Equality scale-based and sound horizon-based analysis of the Hubble tension

Abstract

The Hubble horizon at matter-radiation equality (${k}_{\mathrm{eq}}^{\ensuremath{-}1}$) and the sound horizon at the last scattering surface [${r}_{s}({z}_{*})$] provides an interesting consistency check for the standard $\mathrm{\ensuremath{\Lambda}}$ Cold Dark Matter ($\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$) model and its extensions. It is well known that the reduction of ${r}_{s}$ can be compensated by the increase of ${H}_{0}$, while the same is true for the standard rulers ${k}_{\mathrm{eq}}$. Adding extra radiational component to the early Universe can reduce ${k}_{\mathrm{eq}}$. The addition of early dark energy (EDE), however, tends to increase ${k}_{\mathrm{eq}}$. We perform ${k}_{\mathrm{eq}}$- and ${r}_{s}$-based analyses in both the EDE model and the Wess-Zumino dark radiation (WZDR) model. In the latter case, we find $\mathrm{\ensuremath{\Delta}}{H}_{0}=0.4$ between the ${r}_{s}$- and ${k}_{\mathrm{eq}}$-based datasets, while in the former case, we find $\mathrm{\ensuremath{\Delta}}{H}_{0}=1.2$. This result suggests that the dark radiation scenario is more consistent in the fit of the two standard rulers (${k}_{\mathrm{eq}}$ and ${r}_{s}$). As a forecast analyses, we fit the two models with a mock ${k}_{\mathrm{eq}}$ prior derived from Planck best-fit $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model. Compared with the best-fit ${H}_{0}$ in baseline $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model, we find $\mathrm{\ensuremath{\Delta}}{H}_{0}=1.1$ for the WZDR model and $\mathrm{\ensuremath{\Delta}}{H}_{0}=\ensuremath{-}2.4$ for EDE model.