Constraining holographic dark energy and analyzing cosmological tensions

We investigate cosmological constraints on the holographic dark energy (HDE) using the state-of-the-art cosmological datasets: Planck CMB angular power spectra and weak lensing power spectra, Atacama Cosmology Telescope (ACT) temperature power spectra, baryon acoustic oscillation (BAO) and redshift-space distortion (RSD) measurements from six-degree-field galaxy survey and Sloan Digital Sky Survey (DR12 & DR16) and the Cepheids-Supernovae measurement from SH0ES team (R22). We also examine the HDE model and $\Lambda$CDM with and without $N_{\mathrm{eff}}$ (effective number of relativistic species) being treated as a free parameter. We find that the HDE model can relieve the tensions of $H_0$ and $S_8$ to certain degrees. With “Planck+ACT+BAO+RSD” datasets, the constraints are $H_0=69.70\pm 1.39{\text{km s}}^{-1}{\mathrm{Mpc}}^{-1}$ and $S_8=0.823\pm 0.011$ in HDE model, which brings down the Hubble tension down to $1.92\sigma$ confidence level (C.L.) and the $S_8$ tension to $(1-2)\sigma$ C.L. By adding the R22 data, their values are improved as $H_0=71.86\pm 0.93{\text{km s}}^{-1}{\mathrm{Mpc}}^{-1}$ and $S_8=0.813\pm 0.010$, which further brings the Hubble tension down to $0.85\sigma$ C.L. and relieves the $S_8$ tension. We also quantify the goodness-of-fit of different models with Akaike information criterion (AIC) and Bayesian information criterion (BIC), and find that the HDE agrees with the observational data better than the $\Lambda$CDM and other extended models (treating $N_{\mathrm{eff}}$ as free for fitting).