The $H_0$ tension: $\Delta G_N$ vs. $\Delta N_{\rm eff}$
  We investigate whether the $4.4\sigma$ tension on $H_0$ between SH$_{0}$ES
2019 and Planck 2018 can be alleviated by a variation of Newton's constant
$G_N$ between the early and the late Universe. This changes the Hubble rate
before recombination, similarly to adding $\Delta N_{\rm eff}$ extra
relativistic degrees of freedom. We implement a varying $G_N$ in a
scalar-tensor theory of gravity, with a non-minimal coupling $(M^2+\beta
\phi^2)R$. If the scalar $\phi$ starts in the radiation era at an initial value
$\phi_I \sim 0.5~M_p$ and with $\beta<0$, a dynamical transition occurs
naturally around the epoch of matter-radiation equality and the field evolves
towards zero at late times. As a consequence, the $H_0$ tension between
SH$_{0}$ES (2019) and Planck 2018+BAO slightly decreases, as in $\Delta N_{\rm
eff}$ models, to the 3.8$\sigma$ level. We then perform a fit to a combined
Planck, BAO and supernovae (SH$_0$ES and Pantheon) dataset. When including
local constraints on Post-Newtonian (PN) parameters, we find
$H_0=69.08_{-0.71}^{+0.6}~\text{km/s/Mpc}$ and a marginal improvement of
$\Delta\chi^2\simeq-3.2$ compared to $\Lambda$CDM, at the cost of 2 extra
parameters. In order to take into account scenarios where local constraints
could be evaded, we also perform a fit without PN constraints and find
$H_0=69.65_{-0.78}^{+0.8}~\text{km/s/Mpc}$ and a more significant improvement
$\Delta\chi^2=-5.4$ with 2 extra parameters. For comparison, we find that the
$\Delta N_{\rm eff}$ model gives $H_0=70.08_{-0.95}^{+0.91}~\text{km/s/Mpc}$
and $\Delta\chi^2=-3.4$ at the cost of one extra parameter, which disfavors the
$\Lambda$CDM limit just above 2$\sigma$, since $\Delta N_{\rm
eff}=0.34_{-0.16}^{+0.15}$. Overall, our varying $G_N$ model performs similarly
to the $\Delta N_{\rm eff}$ model in respect to the $H_0$ tension, if a
physical mechanism to remove PN constraints can be implemented.
