KiDS-Legacy: Constraints on Horndeski gravity from weak lensing combined with galaxy clustering and cosmic microwave background anisotropies

We present constraints on modified gravity from a cosmic shear analysis of the final data release of the Kilo-Degree Survey (KiDS-Legacy) in combination with DESI measurements of baryon acoustic oscillations, eBOSS observations of redshift space distortions, and cosmic microwave background anisotropies from Planck. We study the Horndeski class of modified gravity models in an effective field theory framework employing a parameterisation that satisfies stability conditions by construction and, for the first time, present a cosmological analysis in this inherently stable parameter basis. Cosmic shear constrains the Horndeski parameter space significantly, matching or surpassing the CMB contribution. Adopting the de-mixed kinetic term of the scalar field perturbation, $D_{\rm kin}$, and the deviation of the Planck mass from its fiducial value, $ΔM_*^2\equiv M_*^2-1$, as model parameters, we constrain their present values to be $Δ\hat{M}_*^2=0.32^{+0.07}_{-0.21}$ and $\hat{D}_{\rm kin} = 3.74^{+0.69}_{-1.92}$, which deviate from general relativity at $1.5σ$ and $1.9σ$, respectively. We derive constraints on the structure growth parameter $S_8=0.813^{+0.008}_{-0.011}$, which is compatible with the $Λ$CDM constraint at $0.54σ$. We obtain the deviation of the effective Newtonian coupling from the GR value as $Δμ_{\infty,{\rm eff}}=0.066\pm0.023$, corresponding to a $2.9σ$ significance. Although modified gravity provides a slightly better fit to the data, a model comparison shows only a weak preference for modified gravity at the $1.4σ$ level. When adopting a dynamical dark energy model of the background cosmology, the inferred modified gravity parameter constraints are stable with respect to a $Λ$CDM background, while a mild preference at $1.57σ$ for dynamical dark energy remains.