Upala Mukhopadhyay, Purba Mukherjee, Alexandre Tkatchenko

The $Λ$CDM model successfully explains a wide range of cosmological observations; however, persistent discrepancies most notably the $H_0$ tension between early and late time measurements challenge its completeness. No proposed extension has yet resolved this tension while retaining the overall success of $Λ$CDM. In this work, we investigate whether the $H_0$ tension can be associated with a specific epoch in the cosmic expansion history and identify the redshift range most relevant for understanding its origin. In addition to the cosmological constant, we consider three phenomenological models based on general parametrizations of key quantities governing cosmic expansion: the dark energy (DE) equation of state, the DE pressure density, and the scale factor. Using early time Planck data and late time Pantheon+ (with and without SH0ES calibration) and DESI measurements, we constrain model parameters and examine the evolution of the Hubble parameter $H(z)$. We find that $Λ$CDM exhibits discrepancies across all redshifts, whereas the other models shift the dominant deviations toward low redshifts. Among the models considered, the pressure density parametrization alleviates the $H_0$ tension, reducing it to $\sim 2.7σ$, while the other models do not provide significant improvement. A detailed analysis of DESI DR2 data further reveals notable deviations in $H(z)$ at $z=0.51$ and 0.706, whereas higher redshift measurements remain consistent within $1σ$. These results suggest that late-time modifications primarily reshape the redshift dependence of the mismatch in $H(z)$ rather than fully resolve it, in the absence of systematic effects. Furthermore, the reconstructed DE dynamics exhibit qualitatively distinct behaviors across parametrizations, highlighting a persistent inconsistency between early and late Universe probes in describing the nature of DE.