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 (CMB) anisotropies from Planck . We studied the Horndeski class of modified gravity models in an effective field theory framework, employing a parametrisation that satisfies stability conditions by construction. In this work, for the first time, we 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_̊m kin, and the deviation of the Planck mass from its fiducial value, Δ M_*^2≡ M_*^2-1, as model parameters, we constrained their present values as Δ _*^2=0.32^ M +0.07 _ -0.21 and hat D _ ̊m kin = 3.74^ +0.69 _ -1.92 , representing a deviation from general relativity (GR) at 1.5σ and 1.9σ, respectively. We derived constraints on the structure growth parameter, S_8=0.813^ +0.008 _ -0.011 , which is compatible with the constraint at 0.54σ. We obtained the deviation of the effective Newtonian coupling from the GR value as Δ μ_ ∞, ̊m eff =0.066 cosmological background, while a mild preference at 1.57σ for dynamical dark energy remains. corresponding to a 2.9σ significance. Although modified gravity provides a slightly better fit to the data, a model comparison only reveals 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 the Λ cold dark matter ( )