GPAge: a Global Parameterization based on cosmic Age for some unified early-time and late-time solutions to the Hubble tension

The over $5\sigma$ Hubble tension continues to motivate modifications of the standard $\mathrm{\Lambda }$-cold-dark-matter ($\mathrm{\Lambda }\mathrm{C}\mathrm{D}\mathrm{M}$) model. The large number of proposed resolutions for the Hubble tension, broadly divided into early- and late-time solutions, motivates phenomenological parametrizations to constrain the viable solution space. Recently, there is a growing interest in unifying early- and late-time solutions. In this work, we extend the original Parametrization based on cosmic Age (PAge) to cover both the early and late Universe. Specifically, we incorporate the radiation-dominated epoch at the background level and complete the perturbation sector within a unified-dark-fluid framework. The resulting parametrization can accurately reproduce target cosmological models at both the background and perturbation levels. Confronting this global PAge with cosmological observations, we find that it improves the best fit relative to $\mathrm{\Lambda }\mathrm{C}\mathrm{D}\mathrm{M}$, but does not raise $H_0$ significantly enough to alleviate the Hubble tension. Our parametrizations indicate that varying the early-time matter-like density, even together with a flexible late-time expansion history, is insufficient to resolve the Hubble tension. These results suggest that resolving the Hubble tension requires pre-recombination physics beyond a simple retuning of the matter-like density.