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.
