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

The Hubble horizon at matter-radiation equality ($k_{\mathrm{e}\mathrm{q}}^{-1}$) and the sound horizon at the last scattering surface ($r_s(z_{\ast })$) provides interesting consistency check for the $\mathrm{\Lambda }$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 $k_{\mathrm{e}\mathrm{q}}$. Adding extra radiational component to the early universe can reduce $k_{\mathrm{e}\mathrm{q}}$. The addition of early dark energy (EDE), however, tends to increase $k_{\mathrm{e}\mathrm{q}}$. We perform $k_{\mathrm{e}\mathrm{q}}$- and $r_s$-based analyses in both the EDE model and the Wess-Zumino Dark Radiation (WZDR) model. In the latter case we find $\mathrm{\Delta }{H}_0=0.4$ between the $r_s$- and $k_{\mathrm{e}\mathrm{q}}$-based datasets, while in the former case we find $\mathrm{\Delta }{H}_0=1.2$. This result suggests that the dark radiation scenario is more consistent in the fit of the two standard rulers ($k_{\mathrm{e}\mathrm{q}}$ and $r_s$). As a forecast analyses, we fit the two models with a mock $k_{\mathrm{e}\mathrm{q}}$ prior derived from \emph{Planck} best-fit $\mathrm{\Lambda }$CDM model. Compared with the best-fit $H_0$ in baseline $\mathrm{\Lambda }$CDM model, we find $\mathrm{\Delta }{H}_0=1.1$ for WZDR model and $\mathrm{\Delta }{H}_0=-2.4$ for EDE model.