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Investigating the $H_0$ Tension and Expansion-History Mismatch with Diverse Dark Energy Parametrization Frameworks

Upala Mukhopadhyay, Purba Mukherjee, Alexandre Tkatchenko

Abstract

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.

Investigating the $H_0$ Tension and Expansion-History Mismatch with Diverse Dark Energy Parametrization Frameworks

Abstract

The CDM model successfully explains a wide range of cosmological observations; however, persistent discrepancies most notably the tension between early and late time measurements challenge its completeness. No proposed extension has yet resolved this tension while retaining the overall success of CDM. In this work, we investigate whether the 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 . We find that 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 tension, reducing it to , while the other models do not provide significant improvement. A detailed analysis of DESI DR2 data further reveals notable deviations in at and 0.706, whereas higher redshift measurements remain consistent within . These results suggest that late-time modifications primarily reshape the redshift dependence of the mismatch in 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.

Paper Structure

This paper contains 6 sections, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Models and Parameterizations
  3. Observational Data Sets
  4. Evolution of the Hubble Parameter
  5. Dark Energy Density and Equation of State
  6. Summary and Discussions

Figures (5)

  • Figure 3: Plots for the PP model. (a) One-dimensional posterior distributions and two-dimensional marginalized contours for PP (upper panel). (b) Reconstructed $H(z)/(1+z)$ vs $z$ for PP in the full redshift range $0 \leq z \leq 1100$ (lower panel). The inset highlights the lower redshift range $0 \leq z \leq 10$. The tomato and blue regions correspond to the allowed spaces for Pantheon+SH0ES+DESI and Planck+DESI, respectively.
  • Figure 4: Plots for the GM parametrization. (a) One-dimensional posterior distributions and two-dimensional marginalized contours for GM (upper panel). (b) Reconstructed $H(z)/(1+z)$ vs $z$ for GM in the full redshift range $0 \leq z \leq 1100$ (lower panel). The inset highlights the lower redshift range $0 \leq z \leq 10$. The tomato and blue regions correspond to the allowed spaces for Pantheon+SH0ES+DESI and Planck+DESI respectively.
  • Figure 5: Heatmaps showing the statistical significance of deviations between theoretical predictions of the Hubble parameter $H(z)$ and DESI measurements at redshifts $z = 0.51, 0.706, 0.934, 1.321, 1.484,$ and $2.33$. Predictions are derived within the $\Lambda$CDM, CPL, PP, and GM models. The color scale represents the deviation in units of $\sigma$.
  • Figure 6: Reconstructed DE equation of state $w_{\rm DE}(z)$ for the CPL, PP, and GM models. The upper panel shows the reconstruction obtained using low-$z$ data, while the lower panel corresponds to high-$z$ data.
  • Figure 7: Evolution of the DE density normalized to the present-day critical energy density, $\rho_{\rm DE}/\rho_{c0}$, reconstructed for the $\Lambda$CDM, CPL, PP, and GM models. The upper panel shows the reconstruction using low-$z$ data, while the lower panel corresponds to high-$z$ data.