Linear perturbation theory and structure formation in a Horndeski theory of gravity without dark matter

We study the formation and evolution of the cosmic large-scale structure (LSS) in a scalar-tensor Horndeski theory of gravity without dark matter. We impose that this theory reduces to Refracted Gravity (RG) in the weak-field limit. RG is a non-relativistic theory where the gravitational permittivity $ε(ρ)$, with $ρ$ the baryonic matter density, provides the gravitational boost required to describe both the dynamics of galaxies and galaxy clusters without dark matter. We show that $ε(ρ)$ equals the function $G_4(φ)$ of the scalar field $φ$ of our Horndeski model. The background evolution of a flat, matter-dominated, homogeneous and isotropic universe shows that $φ$ is also responsible for the accelerated expansion of the universe and that the model is capable of reproducing the observed redshift evolution of the Hubble-Lemaître parameter. We finally develop the linear perturbation theory and study the growth factor equation. The model fails to correctly describe the LSS by predicting a late onset of the formation of cosmic structure at redshift $z\lesssim 1$.