Can conformally coupled modified gravity solve the Hubble tension?

The discrepancy between early-Universe inferences and direct measurements of the Hubble constant, known as the Hubble tension, recently became a pressing subject in high precision cosmology. As a result, a large variety of theoretical models have been proposed to relieve this tension. In this work we analyze a conformally coupled modified gravity (CCMG) model of an evolving gravitational constant due to the coupling of a scalar field to the Ricci scalar, which becomes active around matter-radiation equality, as required for solutions to the Hubble tension based on increasing the sound horizon at recombination. The model is theoretically advantageous as it has only one free parameter in addition to the baseline $\mathrm{\Lambda }$ cold dark matter ones. Inspired by similar recent analyses of so-called early dark energy models, we constrain the CCMG model using a combination of early- and late-Universe cosmological datasets. In addition to the Planck 2018 cosmic microwave background (CMB) anisotropies and weak lensing measurements, baryon acoustic oscillations, and the Supernova H0 for the Equation of State datasets, we also use large-scale structure (LSS) datasets such as the Dark Energy Survey Year 1 and the full-shape power spectrum likelihood from the Baryon Oscillation Spectroscopic Survey, including its recent analysis using effective field theory, to check the effect of the CCMG model on the (milder) $S_8$ tension between the CMB and LSS. We find that the CCMG model can slightly relax the Hubble tension, with $H_0=69.6\pm 1.6\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ at 95% confidence level, while barely affecting the $S_8$ tension. However, current data does not exhibit a strong preference for CCMG over the standard cosmological model. Lastly, we show that the planned CMB-S4 experiment will have the sensitivity required to distinguish between the CCMG model and the more general class of models involving an evolving gravitational constant.

Figure 1

Evolution of the relative deviation of Newton’s constant $\mathrm{\Delta }{G}_N(\%)=100\times |G_N^{*}/G_N-1|$ (solid green) and the energy fraction associated with $\varphi$ (dashed green), in the CCMG scenario ($\xi =-1/6$), using the parameters specified in Table 7. For comparison, we also plot the energy fraction associated with the EDE field (dashed blue), corresponding to the best-fit values from Table IV in Ref. [7]. We also mark the matter-radiation equality (dashed red), the critical redshift of the EDE model ($z_c$ parameter, vertical dashed blue), in which the EDE field is most dominant, and the recombination epoch (gray band). Both models become dynamic just prior to recombination (around matter-radiation equality). The EDE component is transiently dominant near $z_c$ (peaks to over 10%), in contrast to the CCMG component which is present during the entire early-Universe era and is less significant (about 6%). The CCMG and EDE models were simulated by a modified version of hi-class and the code of class-EDE, respectively.

Figure 2

Cosmological parameter constraints from the combination of Planck 2018 primary CMB data ($\mathrm{TT}+\mathrm{TE}+\mathrm{EE}$); Planck 2018 CMB lensing data; BAO data from 6dF, SDSS DR7, and SDSS DR12; Pantheon SNIa data; the latest SH0ES $H_0$ constraint; and SDSS DR12 RSD data. We do not plot $\tau$, as it is essentially unchanged in the CCMG fit. Some parameters shift by a non-negligible amount in the CCMG fit (compared to $\mathrm{\Lambda }\mathrm{CDM}$), including increases in ${\mathrm{\Omega }}_c{h}^2$, $n_s$, and ${\sigma }_8$, as well as broadening of the error bars on these parameters. The increase in $H_0$ is not large enough to reconcile the tension with the SH0ES-only constraint (shown in the grey bands), but it does reduce the tension significantly.

Figure 3

Constraints on MG parameters and $H_0$, expected by the planned CMB-S4 experiment, after maximizing over all the cosmological parameters. We find the constraints $\xi =-1/6\pm 0.0115$ and ${\varphi }_i=0.297\pm 0.005$ at 68% CL, where for ${\varphi }_i$ we chose the best-fit value from the analysis of the joint dataset in Table 3. We can expect CMB-S4 constraints on MG parameters to be at least an order of magnitude smaller than the values themselves, thus advocating whether the CC scenario is favorable or not.

Figure 4

Constraints on the CCMG parameter from various datasets: primary Planck 2018, CMB lensing, BAO, RSD, SNIa, SH0ES, DES-Y1, and a combined $\mathrm{BAO}+\mathrm{EFT}$. Here we present a subset of the parameters: the initial condition ${\varphi }_i$ for the CCMG, along with $H_0$ [$\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$] and ${\sigma }_8$. The contours show $1\sigma$ and $2\sigma$ posteriors for various dataset combinations computed with getdist [50]. The P18 dataset alone (green) yields a posterior for ${\varphi }_i$, which tends to zero, thus disfavoring a significant CCMG component in contrast to the combined datasets (purple and blue), which feature a significant CCMG component. The EFT dataset seems to put lower constraints on ${\sigma }_8$ than DES-Y1, as shown in its posteriors for each dataset, which explains the more significant CCMG component when using the EFT instead of DES-Y1 (blue on each side). Here we also include datasets without SH0ES (salmon), in which the CCMG significance is almost completely erased, indicating the low preference of the model by all other datasets.