Constraints on deviations from ${\Lambda}$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 $\Lambda$CDM. Of particular interest are models where general
relativity (GR) is modified on large cosmological scales. Here we consider
deviations from $\Lambda$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 $\alpha_i(t)$ are proportional to the cosmological density of
dark energy $\Omega_{DE}(t)$. 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 $\Lambda$CDM+GR
when performing a global fit to recent cosmological data and thus there is no
evidence against $\Lambda$CDM+GR from an analysis of the value of the Bayesian
evidence ratio of the modified gravity models with respect to $\Lambda$CDM,
despite introducing extra parameters. The posterior distribution of these extra
parameters that we derive return strong constraints on any possible deviations
from $\Lambda$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.
