Constraints on the phase transition of Early Dark Energy with the CMB anisotropies

Abstract

Early dark energy (EDE) models have attracted attention in the context of the recent problem of the Hubble tension. Here we extend these models by taking into account the new density fluctuations generated by the EDE which decays around the recombination phase. We solve the evolution of the density perturbations in dark energy fluid generated at the phase transition of EDE as isocurvature perturbations. Assuming that the isocurvature mode is characterized by a power-law power spectrum and is uncorrelated with the standard adiabatic mode, we calculate the CMB angular power spectra. By comparing them to the Planck data using the Markov-Chain Monte Carlo method, we obtained zero-consistent values of the EDE parameters and $H_0=67.56^{+0.65}_{-0.66}~\mathrm{km} \, \mathrm{s}^{-1} \mathrm{Mpc}^{-1}$ at $68 \%$ CL. This $H_0$ value is almost the same as the Planck value in the $\Lambda$CDM model, $H_0=67.36 \pm 0.54~\mathrm{km} \, \mathrm{s}^{-1} \mathrm{Mpc}^{-1}$, and there is still a $\sim 3.5 \sigma$ tension between the CMB and Type Ia supernovae observations. Including CMB lensing, BAO, supernovae and SH0ES data sets, we find $H_0=68.94^{+0.47}_{-0.57}~\mathrm{km} \, \mathrm{s}^{-1} \mathrm{Mpc}^{-1}$ at $68 \%$ CL. The amplitude of the fluctuations induced by the phase transition of the EDE is constrained to be less than $1$--$2$ percent of the amplitude of the adiabatic mode. This is so small that such non-standard fluctuations cannot appear in the CMB angular spectra. In conclusion, the isocurvature fluctuations induced by our simplest EDE phase transition model do not explain the Hubble tension well.