndeski gravity, which is the most general theory of gravity with second derivatives in the equations of motion. We adopt a parametrization in which the four additional Horndeski functions of time $α_i(t)$ are proportional to the cosmological density of dark energy $Ω_{DE}(t)$. Constraints on this extended parameter space using a suite of state-of-the art cosmological observations are presented for the first time. Although the theory is able to accommodate the low multipoles of the Cosmic Microwave Background and the low amplitude of fluctuations from redshift space distortions, we find no significant tension with $Λ$CDM+GR when performing a global fit to recent cosmological data and thus there is no evidence against $Λ$CDM+GR from an analysis of the value of the Bayesian evidence ratio of the modified gravity models with respect to $Λ$CDM, despite introducing extra parameters. The posterior distribution of these extra parameters that we derive return strong constraints on any possible deviations from $Λ$CDM+GR in the context of Horndeski gravity. We illustrate how our results can be applied to a more general frameworks of modified gravity models.