Phenomenological gravitational phase transition: Reconciliation between the late and early Universe

In this work we investigate whether a certain phenomenological extension to the general relativity (GR), in the form of a gravitational phase transition (GPT) in the Universe, can reduce the external $Planck$ tensions with the local Hubble measurements and the distribution of matter, characterized by $\sigma_8$, as well as its internal inconsistencies in the lensing amplitude and the low--high $\ell$ parameter estimates. We introduce new degrees of freedom into the background and the two scalar perturbation equations in the Newtonian gauge, with simultaneous transitions from an early gravitational phase equivalent to GR toward a late phase. We model the transition as a "tanh" parametrized by the transition redshift $z_{\rm t}$ and width $\alpha$, with amplitudes $\Lambda(z)$ and $(\mu(z),\gamma(z))$ for the background and perturbations respectively. We verify the consistency of the datasets used in this work in the GPT framework and confirm that the individual tensions do not require conflicting transitions. We find that the joint datasets prefer a recent transition at $z_{\rm t}\approx 0.9$ in the background and perturbed Einstein equations, driven mainly by the local Hubble measurement. This transition relaxes all the tensions considered in this work.