Phantom scalar-tensor models and cosmological tensions

We study three different extended scalar-tensor theories of gravity by also allowing a negative sign for the kinetic term for the scalar field in the Jordan frame. Our scope is to understand how the observational constraints for these models cope with the volume of the parameter space in which the theory is healthy. Models with a negative kinetic term lead to decreasing effective gravitational constant with redshift and behave as an effective relativistic component with a negative energy density as opposite to their corresponding version with a standard kinetic term. As a consequence, we find that the extended branch with a negative sign for the kinetic term correspond in general to lower $H_0$ and $\sigma_8$ compared to $\Lambda$CDM. We find that in all the cases with a negative sign for the kinetic term studied here, cosmological observations constrain these models around GR and prefer a volume of the parameter space in which the theory is not healthy since the scalar field behave as a ghost also in the related Einstein frame. We show that also in the phantom branch early modify gravity with a quartic coupling can substantially reduce the $H_0$ tension fitting the combination of cosmic microwave background data from Planck, baryon acoustic oscillations from BOSS and eBOSS, and Supernovae from the Pantheon sample with calibration information by SH0ES.