Brans-Dicke-like field for co-varying $G$ and $c$: observational constraints

This paper constrains a Brans-Dicke-like framework where the gravitational constant $G$ and speed of light $c$ co-vary ($c^3/G=\text{constant}$) using SN Ia, BAO, and CMB data, finding that Pantheon+ combined with DESI strongly favors a variable speed of light at over $3\sigma$ confidence due to a correlation with $H_0$, while Union2.1 data suggests no variation.

The Gist

The Big Idea: The Universe's "Ruler" Might Be Stretching

Imagine you are trying to measure the distance across a room. You grab a ruler, but you realize that the ruler itself is made of rubber. Sometimes it stretches, sometimes it shrinks. If you don't know how the ruler is changing, your measurements of the room will be wrong.

For over a century, physicists have assumed that the "rulers" of the universe are made of solid steel. Specifically, they assumed two fundamental constants never change:

1. $G$ (Gravity): How strongly things pull on each other.
2. $c$ (The Speed of Light): How fast light zips through space.

This paper asks a bold question: What if these "rulers" are actually made of rubber? What if the speed of light and the strength of gravity have been slowly changing as the universe has aged?

The Theory: A Tug-of-War Between Gravity and Light

The authors are building on an old idea called the Brans-Dicke theory. Think of the universe as a giant rubber sheet. In standard physics, the tension of that sheet (gravity) is fixed. In this new model, the tension can change.

The authors propose a specific "dance" between gravity and light:

• They suggest that if the speed of light ($c$) changes, gravity ($G$) must change too to keep the universe balanced.
• They use a mathematical rule: $c^3 / G = \text{constant}$.
• The Analogy: Imagine a seesaw. If one side (the speed of light) goes up, the other side (gravity) must go down in a very specific way to keep the seesaw level. They are "co-varying" (changing together).

The Experiment: Checking the Cosmic Map

To test this, the authors looked at the most reliable "map" we have of the universe: Type Ia Supernovae. These are exploding stars that act like "standard candles." Because we know exactly how bright they should be, we can tell how far away they are by how dim they look.

They combined this with two other datasets:

1. BAO (Baryon Acoustic Oscillations): Think of these as "fossilized sound waves" from the baby universe, acting like a standard ruler for the size of the cosmos.
2. CMB (Cosmic Microwave Background): The "baby photo" of the universe, showing us the conditions shortly after the Big Bang.

The Twist: It Depends on Which Map You Use

Here is where the story gets interesting. The authors ran their "rubber ruler" model against the data using two different catalogs of supernovae: Pantheon+ and Union2.1.

The Result was a split decision:

1. The Pantheon+ Team (The "Stretchy" Ruler):
   When they used the Pantheon+ data (which is very precise and suggests the universe is expanding faster, meaning a higher "Hubble Constant"), the model said: "Yes! The speed of light is changing!"

   • The data strongly favored a universe where light traveled slightly differently in the past. The confidence was high (over 99.7%, or 3-sigma).

2. The Union2.1 Team (The "Solid" Ruler):
   When they used the older Union2.1 data (which has slightly more uncertainty and suggests a slower expansion rate), the model said: "Nope. The speed of light is constant, just like Einstein said."

The "Aha!" Moment: The Hubble Tension Connection

Why the difference? The paper reveals a hidden correlation.

Imagine you are trying to guess the speed of a car ($H_0$, the expansion rate) and the length of its tire ($c$, the speed of light).

• If you think the car is going fast, you might assume the tire is smaller than you thought to make the math work.
• If you think the car is going slow, you assume the tire is normal size.

In this paper, the "Pantheon+" data suggests the universe is expanding faster than the "Union2.1" data. To make the math work with a faster expansion, the model needs the speed of light to have been different in the past. If the expansion is slower (Union2.1), the model doesn't need the speed of light to change; it fits perfectly with the standard "solid ruler" theory.

The Conclusion: A Work in Progress

The authors conclude that their "rubber ruler" theory is a viable candidate for explaining the universe, but it is currently stuck in a traffic jam caused by the Hubble Tension (the disagreement between how fast different parts of the universe seem to be expanding).

• If the universe is expanding fast: The speed of light likely changed.
• If the universe is expanding slow: The speed of light is constant.

The Takeaway:
This paper doesn't prove that the speed of light changed. Instead, it shows that our uncertainty about how fast the universe is expanding is directly linked to whether we think the speed of light is constant. Until we get a better, unified measurement of the universe's expansion rate, we can't be sure if the "rubber ruler" is real or just an illusion caused by our measurement errors.

It's like trying to solve a puzzle where two pieces don't quite fit; the authors are showing us that the shape of one piece (the speed of light) depends entirely on how we force the other piece (the expansion rate) to fit.

Technical Summary

1. Problem Statement

The paper addresses the theoretical possibility of varying fundamental physical constants, specifically the speed of light ($c$) and the gravitational constant ($G$), within a cosmological framework. While General Relativity (GR) and the standard $\Lambda$CDM model assume these constants are invariant, alternative theories (such as Brans-Dicke theory or Varying Speed of Light - VSL models) propose they may evolve over cosmic time.

The specific problem tackled is the lack of observational constraints on a Brans-Dicke-like model where $G$ and $c$ are not independent but co-vary. In this framework, the scalar field $\phi$ is defined as $\phi = c^3/G$. The authors aim to determine if current observational data supports a varying speed of light under the constraint that $G$ and $c$ vary such that $c^3/G$ remains constant (or evolves toward a constant equilibrium), and to investigate how different supernova datasets influence these constraints.

2. Methodology

Theoretical Framework

The authors utilize a modified gravity action based on the Brans-Dicke formalism but generalized for co-varying constants:
$$S = \frac{1}{16\pi} \int d^4x \sqrt{-g} \left[ \phi R - \frac{\omega}{\phi} \nabla_\mu \phi \nabla^\mu \phi - V(\phi) \right] + \int d^4x \sqrt{-g} \frac{1}{c} \mathcal{L}_m$$
where $\phi = c^3/G$.

• Dynamics: The system evolves toward an equilibrium where $\phi$ is constant, implying a strict relationship between $G$ and $c$: $G = G_0 (c/c_0)^3$.
• Key Assumption: The standard redshift-scale factor relation $(1+z) = a^{-1}$ (Lemaître formula) is preserved, distinguishing this model from other VSL theories where redshift depends explicitly on $c$.
• Observables: The authors derive expressions for cosmological distances. Crucially, they find that while the proper distance, angular diameter distance, and sound horizon remain identical to standard $\Lambda$CDM, the luminosity distance ($d_L$) acquires a multiplicative factor of $c(z)/c_0$.
  $$d_L(z) = d_L^{\Lambda CDM}(z) \times \frac{c(z)}{c_0}$$
  This makes Type Ia Supernovae (SNe Ia) the primary probe for detecting deviations in this model.

Parameterizations

Since the field equations do not uniquely determine the functional form of $c(z)$, the authors test three distinct parameterizations:

1. Power-law: $c(z) = c_0(1+z)^n$.
2. Gupta's Ansatz (CCC framework): An exponential time dependence $c(t) = c_0 e^{\alpha(t-t_0)}$, converted to redshift space.
3. Continuous Parameterization: A function that smoothly interpolates between $c_0$ at $z=0$ and $c_0$ at high $z$ (avoiding ad-hoc cut-offs), approximating the power-law at low redshifts.

Data and Analysis

The models were constrained using a combination of datasets via Markov Chain Monte Carlo (MCMC) analysis:

• SNe Ia: Pantheon+ (high precision) and Union2.1 (larger uncertainties).
• Baryon Acoustic Oscillations (BAO): DESI DR2 data.
• CMB: The angular acoustic scale $\theta_*$ from Planck.

3. Key Contributions

1. Derivation of Observables: The paper rigorously demonstrates that in this specific co-varying $G$-$c$ framework, the sound horizon ($r_s$) and proper distance remain unchanged from standard cosmology due to the cancellation of $c(z)$ terms in the integrals. The only observable deviation is in the luminosity distance.
2. Resolution of Redshift Ambiguity: The authors prove that under their specific assumptions, the standard relation $(1+z) = a^{-1}$ holds, countering claims in other VSL literature that redshift must be modified by varying $c$.
3. Identification of the $H_0$-VSL Degeneracy: The study highlights a strong correlation between the Hubble constant ($H_0$) and the variation of the speed of light. Because $d_L \propto (c/c_0) H_0^{-1}$, a shift in the inferred $H_0$ can be compensated by a shift in $c(z)$.

4. Results

The constraints on the variation of $c$ are highly sensitive to the choice of the SNe Ia dataset:

• Pantheon+ + DESI: This combination favors a variable speed of light with high statistical significance.
  • For all three parameterizations, the data prefers $c(z) < c_0$ (light was slower in the past).
  • The significance exceeds 3$\sigma$ (specifically 3.3$\sigma$ for power-law, 3.8$\sigma$ for Gupta and continuous models).
  • The inferred $H_0$ is high ($\approx 72.7$ km/s/Mpc).
• Union2.1 + DESI: This combination favors no variation of the speed of light ($c(z) = c_0$).
  • The inferred $H_0$ is lower ($\approx 69.7$ km/s/Mpc), consistent with early-universe physics (CMB/BAO).
  • The variation parameter ($n$ or $\beta$) is consistent with zero.

Interpretation of the Discrepancy:
The apparent conflict is not due to a failure of the model but rather a degeneracy between $H_0$ and VSL.

• Pantheon+ data, having smaller uncertainties, allows the BAO data to anchor the distance scale, resulting in a higher $H_0$. In the VSL model, this high $H_0$ is "explained" by a varying $c(z)$.
• Union2.1 data, with larger uncertainties, relies more heavily on the BAO scale, pulling $H_0$ down to values consistent with standard $\Lambda$CDM, thereby rendering $c(z)$ constant.

5. Significance

• Model Discrimination: The paper demonstrates that the Brans-Dicke-like co-varying model is observationally viable but currently limited by the precision and consistency of SNe Ia datasets.
• Hubble Tension Context: The results suggest that the "Hubble Tension" (the discrepancy between early and late-universe $H_0$ measurements) might be partially absorbed by VSL models. If $H_0$ is higher (as suggested by Pantheon+), a varying $c$ becomes a statistically preferred solution.
• Future Directions: The authors conclude that to definitively test this model, one must break the $(c/c_0)H_0^{-1}$ degeneracy. This requires independent distance anchors (e.g., strong lensing time delays, standard sirens, or water megamasers) that do not rely on the SNe Ia luminosity distance ladder.

In summary, the paper provides a robust theoretical framework for co-varying $G$ and $c$, showing that current data is inconclusive due to dataset-dependent tensions in $H_0$, but strongly suggests that if the high-$H_0$ Pantheon+ results are correct, a varying speed of light is a compelling physical explanation.