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

Abstract

The Hubble horizon at matter-radiation equality ($k_{\mathrm{eq}}^{-1}$) and the sound horizon at the last scattering surface ($r_s(z_{\ast })$) provides interesting consistency check for the $\Lambda$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 $\Delta H_0=0.4$ between the $r_s$- and $k_{\mathrm{eq}}$-based datasets, while in the former case we find $\Delta H_0=1.2$. 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 \emph{Planck} best-fit $\Lambda$CDM model. Compared with the best-fit $H_0$ in baseline $\Lambda$CDM model, we find $\Delta H_0=1.1$ for WZDR model and $\Delta H_0=-2.4$ for EDE model.

12 pages, 7 figures

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