KiDS+GAMA: Constraints on Horndeski gravity from combined large-scale
  structure probes
  We present constraints on Horndeski gravity from a combined analysis of
cosmic shear, galaxy-galaxy lensing and galaxy clustering from
$450\,\mathrm{deg}^2$ of the Kilo-Degree Survey (KiDS) and the Galaxy And Mass
Assembly (GAMA) survey. The Horndeski class of dark energy/modified gravity
models includes the majority of universally coupled extensions to $\Lambda$CDM
with one scalar field in addition to the metric. We study the functions of time
that fully describe the evolution of linear perturbations in Horndeski gravity.
Our results are compatible throughout with a $\Lambda$CDM model. By imposing
gravitational wave constraints, we fix the tensor speed excess to zero and
consider a subset of models including e.g. quintessence and $f(R)$ theories.
Assuming proportionality of the Horndeski functions $\alpha_B$ and $\alpha_M$
(kinetic braiding and the Planck mass run rate, respectively) to the dark
energy density fraction $\Omega_{\mathrm{DE}}(a) = 1 - \Omega_{\mathrm{m}}(a)$,
we find for the proportionality coefficients $\hat{\alpha}_B =
0.20_{-0.33}^{+0.20} \,$ and $\, \hat{\alpha}_M = 0.25_{-0.29}^{+0.19}$. Our
value of $S_8 \equiv \sigma_8 \sqrt{\Omega_{\mathrm{m}}/0.3}$ is in better
agreement with the $Planck$ estimate when measured in the enlarged Horndeski
parameter space than in a pure $\Lambda$CDM scenario. In our joint three-probe
analysis we report a downward shift of the $S_8$ best fit value from the
$Planck$ measurement of $\Delta S_8 = 0.016_{-0.046}^{+0.048}$ in Horndeski
gravity, compared to $\Delta S_8 = 0.059_{-0.039}^{+0.040}$ in $\Lambda$CDM.
Our constraints are robust to the modelling uncertainty of the non-linear
matter power spectrum in Horndeski gravity. Our likelihood code for multi-probe
analysis in both $\Lambda$CDM and Horndeski gravity is publicly available at
http://github.com/alessiospuriomancini/KiDSHorndeski .
