Information-Geometric Perspective on the Hubble Tension: Eigenmode Rotation and Curvature Suppression in wCDM

Abstract

The Hubble tension is shaped not only by shifts between early- and late-time parameter estimates, but also by the stiffness of the constraints that define them. In this work, we analyze this geometric structure in the wCDM model by separating the discrepancy into two components: a parameter displacement and a directional Fisher curvature. Within the local Gaussian approximation, the quadratic tension along a given direction factorizes into the squared shift and the combined directional curvature contributed by the datasets. Applying this framework to Planck, DESI DR2, and SH0ES, we show that extending \LambdaCDM to wCDM primarily reshapes the Fisher geometry of the CMB constraint rather than opening a genuinely new route to concordance. Allowing the dark-energy equation-of-state parameter w to vary suppresses the leading Planck Fisher eigenvalue to only \sim 2.7 % of its \LambdaCDM value, while producing only a modest rotation of the dominant acoustic-scale eigenmode. The net effect is a strong softening of the effective acoustic rigidity. At the same time, high-precision late-time data, especially from DESI DR2, inject substantial curvature along the expansion-rate direction. This added stiffness acts as a geometric wall, closing off phantom-like escape routes and sharply limiting tension relief within the extended parameter space. Our results indicate that changes in the inferred H_0 tension under model extension are best understood as a reconfiguration of the constraint manifold rather than as evidence for new physical agreement. The shift-curvature decomposition thus offers a simple, fast, and physically transparent way to diagnose cosmological tensions.

Figures (6)

• Figure 1: Redistribution of the Planck Fisher eigenvalue spectrum under the $\Lambda$CDM $\to$$w$CDM extension. Left: Comparison of leading eigenvalues. $\Lambda$CDM (filled circles) and $w$CDM (filled squares) illustrate the principal stiffness suppression by a factor $\sim 37.5$. Right: Full three-dimensional spectrum in $w$CDM (filled circles), where the third eigenvalue indicates the emergence of a new degeneracy direction. The Fisher curvature is redistributed, weakening the acoustic rigidity without introducing a new stiff direction aligned with $w$.
• Figure 2: Validation of the local Gaussian approximation in Planck $w$CDM. The shaded density map shows the full MCMC posterior in the $(\Omega_{\rm{m}0}, H_0)$ plane. Solid and dashed ellipses denote the 68% and 95% confidence regions derived from the marginalized Fisher matrix, while the cross marks the maximum-likelihood point. The Fisher ellipse faithfully reproduces the local orientation and curvature scale near the maximum, justifying its use in the present analysis.
• Figure 3: Directional curvature fractions in the rdFREE configuration. Left: projection along $\ln H_0$. Right: projection along $H_0$. In both parameter bases DESI contributes zero curvature along the expansion-rate axis. The effective stiffness reduces to a binary Planck-SH0ES balance, with relative contributions of approximately $60{:}40$ in the $\ln H_0$ basis and $61{:}39$ in the $H_0$ basis.
• Figure 4: Directional curvature fractions in the rdFIX configuration. Left: projection along $\ln H_0$. Right: projection along $H_0$. Fixing the sound horizon breaks the $(h,r_d)$ degeneracy present in rdFREE, allowing DESI to contribute directly to the expansion-rate curvature. In both parameter bases, DESI supplies nearly $90\%$ of the total stiffness along the expansion-rate axis, while Planck and SH0ES provide only minor contributions.
• Figure 5: Illustrative geometric representation of the three–probe constraint structure in the $(w,H_0)$ plane. The solid curve shows the Planck degeneracy obtained by scanning the dark energy equation of state parameter $w$. The horizontal band indicates the SH0ES measurement of $H_0$, while the ellipses represent the DESI+CMB and DESI+CMB+SNe constraints. The intersection illustrates how late-time datasets relocate the preferred cosmological solution within the Planck degeneracy manifold.