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The Hubble Tension in Light of the Symmetry of Scale Invariance

This paper proposes that the Hubble tension between early and late Universe measurements can be resolved by adopting a scale-invariant vacuum (SIV) model with a matter density of $\Omega_m \simeq 0.20$ , which naturally reconciles the locally measured $H_0 \approx 74$ km/s/Mpc with cosmic microwave background data by accounting for scale invariance effects ignored in the standard $\Lambda$ CDM model.

Original authors: Frederic Courbin, Andre Maeder

Published 2026-02-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Frederic Courbin, Andre Maeder

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

The Big Problem: Two Different Speedometers

Imagine the Universe is a giant car driving away from us. Astronomers want to know exactly how fast it is going right now (this speed is called the Hubble Constant, or $H_0$ ).

The problem is that the car has two different speedometers, and they don't agree:

The "Local" Speedometer: If you look at nearby stars and galaxies (the "late" Universe), they tell you the car is going 74 km/s.
The "Remote" Speedometer: If you look at the "baby picture" of the Universe (the Cosmic Microwave Background, or CMB, from the "early" Universe) and try to calculate how fast the car should be going today, the math says it should only be 67.4 km/s.

This difference is huge in physics terms (a "6-sigma" tension). It's like if your GPS said you were driving 74 mph, but your engine computer said you were only doing 67 mph, and both claimed to be 100% accurate. Most scientists think the standard model of the Universe (called $\Lambda$ CDM) is missing something, but they haven't found what it is yet.

The New Idea: The Universe Has a "Ruler" That Shrinks

The authors of this paper, Frédéric Courbin and André Maeder, propose a different way to look at the Universe. They suggest we add a rule called "Scale Invariance" to our physics.

The Analogy:
Imagine you have a rubber ruler. In our standard view ( $\Lambda$ CDM), this ruler never changes size. If you measure a galaxy today and measure it a billion years ago, the ruler is the same length.

In the authors' new view (SIV model), the ruler does change. Specifically, the "ruler" of empty space expands slightly as time goes on. It's as if the fabric of space itself has a tiny, built-in stretching mechanism that the standard model ignores.

How They Tested It

To see if this new "stretching ruler" theory works, they ran two major tests:

1. The Supernova Test (The "Standard Candles")
Type Ia supernovae are exploding stars that act like lightbulbs of known brightness. By seeing how dim they look, we know how far away they are.

The Result: When the authors used their new "stretching ruler" math to fit the data from these exploding stars, the best fit required the Universe to have a specific amount of matter (about 20% of the critical density, written as $\Omega_m = 0.20$ ).
Comparison: The standard model usually needs about 30% matter to fit this data. The new model needs less.

2. The Age Test (The "Grandfather Clock")
They also looked at the relationship between three things: the speed of expansion ( $H_0$ ), the amount of matter ( $\Omega_m$ ), and the age of the Universe.

The Result: If the Universe is about 13.9 billion years old (the current accepted age), and the matter density is 20% (as found in the supernova test), the math in the new model points directly to a speed of 74 km/s.

The "Aha!" Moment: Solving the Tension

Here is the magic trick the paper reveals.

Imagine the Universe at the moment of Recombination (when the "baby picture" was taken, 380,000 years after the Big Bang). At that exact moment, both the Standard Model and the New Model agree on the conditions:

The temperature was 3,000 Kelvin.
The expansion speed was roughly 1.38 million km/s.

The Divergence:

The Standard Model ( $\Lambda$ CDM): It takes that starting point and runs the "non-stretching ruler" math forward to today. It concludes the speed must have slowed down to 67.4 km/s.
The New Model (SIV): It takes that exact same starting point but runs the "stretching ruler" math forward. Because the ruler stretches slightly over time, the math concludes the speed today is 74 km/s.

The Conclusion:
The "Hubble Tension" isn't because the early Universe data is wrong, or the local data is wrong. It's because the standard model is using the wrong "ruler" to connect the past to the present.

By adding the tiny effect of scale invariance (the stretching ruler), the authors show that you can start with the same "baby picture" (CMB) and end up with the faster speed (74 km/s) that we actually see today.

Summary

The paper claims that the conflict between the "early" and "late" Universe speeds disappears if we accept that empty space has a slight, measurable property called scale invariance. This property causes the Universe to expand slightly differently than we thought, allowing the "baby picture" of the Universe to perfectly match the "adult picture" we see today, all without needing to invent new, mysterious forces.

Key Takeaway: The tension isn't a measurement error; it's a physics error. The standard model is ignoring a small but real effect of how space scales over time.

Technical Summary: The Hubble Tension in Light of the Symmetry of Scale Invariance

Problem Statement
The paper addresses the "Hubble tension," a significant discrepancy between the local measurement of the Hubble constant ( $H_0 \approx 74$ km s $^{-1}$ Mpc $^{-1}$ ) derived from the late Universe (e.g., Type Ia supernovae, Cepheids) and the value inferred from the Cosmic Microwave Background (CMB) in the early Universe ( $H_0 \approx 67.4$ km s $^{-1}$ Mpc $^{-1}$ ) under the standard $\Lambda$ CDM model. The authors posit that this tension may not be an observational error but a consequence of the $\Lambda$ CDM model ignoring the physical effects of scale invariance, a symmetry present in empty space but suppressed by matter.

Methodology
The authors explore the "Scale Invariant Vacuum" (SIV) model, a cosmological framework based on the scale covariance of the vacuum. Unlike $\Lambda$ CDM, which introduces a cosmological constant ( $\Lambda$ ) to explain acceleration, SIV derives acceleration naturally from the scale transformation of the metric ( $ds' = \lambda ds$ ) and a gauge condition where macroscopic empty space is scale-invariant ( $\lambda \sim 1/t$ ).

The study employs the following methodological steps:

Analytical Derivation: The authors utilize the SIV field equations to derive the expansion history $H(z)$ , the evolution of matter density, and temperature $T(z)$ , noting that SIV introduces time-dependent terms ( $t^{-1/2}$ ) absent in standard Friedmann equations.
Observational Constraints (Two Major Tests):
- Type Ia Supernovae (SNIa): A Bayesian fit is performed on the Pantheon+SH0ES dataset (extended to $z=2.9$ ) to constrain the matter density parameter $\Omega_m$ within the SIV framework.
- Cosmic Age Consistency: The relationship between $H_0$ , $\Omega_m$ , and the age of the Universe ( $\tau_0$ ) is analyzed to ensure consistency with the estimated age of the oldest globular clusters and the current epoch.
Recombination Analysis: The authors compare the thermal and dynamical states at recombination ( $T \approx 3000$ K) between $\Lambda$ CDM and SIV. They calculate the expansion rate $H(z^*_{rec})$ and redshift $z^*_{rec}$ required to connect these early states to the present epoch under both models.
CMB Acoustic Peak Check: Preliminary calculations are performed on the angular position of the first acoustic peak ( $\theta$ ) to ensure the SIV model does not violate CMB power spectrum constraints.

Key Contributions and Results

Constrained Matter Density ( $\Omega_m$ ): The analysis of the SNIa Hubble diagram within the SIV framework yields a best-fit matter density of $\Omega_m = 0.18 \pm 0.03$ . This is consistent with independent constraints from the $H_0$ - $\Omega_m$ -age relation, which favors $\Omega_m \approx 0.20$ for a Universe age of $\tau_0 \approx 13.92$ Gyr.
Resolution of Hubble Tension: The core finding is that for $\Omega_m \approx 0.20$ $Ω_{m} \approx 0.20$ in SIV, the expansion rate at recombination ( $H(z^*_{rec})$ $H (z_{r ec}^{*})$ ) and the temperature ( $T=3000$ $T = 3000$ K) are nearly identical to those in the standard $\Lambda$ $Λ$ CDM model ( $\Omega_m = 0.315$ $Ω_{m} = 0.315$ ). However, due to the different expansion histories ( $H(z)$ $H (z)$ ) and the higher recombination redshift in SIV ( $z^*_{rec} \approx 1438$ $z_{r ec}^{*} \approx 1438$ vs. $1100$ in $\Lambda$ $Λ$ CDM), the integration of these identical early conditions to the present day yields different $H_0$ $H_{0}$ values:
- $\Lambda$ CDM integration leads to $H_0 \approx 67.4$ km s $^{-1}$ Mpc $^{-1}$ .
- SIV integration leads to $H_0 \approx 74$ km s $^{-1}$ Mpc $^{-1}$ .
Recombination Redshift Shift: In SIV, the recombination redshift is density-dependent ( $z^*_{rec} \propto \Omega_m^{-1/6}$ ). For $\Omega_m = 0.20$ , $z^*_{rec}$ is calculated to be 1438.08, significantly higher than the standard $\sim 1100$ , yet the physical conditions (temperature and expansion rate) at that epoch match the CMB constraints.
CMB Acoustic Peak Compatibility: Preliminary results indicate that for $\Omega_m \approx 0.18$ (the SNIa value), the predicted angular scale of the first acoustic peak is $\theta_{SIV} \approx 0.588^\circ$ , which is in even better agreement with CMB observations than the $\Lambda$ CDM value of $0.594^\circ$ .

Significance and Claims
The paper claims that the Hubble tension may be an artifact of the $\Lambda$ CDM model's failure to account for scale invariance. The authors argue that:

Scale Invariance is Measurable: Even in a matter-dominated universe, the effects of scale invariance do not vanish completely but are reduced. These "small but still measurable effects" alter the expansion and temperature histories sufficiently to resolve the tension.
No New Physics Required: The solution does not require "Early Dark Energy" or new scalar fields; rather, it arises from a fundamental symmetry (scale invariance) already present in the theory of gravitation, which is suppressed but not eliminated by the presence of matter.
Consistency: The SIV model with $\Omega_m \approx 0.20$ satisfies the requirement of not performing worse than $\Lambda$ CDM in describing observational data (SNIa, cosmic ages, and preliminary CMB peak data) while naturally accommodating the locally measured high value of $H_0$ without discarding the CMB anchor.

The authors conclude that the discrepancy between early and late Universe $H_0$ values likely stems from the different ways $\Lambda$ CDM and SIV connect the thermal/dynamical state of recombination to the present epoch, with SIV providing a consistent framework where $H_0 = 74$ km s $^{-1}$ Mpc $^{-1}$ is the natural outcome.