Stringent constraint on the CCC+TL cosmology with $H(z)$ Measurements

This study demonstrates that the CCC+TL cosmological model, proposed to explain JWST's high-redshift galaxy observations, is strongly disfavored by model-independent Hubble parameter measurements due to severe internal tensions between its supernova-optimized parameters and cosmic chronometer data, suggesting that the observed galaxy anomalies likely stem from intrinsic early-universe galaxy evolution rather than new physics.

The Gist

The Big Picture: A Cosmic Detective Story

Imagine the universe is a giant, expanding balloon. For decades, astronomers have been using a standard "instruction manual" called $\Lambda$CDM (Lambda Cold Dark Matter) to understand how this balloon inflates. It's a bit like a trusted recipe for baking a cake that has worked perfectly for every occasion for the last 50 years.

However, recently, the James Webb Space Telescope (JWST) came along and took some photos of the "baby cakes" of the universe (galaxies from very early times). These photos showed something weird: the galaxies looked incredibly small and compact, much smaller than our "trusted recipe" predicted they should be.

This caused a panic. Some scientists thought, "Maybe our recipe is wrong! Maybe the universe isn't expanding the way we think."

The New Contender: The "CCC+TL" Model

Enter a new theory called CCC+TL (Covarying Coupling Constants + Tired Light). Think of this as a radical new recipe proposed to fix the "small galaxy" problem.

Instead of the universe expanding normally, this new theory suggests two wild ideas:

1. The Speed of Light is Changing: Imagine light is a runner. In the old model, the runner's speed is constant. In this new model, the runner was much faster in the past and has been slowing down.
2. Tired Light: Imagine light is a marathon runner who gets tired as they run. As they travel across the universe, they lose energy, making the light look "redder" (which we usually think is caused by expansion).

By combining these two ideas, the CCC+TL model claims it can explain why those early galaxies look so small without breaking the laws of physics. It's a clever fix, and it fits the "small galaxy" photos perfectly.

The Test: The "Cosmic Chronometer"

But here is the catch: A good theory has to explain everything, not just one weird photo. You can't just fix the cake size; you also have to explain how long the oven has been on and how fast the heat is rising.

The authors of this paper decided to put the CCC+TL model to the ultimate test using something called Cosmic Chronometers.

• The Analogy: Imagine you are trying to figure out how fast a car is driving.
  • Method A (Supernovae): You look at how bright the car's headlights are. This is what the CCC+TL model was built to fit.
  • Method B (Cosmic Chronometers): You look at the car's speedometer directly. This method uses "passive" galaxies (old stars that aren't changing much) to measure the expansion rate of the universe ($H(z)$) directly, without relying on any assumptions about how the universe works. It's the "speedometer" of the cosmos.

The Results: The Model Crashes

The authors took the "speedometer" data (32 measurements of the universe's expansion rate) and tried to fit the CCC+TL model to it.

The result was a disaster for the new theory.

1. The "Speedometer" vs. The "Headlights": The parameters (the settings) that made the CCC+TL model fit the "headlight" data (Supernovae) completely failed when they tried to fit the "speedometer" data.
   • Analogy: It's like a car that drives perfectly on a straight highway (Supernovae) but immediately crashes when you try to drive it on a winding mountain road (Cosmic Chronometers).
2. The "Light Speed" Contradiction: The CCC+TL model requires the speed of light to be changing rapidly to fit the galaxy sizes. But when they looked at the expansion rate data, the speed of light seemed to be almost perfectly constant. The two datasets are screaming at each other.
3. The Verdict: The standard $\Lambda$CDM model (the old recipe) fit the "speedometer" data perfectly. The new CCC+TL model was rejected with a statistical certainty so high it's like flipping a coin and getting "heads" 50 times in a row by pure luck.

The Conclusion: It's Not the Universe, It's the Galaxies

So, what does this mean?

The paper concludes that the "CCC+TL" model is likely wrong. The fact that the universe's expansion history (measured by the speedometer) doesn't match the new theory suggests that the problem isn't with our understanding of the universe's expansion.

Instead, the "small galaxy" problem observed by JWST is likely because early galaxies were just naturally smaller and more compact than we thought. They grew and evolved differently in the early universe.

The Takeaway:
We don't need to rewrite the laws of physics (like changing the speed of light) to explain the JWST photos. We just need to update our understanding of how baby galaxies grow up. The universe is expanding exactly as we thought; the galaxies just had a different childhood than we expected.

Technical Summary

1. Problem Statement

The paper addresses a significant tension in modern cosmology arising from observations by the James Webb Space Telescope (JWST). JWST has detected high-redshift ($z \sim 10$) galaxies with unexpectedly small angular diameters and high stellar masses, which are difficult to reconcile with the standard $\Lambda$CDM model.

To resolve this, Gupta (2023) proposed the Covarying Coupling Constants and Tired Light (CCC+TL) hybrid model. This model attempts to explain the observations by:

1. Covarying Coupling Constants (CCC): Introducing a varying speed of light ($c$) and gravitational constant ($G$) that evolve with the scale factor.
2. Tired Light (TL): Reinterpreting cosmological redshift as a gradual energy loss of photons during propagation, rather than metric expansion.

The CCC+TL model claims to fit Type-Ia Supernova (SN Ia) data (Pantheon+) and Baryon Acoustic Oscillation (BAO) data while extending the age of the Universe to $\approx 26.7$ Gyr, thereby alleviating the "early mass growth" problem. However, the model's validity had not been rigorously tested against model-independent measurements of the Hubble parameter, $H(z)$, derived from Cosmic Chronometers (CC).

2. Methodology

The authors conducted a rigorous statistical test of the CCC+TL model using 32 independent $H(z)$ data points obtained from Cosmic Chronometers (passively evolving galaxies) covering the redshift range $0.01 < z < 2$.

• Model Formulation:
  • The authors adopted the mathematical framework from Gupta (2023).
  • The Hubble parameter in CCC+TL is a function of the local Hubble constant ($H_{0,CCC}$) and a speed-of-light variation exponent ($\alpha$).
  • Crucially, the model distinguishes between the total observed redshift ($z_{CCC+TL}$) and the redshift component due to expansion ($z_{CCC}$), requiring numerical inversion of transcendental equations to map observed redshifts to the model's expansion history.
• Data Analysis:
  • Dataset: 32 $H(z)$ measurements, including systematic error budgets and covariance matrices where available (specifically for 15 points from Moresco et al.).
  • Fitting Procedure: Bayesian analysis using the emcee package (MCMC) to derive posterior distributions for parameters.
  • Comparative Models:
    1. CCC+TL (Fixed): Parameters fixed to the SN Ia best-fit values ($H_{0,CCC} = 59.51$, $\alpha = -47.59$).
    2. CCC+TL (Free): Parameters ($H_{0,CCC}$ and $\alpha$) freely fitted to the $H(z)$ data.
    3. $\Lambda$CDM: Standard model with free parameters ($H_0$, $\Omega_{0,m}$).
• Statistical Metrics: Goodness-of-fit was evaluated using $\chi^2$, maximum log-likelihood ($\ln \mathcal{L}_{max}$), Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC).

3. Key Contributions

• First Model-Independent Test: This is the first study to confront the CCC+TL model with Cosmic Chronometer data, a probe that does not rely on the "distance ladder" or standard candles (SN Ia).
• Revealing Internal Tension: The study demonstrates a fundamental incompatibility within the CCC+TL framework: the parameters required to fit SN Ia data are statistically mutually exclusive with those required to fit $H(z)$ data.
• Quantitative Exclusion: The authors provide a rigorous likelihood ratio showing that the SN Ia-optimized CCC+TL model is rejected by $H(z)$ data with extreme statistical significance.

4. Results

The analysis yielded three primary findings:

1. Failure of SN Ia Parameters: When the CCC+TL model uses the parameters optimized for SN Ia data ($\alpha = -47.59$), it fails catastrophically to reproduce the $H(z)$ observations.

   • $\Delta \chi^2 = 61.52$ relative to the $\Lambda$CDM model.
   • Likelihood Ratio: The rejection of the SN Ia-optimized $\alpha$ by the $H(z)$ dataset yields a likelihood ratio of $R \approx 1.7 \times 10^{-14}$. This indicates the two datasets are incompatible within the CCC+TL framework at a very high confidence level.

2. Internal Inconsistency: Even when the CCC+TL parameters are freely fitted to the $H(z)$ data, the resulting best-fit $\alpha$ is $13.03^{+2.07}_{-5.76}$. This value is in stark contrast to the SN Ia best-fit value of $\alpha = -47.59 \pm 0.83$.

   • The SN Ia data suggests a rapidly decreasing speed of light over cosmic time.
   • The $H(z)$ data suggests minimal evolution (or a slight increase) in the speed of light.
   • This creates a severe internal tension where the model cannot simultaneously explain the luminosity distance (SN Ia) and the expansion rate ($H(z)$).

3. Superiority of $\Lambda$CDM:

   • The standard $\Lambda$CDM model fits both the SN Ia and $H(z)$ datasets consistently.
   • The $\Lambda$CDM parameters derived from SN Ia data fall comfortably within the $1\sigma$ credible interval of the parameters derived from $H(z)$ data.
   • $\Lambda$CDM yields the lowest AIC and BIC values, indicating it is the preferred model for the $H(z)$ dataset.

Visual Evidence:

• Figure 1: Shows the CCC+TL prediction (using SN Ia parameters) deviating significantly from the observed $H(z)$ points, while $\Lambda$CDM tracks the data well.
• Figure 4: Illustrates the divergent behavior of the speed-of-light variation function $f(z) = c/c_0$. The SN Ia fit implies a steep decline, whereas the $H(z)$ fit implies a nearly flat evolution.

5. Significance and Conclusion

The paper concludes that the CCC+TL hybrid model is not a viable alternative to $\Lambda$CDM because it fails the critical test of multi-probe consistency.

• Implication for JWST Tensions: The unexpectedly small angular diameters of high-$z$ galaxies observed by JWST are likely not caused by a breakdown of the standard cosmological model (i.e., the need for Tired Light or varying constants). Instead, the tension likely stems from:
  • Incomplete understanding of the intrinsic properties and evolution of early galaxies (e.g., compactness, star formation efficiency).
  • Astrophysical systematics in interpreting high-redshift galaxy sizes.
• Methodological Importance: The study underscores the necessity of using model-independent probes (like Cosmic Chronometers) to validate any new cosmological theory. A viable alternative to $\Lambda$CDM must simultaneously explain the Hubble diagram (SN Ia) and the expansion history ($H(z)$) with a single, self-consistent parameter set.

In summary, while CCC+TL offers a mathematical solution to specific JWST anomalies, it introduces a fatal inconsistency with the expansion history of the Universe, reinforcing the robustness of the standard $\Lambda$CDM model against this specific class of alternative theories.