The $H_0$ tension: $ΔG_N$ vs. $ΔN_{\rm eff}$

TL;DR

The paper investigates whether a time-varying Newton constant $G_N$, realized through a Brans-Dicke–like scalar-tensor theory with a non-minimal coupling to gravity, can alleviate the $H_0$ tension between Planck and SH$_0$ES. It introduces a minimal two-parameter model in the Jordan frame, where the coupling $f(\phi)=1+\beta \phi^2/M^2$ drives a natural transition around matter-radiation equality, producing a early-time expansion change analogous to extra relativistic degrees of freedom $\Delta N_{\rm eff}$. Fitting Planck2018+BAO+Pantheon+SH$_0$ES data with and without Post-Newtonian constraints, the model yields $H_0$ values around $69$ km/s/Mpc and shows a comparable improvement in $\chi^2$ to the $\Delta N_{\rm eff}$ scenario, albeit with different parameter costs and potential tension with local gravity tests. The work highlights that varying-$G_N$ models can mimic early-universe effects and reduce the Hubble tension, especially if screening mechanisms allow evading PN constraints, offering a gravity-based alternative to early dark energy. This has implications for future tests of gravity on Solar-System scales and for reconciling cosmological observations with local constraints.

Abstract

We investigate whether the $4.4σ$ 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 $ΔN_{\rm eff}$ extra relativistic degrees of freedom. We implement a varying $G_N$ in a scalar-tensor theory of gravity, with a non-minimal coupling $(M^2+βφ^2)R$. If the scalar $φ$ starts in the radiation era at an initial value $φ_I \sim 0.5~M_p$ 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 $H_0$ tension between SH$_{0}$ES (2019) and Planck 2018+BAO slightly decreases, as in $ΔN_{\rm eff}$ models, to the 3.8$σ$ 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 $Δχ^2\simeq-3.2$ compared to $Λ$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 $Δχ^2=-5.4$ with 2 extra parameters. For comparison, we find that the $ΔN_{\rm eff}$ model gives $H_0=70.08_{-0.95}^{+0.91}~\text{km/s/Mpc}$ and $Δχ^2=-3.4$ at the cost of one extra parameter, which disfavors the $Λ$CDM limit just above 2$σ$, since $ΔN_{\rm eff}=0.34_{-0.16}^{+0.15}$. Overall, our varying $G_N$ model performs similarly to the $ΔN_{\rm eff}$ model in respect to the $H_0$ tension, if a physical mechanism to remove PN constraints can be implemented.

Figures (7)

• Figure 1: Evolution of the scalar $\phi$ in units of $M_p$ (solid blue) and the quantity $\Delta G_\%\equiv 100 |1-G_N/G_N(t_0)|$ (dashed blue) as a function of the redshift $z$. On the left: $\beta=-0.05$ and $\phi_{I}\simeq 0.7 M_p$. On the right: $\beta=-0.45$ and $\phi_{I}\simeq 0.3~M_p$. The orange vertical line denotes the redshift of matter-radiation equality. These plots are obtained using the best-fit values reported for our model in Table \ref{['tab:cosmoparametersl']}: the left plot corresponds to the column 'w/o PN', whereas the right plot corresponds to 'w/ PN'.
• Figure 2: Contribution to the background energy density due to the scalar field, according to \ref{['eq:endensity']}, as a function of $z$, normalized to the total energy density at matter-radiation equality. These plots are obtained using the best-fit values reported for our model in Table \ref{['tab:cosmoparametersl']}: the left plot corresponds to the column 'w/o PN', whereas the right plot corresponds to 'w/ PN'. The orange vertical line denotes the redshift of matter-radiation equality. The dashed and dotted lines show different scalings of $\rho_\phi$ with $a$.
• Figure 3: Ratio of the energy density due to the scalar field to the total energy density, as a function of $z$. These plots are obtained using the best-fit values reported for our model in Table \ref{['tab:cosmoparametersl']}: the left plot corresponds to the column 'w/o PN', whereas the right plot corresponds to 'w/ PN'. The orange vertical line denotes the redshift of matter-radiation equality. The dashed, dotted and dot-dashed lines correspond to the different contributions to $\rho_\phi$ in \ref{['eq:endensity']}.
• Figure 4: Constraints on parameters for our $\Delta G_N$ model vs. the $\Delta N_{\rm eff}$ model and the base $\Lambda$CDM model, using Planck 2018 high$-\ell$ TT,TE,EE+low$-\ell$ EE+ low$-\ell$ TT+lensing, BAO, Pantheon and SH$_{0}$ES 2019 data. PN constraints are included. Parameters are our sampled MCMC parameters with flat priors. In particular a prior range has been set for the extra parameters: $-0.95<\beta \phi_I^2<0$ and $0<\phi_I<0.95$. Here $H_0$ is in km/s/Mpc. Contours contain $68 \%$ and $95 \%$ of the probability.
• Figure 5: Constraints on parameters for our $\Delta G_N$ model vs. the $\Delta N_{\rm eff}$ model and the base $\Lambda$CDM model, using Planck 2018 high$-\ell$ TT,TE,EE+low$-\ell$ EE+ low$-\ell$ TT+lensing, BAO, Pantheon and SH$_{0}$ES 2019 data, without PN constraints. Parameters are our sampled MCMC parameters with flat priors. In particular a prior range has been set for the extra parameters: $-0.95<\beta \phi_I^2<0$ and $0<\phi_I<0.95$. Here $H_0$ is in km/s/Mpc. Contours contain $68 \%$ and $95 \%$ of the probability.