The $H_0$ tension: $\Delta G_N$ vs. $\Delta N_{\mathrm{eff}}$

Abstract

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 $\Delta N_{\mathrm{eff}}$ extra relativistic degrees of freedom. We implement a varying $G_N$ in a scalar-tensor theory of gravity, with a non-minimal coupling $(M^2+\beta {\varphi }^2)R$. If the scalar $\varphi$ starts in the radiation era at an initial value ${\varphi }_I\sim 0.5M_p$ 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 $\Delta N_{\mathrm{eff}}$ 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 $\Delta N_{\mathrm{eff}}$ 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 $\Delta N_{\mathrm{eff}}=0.34_{-0.16}^{+0.15}$. Overall, our varying $G_N$ model performs similarly to the $\Delta N_{\mathrm{eff}}$ model in respect to the $H_0$ tension, if a physical mechanism to remove PN constraints can be implemented.

20 pages, 7 figures. v2: Pantheon dataset added, additional new results for model without PN constraints, additional short discussion of S8. Version accepted for publication in JCAP

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