Ups and Downs in Dark Energy: phase transition in dark sector as a proposal to lessen cosmological tensions

Based on tensions between the early and late time cosmology, we proposed a double valued cosmological constant which could undergo a phase transition in its history. It is named "double-$\Lambda$ Cold Dark Matter": $\Lambda\Lambda$CDM. An occurred phase transition results in (micro-) structures for the dark sector with a proper (local) interaction. In this paper, inspired by the physics of critical phenomena, we study a simplified model such that the cosmological constant has two values before a transition scale factor, $a_t$, and afterwards it becomes single-valued. We consider both the background and perturbation data sets including CMB, BAO distances and R19 data point. $\Lambda\Lambda$CDM has its maximum likelihood for $a_t= 0.916^{+0.055}_{-0.0076}$ and $H_0= 72.8\pm 1.6$. This result shows no inconsistency between early and late time measurements of Hubble parameter in $\Lambda\Lambda$CDM model. In comparison to $\Lambda$CDM, our model has better fit to data such that $\Delta\chi^2=-11$ and even if we take care of two additional degrees of freedom we do have better AIC quantity $\Delta$AIC$=-7$. We conclude that a phase transition in the behavior of dark energy can address $H_0$ tension successfully and may be responsible for the other cosmological tensions.