The Hubble horizon at matter-radiation equality ($k_{\mathrm{eq}}^{-1}$) and the sound horizon at the last scattering surface [$r_s(z_{*})$] provides an interesting consistency check for the standard $\mathrm{\Lambda }$ Cold Dark Matter ($\mathrm{\Lambda }\mathrm{CDM}$) 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 $\mathrm{\Delta }{H}_0=0.4$ between the $r_s$- and $k_{\mathrm{eq}}$-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 ($k_{\mathrm{eq}}$ and $r_s$). As a forecast analyses, we fit the two models with a mock $k_{\mathrm{eq}}$ prior derived from Planck best-fit $\mathrm{\Lambda }\mathrm{CDM}$ model. Compared with the best-fit $H_0$ in baseline $\mathrm{\Lambda }\mathrm{CDM}$ model, we find $\mathrm{\Delta }{H}_0=1.1$ for the WZDR model and $\mathrm{\Delta }{H}_0=-2.4$ for EDE model.

• Received 20 October 2022
• Accepted 10 March 2023

DOI:https://doi.org/10.1103/PhysRevD.107.063536