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, including all cross-correlations. The Horndeski class of dark energy/modified gravity models includes the majority of universally coupled extensions to $\Lambda$CDM with one scalar degree of freedom in addition to the metric. We study the functions of time that fully describe the evolution of linear perturbations in Horndeski gravity, and set constraints on parameters that describe their time evolution. Our results are compatible throughout with a $\Lambda$CDM model. Assuming proportionality of the Horndeski functions $\alpha_B$ and $\alpha_M$ (describing the braiding of the scalar field with the metric 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 Horndeski gravity analysis of cosmic shear alone, we report a downward shift of the $S_8$ best fit value from the $Planck$ measurement of $\Delta S_8 = 0.048_{-0.056}^{+0.059}$, compared to $\Delta S_8 = 0.091_{-0.045}^{+0.046}$ in $\Lambda$CDM. In the joint three-probe analysis, we find $\Delta S_8 = 0.016_{-0.046}^{+0.048}$ in Horndeski gravity and $\Delta S_8 = 0.059_{-0.039}^{+0.040}$ in $\Lambda$CDM. Our likelihood code for multi-probe analysis in both $\Lambda$CDM and Horndeski gravity is made publicly available.