Early modified gravity in light of the $H_0$ tension and LSS data

We present a model of Early Modified Gravity (EMG) consisting in a scalar field $\sigma$ with a non-minimal coupling to the Ricci curvature of the type $M^2_{\rm pl}+\xi \sigma^2$ plus a cosmological constant and a small effective mass and demonstrate its ability to alleviate the $H_0$ tension while providing a good fit to Cosmic Microwave Background (CMB) anisotropies and Baryon Acoustic Oscillations (BAO) data. In this model the scalar field, frozen deep in the radiation era, grows around the redshift of matter-radiation equality because of the coupling to non-relativistic matter. The small effective mass, which we consider here as induced by a quartic potential, then damps the scalar field into coherent oscillations around its minimum at $\sigma=0$, leading to a weaker gravitational strength at early times and naturally recovering the consistency with laboratory and Solar System tests of gravity. We analyze the capability of EMG with positive $\xi$ to fit current cosmological observations and compare our results to the case without an effective mass and to the popular early dark energy models with $\xi=0$. We show that EMG with a quartic coupling of the order of $\lambda\sim\mathcal{O}({\rm eV}^4/M_{\rm pl}^4)$ can substantially alleviate the $H_0$ tension also when the full shape of the matter power spectrum is included in the fit in addition to CMB and Supernovae (SN) data.