The H0 tension. ΔGN vs. ΔNeeff

We investigate whether the 4.4(sigma) tension on H-0 between SH0ES 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 expansion rate before recombination, similarly to the addition of Delta N-eff extra relativistic degrees of freedom. We implement a varying G(N) in a scalar-tensor theory of gravity, with a non-minimal coupling of the form (M-2 + beta phi(2))R. If the scalar phi starts in the radiation era at an initial value phi(I) -similar to 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 SH0ES (2019) and Planck 2018+BAO slightly decreases, as in Delta N-eff models, to the 3.8 sigma level. We then perform a fit to a combined Planck, BAO and supernovae (SH0ES and Pantheon) dataset. When including local constraints on Post-Newtonian (PN) parameters, we find H-0 = 69.08(-0.71)(+0.6) km/s/Mpc and a marginal improvement of Delta chi(2) similar or equal to 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) 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(eff )model gives H-0 = 70.08(-0.95)(+0.91) km/s/Mpc and Delta chi(2) = -3.4 at the cost of one extra parameter, which disfavors the Lambda CDM limit just above 2a, since Delta N-eff = 0.34(-0.16)(+0.15). Overall, our varying GN model performs similarly to the AN eff model in respect to the Ho tension, if a physical mechanism to remove PN constraints can be implemented.