⚛️ general relativity

Model-independent late-universe measurements of $H_0$ and $\Omega_K$ with the parametrization based on cosmic age-improved inverse distance ladder

This paper employs cosmic age-based parameterizations (PAge and MAPAge) within an improved inverse distance ladder framework using diverse late-universe datasets to model-independently measure a Hubble constant of $72.20 \pm 1.00$ km s $^{-1}$ Mpc $^{-1}$ , thereby reducing the $H_0$ tension to the $0.6\sigma$ level and confirming a flat universe consistent with current observational data.

Original authors: Guo-Hong Du, Tian-Nuo Li, Jia-Le Ling, Yan-Hong Yao, Jing-Fei Zhang, Xin Zhang

Published 2026-07-08

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CC BY 4.0

Original authors: Guo-Hong Du, Tian-Nuo Li, Jia-Le Ling, Yan-Hong Yao, Jing-Fei Zhang, Xin Zhang

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 as a giant, expanding balloon. For a long time, scientists have been trying to measure two specific things about this balloon: how fast it is currently inflating (the Hubble constant, or $H_0$ ) and whether the surface of the balloon is perfectly flat, slightly curved like a bowl, or curved like a saddle (the curvature parameter, $\Omega_K$ ).

Recently, a major problem has emerged. It's like two groups of surveyors measuring the same road.

Group A (The Early Universe Team): They look at the "baby photos" of the universe (light from the Big Bang) and calculate the speed. They say the road is about 67 units long.
Group B (The Late Universe Team): They look at the "adult photos" (nearby exploding stars and galaxies) and say the road is about 73 units long.

This disagreement is called the "Hubble Tension." It's like if one group said you are 5 feet tall and the other said you are 6 feet tall, and both were sure they were right. Furthermore, there's a disagreement about the shape of the road: some data suggests it's slightly curved, while other data says it's perfectly flat.

The New Approach: A "Model-Independent" Ladder

In this paper, the authors (Guo-Hong Du and colleagues) decided to stop guessing the shape of the universe based on a specific theory (like the standard "Lambda-CDM" model, which assumes a flat universe). Instead, they built a new measuring tool called an "Inverse Distance Ladder."

Think of this like calibrating a ruler:

The Anchor: They used a very reliable "standard ruler" called Baryon Acoustic Oscillations (BAO) from the new DESI data. This is like a known distance marker on the road.
The Calibration: To make sure their ruler works for the whole universe, they didn't just rely on one type of measurement. They used three different "calibrators" to check the ruler:
- Cosmic Chronometers (CC): Measuring the age of old stars (like checking the time on a grandfather clock).
- Strong Gravitational Lensing (SGL): Using massive galaxies as magnifying glasses to measure distances (like using a lens to see how far away a mountain is).
- Gamma-Ray Bursts (GRB): Using massive explosions as bright beacons (like using a lighthouse to measure distance).

The Secret Sauce: "PAge" and "MAPAge"

The tricky part is that the universe's expansion history is complex. Usually, scientists force the data to fit a specific mathematical curve. The authors used two new, flexible mathematical shapes called PAge and MAPAge.

The Analogy: Imagine trying to draw the path of a rollercoaster. The old method (Lambda-CDM) forces you to draw a perfect, smooth parabola. If the real rollercoaster has a weird bump, the perfect parabola doesn't fit, and your measurements are off.
The New Method: PAge and MAPAge are like flexible, bendable rulers. They don't force the data into a perfect shape; they let the data tell them what the shape actually is. This makes the measurement "model-independent"—it doesn't care about the old theories, it just looks at the raw data.

What Did They Find?

Solving the Speed Dispute: When they used their new flexible rulers with the latest data (DESI, Supernovae, and the three calibrators), they found the universe is expanding at a speed of roughly 72 km/s per Megaparsec.
- This result sits almost perfectly between the two warring groups.
- In the most advanced version of their model (MAPAge), the disagreement with the "Baby Photo" group drops from a massive 5-sigma (a huge conflict) to just 0.6 sigma. In statistics, this is like saying, "Hey, our measurements are actually very close; the difference is probably just a tiny bit of noise."
The Shape of the Universe: When they allowed the universe to be curved (not forcing it to be flat), the old standard model suggested the universe might be slightly curved. However, when they used their flexible PAge/MAPAge rulers, the curvature disappeared.
- The result: The universe is flat.
- The authors suggest that the "curved" result from the old model was an illusion caused by forcing the data into the wrong shape. When you let the data speak freely, it says the universe is flat.
Which Model Wins? They ran a statistical "vote" (Bayesian analysis) to see which mathematical shape fits the data best. The flexible PAge and MAPAge models got more votes than the old standard model. The data prefers the flexible rulers.

The Bottom Line

The authors didn't just measure the speed of the universe; they fixed the measuring tape. By using a flexible, theory-free approach and combining the latest data from powerful telescopes and cosmic explosions, they found:

The Hubble Tension (the speed disagreement) is almost gone.
The Curvature Tension (the shape disagreement) is resolved in favor of a flat universe.

It's as if they realized the surveyors were arguing because they were using rigid, broken rulers. Once they switched to flexible, high-tech rulers, everyone agreed on the measurements.

Technical Summary: Model-Independent Late-Universe Measurements of $H_0$ and $\Omega_K$

Problem Statement
The standard $\Lambda$ CDM cosmological model faces significant challenges, most notably the "Hubble tension," where local distance ladder measurements (e.g., SH0ES) yield $H_0 \sim 72-75$ km s $^{-1}$ Mpc $^{-1}$ , while early-universe observations (e.g., Planck CMB) assuming $\Lambda$ CDM yield $\sim 66-68$ km s $^{-1}$ Mpc $^{-1}$ . This discrepancy reaches a significance of $4-6\sigma$ . Additionally, inconsistencies exist regarding the curvature parameter $\Omega_K$ : Planck CMB data alone suggests a deviation from a flat universe ( $\Omega_K \neq 0$ ) at the $3\sigma$ level, whereas combinations with Baryon Acoustic Oscillation (BAO) data favor a flat universe. Most proposed resolutions to the Hubble tension assume a flat universe as a prior, which can introduce degeneracies or exacerbate tensions if curvature is non-zero. Therefore, a joint, model-independent measurement of $H_0$ and $\Omega_K$ using late-time data is crucial.

Methodology
The authors employ the Inverse Distance Ladder (IDL) framework, which calibrates the absolute magnitude of Type Ia supernovae ( $M_B$ ) by anchoring the BAO sound horizon ( $r_d$ ) using late-time observations, thereby avoiding early-universe CMB priors.

Parameterizations: To minimize dependence on specific cosmological models, the study utilizes two cosmic age-based parameterizations:
- PAge: A parameterization based on cosmic age expanded to the quadratic order in $t/t_0$ , defined by parameters $p_{age}$ and $\eta$ .
- MAPAge: An extension of PAge to the cubic order in cosmic time, introducing an additional parameter $\eta_2$ to account for the jerk parameter, offering higher accuracy and flexibility across extended redshift ranges.
Datasets: The analysis combines the latest DESI DR2 BAO data with:
- Supernovae: DESY5 sample and the re-calibrated DES-Dovekie sample.
- Calibrators: Strong Gravitational Lensing (SGL) from TDCOSMO, Cosmic Chronometers (CC), and Gamma-Ray Bursts (GRB) standardized via the Amati correlation.
Statistical Analysis: Bayesian inference is performed using the Cobaya package and PolyChord nested sampler. Model comparisons are conducted using the Bayes factor ( $\ln B_{ij}$ ) interpreted via the Jeffreys scale.

Key Contributions and Results

$H_0$ Measurements:
- Using the combined dataset (DESI + DES-Dovekie + SGL + CC + GRB), the study obtains model-independent $H_0$ values of $72.63 \pm 0.96$ km s $^{-1}$ Mpc $^{-1}$ (PAge) and $72.20 \pm 1.00$ km s $^{-1}$ Mpc $^{-1}$ (MAPAge).
- These results reduce the tension with the SH0ES measurement to $0.3\sigma$ (PAge) and $0.6\sigma$ (MAPAge).
- The precision of these late-universe measurements surpasses that of the SH0ES measurement.
- The PAge and MAPAge models generally yield lower $H_0$ values than $\Lambda$ CDM when fitting the same data, reflecting a preference in current DESI data for lower expansion rates potentially linked to dark energy dynamics.
Curvature ( $\Omega_K$ ) Constraints:
- In the $\Lambda$ CDM + $\Omega_K$ model, the data shows a $\sim 2\sigma$ preference for positive curvature ( $\Omega_K \approx 0.05-0.09$ ), consistent with the "curvature tension" observed in other analyses.
- However, when employing the model-independent PAge and MAPAge frameworks, this preference vanishes. The combined data yields $\Omega_K = 0.005 \pm 0.037$ (MAPAge + $\Omega_K$ ), indicating consistency with a spatially flat universe within the $1\sigma$ range.
- This suggests that the apparent curvature preference in $\Lambda$ CDM may be an artifact of the model's rigidity rather than a physical reality.
Model Comparison:
- Bayesian evidence analysis indicates that the current late-universe data provide weak to moderate evidence in favor of the PAge and MAPAge models over the $\Lambda$ CDM + $\Omega_K$ model.
- Specifically, the MAPAge model is favored over $\Lambda$ CDM + $\Omega_K$ with a Bayes factor of $\ln B \approx 3.30$ (moderate evidence) for the full dataset combination.

Significance
The paper demonstrates that utilizing cosmic age-based parameterizations within an inverse distance ladder framework allows for high-precision, model-independent constraints on $H_0$ and $\Omega_K$ . The primary significance lies in the finding that the apparent curvature tension and the severity of the Hubble tension are sensitive to the underlying cosmological model assumptions. When moving away from the rigid $\Lambda$ CDM prior to more flexible, model-independent descriptions (PAge/MAPAge), the data consistently supports a flat universe ( $\Omega_K = 0$ ) and significantly alleviates the Hubble tension. The authors conclude that current late-universe probes provide $H_0$ measurements competitive with CMB-based results while robustly supporting spatial flatness, independent of early-universe priors.

The New Approach: A "Model-Independent" Ladder

The Secret Sauce: "PAge" and "MAPAge"

What Did They Find?

The Bottom Line

Technical Summary: Model-Independent Late-Universe Measurements of H0H_0H0​ and ΩK\Omega_KΩK​

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Technical Summary: Model-Independent Late-Universe Measurements of $H_0$ and $\Omega_K$