From Scalar $H_0$ to $E(z)$: A Reformulation of the Hubble Tension

Abstract

The Hubble tension is usually expressed as a discrepancy between the low H_0 inferred from Planck CMB data within base \LambdaCDM and the higher value obtained from late-time distance-ladder measurements. This scalar comparison compresses distinct inference problems into one derived parameter: Planck CMB, DESI DR2 BAO, and Pantheon+SH0ES constrain physical densities and acoustic scales, ruler-normalized distances, and calibrated luminosity-distance relations, respectively. We reformulate the comparison in terms of the dimensionless expansion history E(z)=H(z)/H_0. This does not remove the absolute-scale discrepancy, but separates the normalization encoded in $H_0$ from the redshift-dependent shape of the expansion history. Within a common flat-\LambdaCDM framework, each probe posterior is mapped onto posterior-implied E(z) histories. Since the reconstructed values E(z_k) are strongly correlated across redshift, we quantify the global mismatch with a covariance-subspace history displacement S_{hist}, alongside pointwise redshift differences. The histories are not identical, but the discrepancies are moderate: the pointwise significance is typically 1-2\sigma, while S_{hist} simeq 1.65 for DESI DR2 and S_{hist} \simeq 2.55 for Pantheon+SH0ES relative to Planck. With two retained covariance eigenmodes, these correspond to two-sided one-dimensional Gaussian equivalents of approximately 1.1\sigma and 2.1\sigma, both below the conventional \simeq 4.9\sigma Planck-SH0ES scalar-H_0 discrepancy.