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

This paper reformulates the Hubble tension by analyzing the dimensionless expansion history $E(z)$ rather than the scalar $H_0$ , revealing that the discrepancies between Planck, DESI DR2, and Pantheon+SH0ES data are moderate (1–2 $\sigma$ ) and significantly lower than the conventional 4.9 $\sigma$ scalar tension when accounting for the correlated shape of the expansion history.

Original authors: Seokcheon Lee

Published 2026-05-08

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Seokcheon Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is a giant, expanding balloon. For a long time, cosmologists have been arguing about how fast this balloon is currently inflating. This argument is called the "Hubble Tension."

On one side, we have Planck, a satellite that looks at the "baby picture" of the universe (the Cosmic Microwave Background). Based on that picture, it predicts the balloon is inflating at a certain speed. On the other side, we have SH0ES and DESI, which are telescopes looking at the "adult universe" (supernovas and galaxy clusters). They measure the speed directly and say, "No, it's actually inflating much faster!"

The difference between these two speeds is huge—statistically speaking, it's a 5-sigma disagreement. That's like flipping a coin and getting heads 50 times in a row; it suggests something is seriously wrong with our understanding of physics.

The Paper's New Idea: Separating "Speed" from "Shape"

Author Seokcheon Lee argues that we've been comparing apples and oranges by just looking at the final speed number ( $H_0$ ).

Think of it like two runners on a track:

Runner A (Planck) measures the runner's speed by looking at the starting line and the total distance of the track, then doing some math to guess the speed.
Runner B (SH0ES) stands at the finish line with a stopwatch and measures the speed directly.

The paper asks: What if the disagreement isn't about the final speed, but about the shape of the track itself?

To fix this, the author reformulates the problem. Instead of just comparing the final speed, he compares the shape of the expansion history (called $E(z)$ ).

The Analogy: Imagine the expansion of the universe is a song. The "speed" ( $H_0$ ) is the volume of the song. The "shape" ( $E(z)$ ) is the melody or the rhythm.
The author separates the Volume (the absolute scale) from the Melody (how the speed changes over time).

What They Did

The author took the data from Planck, DESI, and SH0ES and forced them all to play the same "song" (using the same standard model of physics, called flat $\Lambda$ CDM). He then asked: If we ignore the volume and just look at the melody, do these three groups agree on the tune?

Planck predicts a specific melody based on the baby picture.
DESI (galaxy data) predicts a melody based on the middle-aged universe.
SH0ES (supernova data) predicts a melody based on the current universe.

The Results: The Melody is Actually Pretty Similar

Here is the surprising part of the paper:

The Old View: If you just compare the final speed numbers, the disagreement is massive (about 5 sigma). It looks like a crisis.
The New View: When you look at the melody (the shape of the expansion history) and account for the fact that the data points are all connected (like notes in a song), the disagreement shrinks dramatically.
- The disagreement between Planck and DESI drops to about 1.1 sigma (a small, manageable difference).
- The disagreement between Planck and SH0ES drops to about 2.1 sigma (a moderate difference, but not a crisis).

The "Volume" Problem Remains

The author is very clear: He did not solve the Hubble Tension.

The "Volume" (the absolute speed of the universe) is still different. The distance ladder (SH0ES) still says the universe is bigger/faster than Planck thinks. The paper just shows that the way the universe expands over time (the shape) is actually much more consistent between the groups than we thought.

The Takeaway

The paper suggests that the "Hubble Tension" might be less about the universe changing its tune (the shape of expansion) and more about a disagreement over the volume knob (the absolute scale).

Before: We thought the whole song was wrong.
Now: We know the melody is mostly the same, but we can't agree on how loud it should be.

This distinction is important because it tells scientists where to look next. Instead of trying to rewrite the laws of physics to change the "melody," they might need to focus on why the "volume" (the calibration of distances) is so different between the early universe and the late universe.

In short: The universe's expansion "song" is consistent across different measurements, but we still can't agree on how loud the song is.

Technical Summary: From Scalar H0 to E(z): A Reformulation of the Hubble Tension

Problem Statement
The Hubble tension is conventionally framed as a discrepancy between the low value of the Hubble constant ( $H_0$ ) inferred from Planck CMB data within the base $\Lambda$ CDM model and the higher values derived from late-time distance-ladder measurements (e.g., Pantheon+SH0ES). The paper argues that this scalar comparison is methodologically limited because it compresses distinct inference problems into a single derived parameter. The three primary probes—Planck CMB, DESI DR2 BAO, and Pantheon+SH0ES—do not constrain the same primitive observable. Instead, they constrain different combinations of physical quantities:

Planck CMB: Constrains physical densities ( $\omega_b, \omega_c$ ) and the acoustic angular scale ( $\theta_*$ ).
DESI DR2 BAO: Constrains standard-ruler-normalized distances ( $D_M/r_d, D_H/r_d, D_V/r_d$ ).
Pantheon+SH0ES: Constrains a calibrated luminosity-distance relation.

In each case, the quoted scalar $H_0$ is a derived quantity obtained only after projecting these probe-level observables through a background cosmological model and, in the case of the distance ladder, an absolute calibration. Consequently, a disagreement in derived $H_0$ values may reflect a mismatch in the reconstructed background history, a difference in projection geometry, or both, rather than a direct conflict in the primary observables.

Methodology
To disentangle the absolute-scale discrepancy from the shape of the expansion history, the authors reformulate the comparison in terms of the dimensionless expansion history $E(z) \equiv H(z)/H_0$ . The analysis proceeds as follows:

Common Model Framework: All reconstructions are performed within a fixed, flat base $\Lambda$ CDM framework. This isolates the comparison to the normalized history shape, removing the absolute normalization encoded in $H_0$ .
Posterior-Implied Histories: Rather than performing a model-independent nonparametric reconstruction, the authors map the posterior distributions of the three probes onto $E(z)$ $E (z)$ histories.
- For Planck, they use published base- $\Lambda$ CDM posterior chains.
- For DESI DR2 and Pantheon+SH0ES, they utilize Gaussian posterior approximations in the flat- $\Lambda$ CDM parameter space (specifically $\Omega_{m0}$ for the shape of $E(z)$ ).
Covariance-Aware Comparison: Because the reconstructed $E(z)$ $E (z)$ values at different redshifts are strongly correlated in flat $\Lambda$ $Λ$ CDM (driven primarily by $\Omega_{m0}$ $Ω_{m 0}$ ), a simple pointwise comparison or a diagonal $\chi^2$ $χ^{2}$ sum would over-count the tension. The authors employ a covariance-subspace analysis:
- They compute the covariance matrix of the history difference ( $\Delta E$ ) between probes.
- They perform an eigenmode decomposition of the covariance spectrum, which is found to be nearly rank-one.
- They retain the leading two eigenmodes ( $N_{kept}=2$ ) to define a regularized covariance subspace.
- The global mismatch is quantified by the history displacement statistic $S_{hist} = \sqrt{\chi^2_E}$ , where $\chi^2_E$ is calculated using the pseudo-inverse of the covariance matrix on the retained subspace.

Key Results
The analysis reveals that while the reconstructed dimensionless histories are not identical, the discrepancies are significantly milder than the conventional scalar $H_0$ tension:

Visual Offset: Relative to Planck, the DESI DR2 history lies slightly below the Planck-implied curve, while the Pantheon+SH0ES history lies above it.
Pointwise Significance: The maximum fractional differences are approximately 3.1% (DESI) and 6.4% (Pantheon+SH0ES). The maximum pointwise significances are roughly $1.68\sigma$ and $2.29\sigma$ , respectively.
Global Mismatch ( $S_{hist}$ ): When redshift covariance is properly accounted for:
- DESI DR2 vs. Planck: $S_{hist} \simeq 1.65$ . This corresponds to a two-sided one-dimensional Gaussian equivalent of approximately $1.1\sigma$ .
- Pantheon+SH0ES vs. Planck: $S_{hist} \simeq 2.55$ . This corresponds to a two-sided one-dimensional Gaussian equivalent of approximately $2.1\sigma$ .
Comparison to Scalar Tension: These values are substantially lower than the conventional $\sim 4.9\sigma$ discrepancy observed in the scalar $H_0$ comparison.

Significance and Claims
The paper claims that the scalar $H_0$ discrepancy and the normalized-history discrepancy represent different statistical questions. The results demonstrate that:

Scale-Shape Decomposition: The Hubble tension can be decomposed into an absolute-scale component (diagnosed by scalar $H_0$ ) and a normalized-shape component (diagnosed by $E(z)$ ). The analysis shows that the tension is localized primarily in the absolute-scale projection rather than in the late-time dynamical shape of the expansion history.
Projection Dependence: The large scalar discrepancy does not automatically imply an equally large mismatch in the reconstructed background history. The difference arises because the same symbol $H_0$ denotes quantities obtained through different inference chains (acoustic scales vs. calibrated distances) projected through the model manifold.
Methodological Caution: The study highlights that treating scalar parameter comparisons as complete diagnostics of cosmological consistency can be misleading. It emphasizes the importance of considering the geometry of the likelihood and the correlations among derived parameters.

Limitations
The authors explicitly state that this result does not resolve the Hubble tension or remove the absolute-scale discrepancy encoded in the distance ladder. The analysis is restricted to the flat base $\Lambda$ CDM model; extending the comparison to models with additional degrees of freedom (e.g., $w$ CDM) would address a different problem involving the rotation and broadening of the posterior in $H_0$ , rather than a direct test of the flat- $\Lambda$ CDM result presented here. The paper concludes that within the specific context of a common flat- $\Lambda$ CDM projection, the probe posteriors show a weaker mismatch in the normalized shape-level history than in the absolute-scale scalar projection.