The H₀ tension: Δ G_N vs. Δ N_eff

We investigate whether the 4.4σ tension on H₀ 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 Δ 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 (M²+β ϕ²)R. If the scalar ϕ starts in the radiation era at an initial value ϕ_I ∼ 0.5 M_p and with β<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 H₀ tension between SH₀ES (2019) and Planck 2018+BAO slightly decreases, as in Δ N_eff models, to the 3.8σ 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 H₀=69.08_−0.71^+0.6 km/s/Mpc and a marginal improvement of Δχ^2≃−3.2 compared to ΛCDM, 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 H₀=69.65_−0.78^+0.8 km/s/Mpc and a more significant improvement Δχ²=−5.4 with 2 extra parameters. For comparison, we find that the Δ N_eff model gives H₀=70.08_−0.95^+0.91 km/s/Mpc and Δχ²=−3.4 at the cost of one extra parameter, which disfavors the ΛCDM limit just above 2σ, since Δ N_eff=0.34_−0.16^+0.15. Overall, our varying G_N model performs similarly to the Δ N_eff model in respect to the H₀ tension, if a physical mechanism to remove PN constraints can be implemented.