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On the Difficulties with Late-Time Solutions for the Hubble Tension

This paper demonstrates that late-time cosmological models cannot simultaneously resolve the Hubble tension while fitting SH0ES, DESI, and supernova data unless they invoke unphysical mechanisms like a sharp low-redshift step in supernova absolute magnitudes or a violation of the distance duality relation, effectively decoupling the datasets rather than providing a genuine physical solution.

Original authors: Prakhar Bansal, Dragan Huterer

Published 2026-02-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Prakhar Bansal, Dragan Huterer

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 balloon being blown up. For decades, scientists have been trying to measure exactly how fast this balloon is inflating. This speed is called the Hubble Constant.

Here's the problem: When we look at the "baby pictures" of the universe (light from the Big Bang), the math says the balloon is inflating at one speed (let's call it Speed A). But when we look at the "teenage pictures" (exploding stars nearby), the math says it's inflating much faster (Speed B).

This disagreement is called the "Hubble Tension." It's like two friends measuring the same car's speed; one says 60 mph, the other says 75 mph. They can't both be right, unless one of them is using a broken speedometer or the car is doing something weird.

This paper, written by Prakhar Bansal and Dragan Huterer, asks a very specific question: Can we fix this by changing how the universe expands recently (in the "late-time" era)?

Here is the breakdown of their findings, using simple analogies:

1. The "Smooth Road" Doesn't Work

Scientists have tried to fix the problem by suggesting the universe's expansion speed changed smoothly over time, like a car gently pressing the gas pedal.

The authors ran the numbers and found that smooth changes don't work. No matter how you tweak the "gas pedal" (the expansion rate) in the recent past, you cannot make the "baby picture" math and the "teenage picture" math agree at the same time. It's like trying to fix a broken speedometer by just driving the car more smoothly; the needle is still stuck.

2. The Only "Fixes" Are Cheating (or Breaking Physics)

The paper finds that you can make the math work, but only if you do one of two very strange things:

Option A: The "Magic Switch" on the Stars
Imagine you have a ruler to measure distance. Suddenly, at a very specific, very recent moment in time (when the universe was almost as old as it is now), the ruler suddenly shrinks or grows.

The Paper's Claim: The only way to fit the data is if the "brightness" of the exploding stars (Type Ia supernovae) suddenly changed at a very low redshift (about $z \sim 0.01$ ).
The Analogy: It's as if the stars decided to suddenly turn off their lights or turn them up to maximum brightness right before our eyes. If you allow the stars to change their brightness abruptly, you can separate the "nearby" measurements from the "faraway" ones.
The Catch: This is considered a "trivial" solution. It doesn't really explain why the tension exists; it just says, "Let's pretend the nearby stars are different from the faraway ones so the math works." It effectively ignores the conflict rather than solving it.

Option B: Breaking the Rules of Geometry
There is a fundamental rule in physics (called the Etherington duality relation) that links how bright an object looks to how big it appears.

The Paper's Claim: The only other way to fix the tension is if this rule is broken.
The Analogy: Imagine a rule that says, "If a ball looks twice as far away, it must look four times smaller." If you break this rule, you can make the measurements fit.
The Catch: This is like saying the laws of geometry are wrong. While possible in theory, it's a huge, radical change to our understanding of the universe.

3. The "Middle Ground" Solution (The Physical Model)

The authors also tested a more "real" physical model involving a mysterious field (a scalar field) that interacts with gravity.

The Result: This model did a better job than the "smooth road" idea, but it still wasn't perfect. It managed to get the math to fit somewhat better, but only because it allowed the stars to change their brightness gradually over a longer period (around $z \sim 0.15$ ).
The Analogy: Instead of a sudden "magic switch," this is like a dimmer switch that slowly turns the lights down over a few years. It helps a little bit, but it doesn't solve the problem as well as the "magic switch" (Option A) because the change wasn't sharp enough to completely separate the conflicting data.

4. What About New Data?

The authors also checked if adding a new type of measurement (Surface Brightness Fluctuations, or SBF—measuring the "fuzziness" of galaxy light) would change their conclusions.

The Result: It didn't change much. Even with this new data, the conclusion remains: you can't fix the Hubble Tension with simple, smooth changes to the universe's expansion history. You still need either a sudden change in star brightness or a break in the laws of physics.

The Bottom Line

The paper concludes that modifying the recent history of the universe's expansion is not a viable solution to the Hubble Tension.

If you want to fix the math, you have to either:

Assume the nearby exploding stars suddenly changed their brightness (which effectively ignores the conflict).
Assume the fundamental rules of how light and distance work are broken.

The authors suggest that the "Hubble Tension" is likely not solved by tweaking the expansion rate, but perhaps by looking for errors in the measurements themselves or by considering physics from the very early universe (which this paper did not focus on).

Technical Summary: On the Difficulties with Late-Time Solutions for the Hubble Tension

Problem Statement
The paper addresses the "Hubble tension," a significant discrepancy ( $\sim 5\sigma$ ) between the local direct measurement of the Hubble constant ( $H_0 \approx 73.04 \pm 1.04$ km s $^{-1}$ Mpc $^{-1}$ ) by the SH0ES collaboration and the value inferred from early-universe physics via the Cosmic Microwave Background (CMB) and Baryon Acoustic Oscillations (BAO) ( $H_0 \approx 67.14 \pm 0.47$ km s $^{-1}$ Mpc $^{-1}$ ). While early-universe solutions (modifying the sound horizon) are theoretically possible but fine-tuned, late-time solutions (modifying the expansion history at $z < 1$ ) have been suggested to be even less feasible. The authors aim to quantitatively demonstrate why phenomenological models modifying the late-time expansion history struggle to simultaneously fit SH0ES measurements, Type Ia supernova (SNIa) distances, and DESI BAO data without invoking trivial or unphysical mechanisms.

Methodology and Models
The authors assume a spatially flat cosmology within General Relativity and utilize a Monte Carlo Markov Chain (MCMC) analysis (Cobaya/GetDist) to constrain parameters against a combined dataset: DESI DR2 BAO, compressed CMB likelihood, and the full PantheonPlus+SH0ES SNIa catalog (including calibrators).

They investigate several phenomenological models designed to alter the late-time expansion history:

$\Lambda$ CDM: The standard baseline.
$H_{step}$ : A model allowing a sharp step transition in the Hubble parameter $H(z)$ at a redshift $z_t$ .
$M_{step}$ : A model allowing a sharp step in the absolute magnitude ( $M$ ) of SNIa at $z_t$ , while keeping $\Lambda$ CDM expansion history.
$H_{step} + M_{step}$ : A combination of both transitions.
$w_0w_a + M_{step}$ : A model using the dynamical dark energy equation of state ( $w_0, w_a$ ) combined with an $M_{step}$ .
Non-minimally coupled scalar field: A physical model (based on Wolf et al.) featuring a time-varying gravitational constant $G_{eff}$ , which is linked phenomenologically to a smooth transition in SNIa absolute magnitude.

The analysis also incorporates Surface Brightness Fluctuation (SBF) distance measurements to test the robustness of conclusions against the specific SH0ES calibration.

Key Results
The study yields the following quantitative findings regarding the goodness of fit ( $\Delta\chi^2$ relative to best-fit $\Lambda$ CDM):

Failure of Expansion History Modifications: Models that only modify the expansion rate (e.g., $H_{step}$ , $w_0w_a$ CDM) fail to resolve the tension. The $H_{step}$ model improves the fit negligibly ( $\Delta\chi^2 \approx -2$ ). Even with spatial curvature ( $\Omega_K$ ) or dynamical dark energy, the improvement is insufficient to reconcile the data. The primary issue is that SNIa data, when anchored to BAO distances, prefer an absolute magnitude $M \approx -19.4$ , which is $4\text{--}5\sigma$ away from the SH0ES calibrator value ( $M \approx -19.25$ ). Modifying $H(z)$ does not decouple these conflicting constraints.
The Necessity of an $M$ -Transition: Significant improvement in fit ( $\Delta\chi^2 \approx -40$ $Δ χ^{2} \approx - 40$ to $-43$) is achieved only by models that introduce a sharp step in the absolute magnitude of SNIa ( $M_{step}$ $M_{s t e p}$ ).
- The optimal transition occurs at a very low redshift ( $z_t \approx 0.01$ ).
- This effectively decouples the SH0ES calibration (Cepheids) from the rest of the Hubble diagram, allowing the high-redshift SNIa to align with BAO/CMB constraints while preserving the local SH0ES value.
- Adding an $H$ -step to an $M$ -step ( $H_{step}+M_{step}$ ) provides no significant additional benefit over the $M_{step}$ alone.
Alternative Trivial Solutions: The authors identify that breaking the Etherington distance duality relation ( $d_L = (1+z)^2 d_A$ ) is another theoretical possibility that would decouple BAO and SNIa normalizations, though this represents a radical departure from standard physics.
Role of SBF Data: Including Surface Brightness Fluctuation (SBF) data (calibrated via TRGB) does not alter the qualitative conclusions. When systematic uncertainties in the SBF zero-point are included, the data becomes less constraining, and the preference for an $M$ -transition remains the dominant factor for improving the fit.
Physical Model Performance: The non-minimally coupled scalar field model, which induces a smooth transition in $G_{eff}$ and consequently in $M$ , achieves a moderate improvement ( $\Delta\chi^2 \approx -27$ ). However, because its transition is gradual and occurs at $z \approx 0.15$ rather than the sharp, low- $z$ ( $z \approx 0.01$ ) transition preferred by the data, it does not resolve the tension as effectively as the phenomenological $M_{step}$ models.

Significance and Conclusions
The paper concludes that late-time solutions to the Hubble tension face a fundamental geometric and observational hurdle: the SH0ES Cepheid calibration and the BAO/SNIa distance ladder are in direct conflict regarding the absolute magnitude of SNIa.

The authors argue that any model modifying the late-time expansion history can only fit the combined dataset satisfactorily if it effectively decouples the SH0ES measurements from the rest of the cosmological data. This is achieved either by:

Introducing a sharp, unphysical step in the absolute magnitude of SNIa at very low redshift ( $z \sim 0.01$ ).
Breaking the Etherington distance duality relation.

The paper posits that these solutions are "trivial" because they essentially remove the tension by discarding the consistency between the local distance ladder and the global cosmological constraints, rather than solving the underlying physical discrepancy. Consequently, the authors suggest that late-time modifications to the expansion history alone are unlikely to provide a compelling, non-trivial resolution to the Hubble tension.

1. The "Smooth Road" Doesn't Work

2. The Only "Fixes" Are Cheating (or Breaking Physics)

3. The "Middle Ground" Solution (The Physical Model)

4. What About New Data?

The Bottom Line

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