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Bayesian Model Comparison of $R_h=ct$ versus $\Lambda$ CDM using HII galaxy Hubble diagram

This paper presents a Bayesian model comparison of the $R_h=ct$ and $\Lambda$ CDM/ $w$ CDM cosmological models using HII galaxy data, revealing that while uniform priors yield no preference between the models, adopting Planck-based normal priors strongly favors $R_h=ct$ over flat $\Lambda$ CDM.

Original authors: Yuva Himanshu Pallam, Shantanu Desai

Published 2026-03-31

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CC BY 4.0

Original authors: Yuva Himanshu Pallam, Shantanu Desai

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 you are a detective trying to solve a mystery about the universe: How is the universe expanding, and what is the "engine" driving it?

For decades, scientists have had two main suspects in the case:

The Standard Suspect (ΛCDM): This is the current "gold standard" theory. It says the universe is filled with invisible stuff called Dark Energy that pushes everything apart, and it has a specific history of how fast it's growing.
The Challenger (Rh = ct): This is a newer, simpler theory. It suggests the universe expands at a perfectly constant speed, like a car on cruise control, without needing complex invisible forces.

Recently, a team of researchers (Wei and Melia) looked at a specific group of cosmic "mile markers" called HII galaxies (giant clouds of gas and young stars). They claimed the data strongly supported the Challenger (Rh = ct) over the Standard Suspect.

But two new detectives, Yuva Himanshu Pallam and Shantanu Desai, decided to double-check the evidence using a different, more rigorous tool. Here is what they found, explained simply.

The Detective's Toolkit: Two Ways to Judge Evidence

To decide which theory is better, you need a way to score them. The previous team used a tool called BIC (Bayesian Information Criterion). Think of BIC like a simple scorecard:

It looks at how well the theory fits the data.
It gives a small penalty if the theory is too complicated (has too many moving parts).
The flaw: It mostly looks at the "best guess" numbers. It doesn't care much about how likely those numbers were to begin with.

The new team used Bayesian Model Comparison. Think of this like a sophisticated courtroom trial:

Instead of just looking at the best guess, it asks: "Considering everything we already know about the universe, how likely is this theory to be true?"
It weighs the evidence against the prior knowledge (what we already believe based on other experiments).

The Experiment: Two Different Juries

The new team ran their "trial" twice, using two different types of "juries" (called priors) to see how the verdict changes.

Trial 1: The "Blank Slate" Jury (Uniform Priors)

In this scenario, the jury knows nothing about the universe. They haven't seen any other data. They just look at the HII galaxy data and say, "Let's assume any speed or any amount of matter is equally possible."

The Verdict: The jury was confused. The score was a tie.
The Result: The Bayes Factor was close to 1. This means the data from the HII galaxies alone cannot decide between the two theories. They are equally likely.
The Takeaway: If you ignore everything else we know, the HII galaxies don't prove the Challenger is better.

Trial 2: The "Expert" Jury (Normal Priors based on Planck)

In this scenario, the jury is made up of experts who have studied the Cosmic Microwave Background (the "baby picture" of the universe) and know the universe's parameters very precisely (from the Planck satellite). They bring this knowledge into the courtroom.

The Verdict: The Challenger (Rh = ct) won big.
The Result: The Bayes Factor was around 50. This is "very strong" evidence.
The Twist: Why did the Challenger win here? It wasn't because the HII galaxies suddenly looked perfect. It was because the Standard Suspect (ΛCDM) looked suspicious when compared to the experts' prior knowledge. The HII galaxy data suggested a universe that was expanding differently than the Planck experts expected. Since the Standard Suspect is tightly bound to those expert expectations, it got penalized. The Challenger, being a simpler, more flexible model, survived the tension better.

The "Hidden Glitch" (Intrinsic Scatter)

The researchers also checked if the HII galaxies were "messy" (having an unknown amount of natural variation, called intrinsic scatter).

When they added this "messiness" to the math, the results for the "Blank Slate" jury stayed the same (a tie).
For the "Expert" jury, the Challenger still won, but the victory was less decisive (from "very strong" to just "strong").

The Big Picture: What Does This Mean?

The Previous Study Wasn't "Wrong," But It Was Incomplete: The earlier study (WM25) used the simple scorecard (BIC) and concluded the Challenger was the clear winner. The new study shows that this conclusion depends heavily on how you look at the data.
The Data is Ambiguous: The HII galaxy data by itself is not strong enough to overthrow the Standard Model. It's a "tie" if you don't bring in outside knowledge.
The Tension is Real: The reason the Challenger looks good when using "Expert" priors is that the HII galaxy data is currently clashing with what we know from the Planck satellite. The Standard Model is getting squeezed in the middle.
We Need More Clues: The authors conclude that we can't rely on just one type of cosmic mile marker. We need to combine HII galaxies with other data (like Supernovae or BAO) to get a clear answer.

The Analogy Summary

Imagine you are trying to guess the speed of a car.

The HII Galaxies are a blurry photo of the car.
The Standard Model (ΛCDM) says the car is a Ferrari that usually goes 150 mph.
The Challenger (Rh = ct) says the car is a truck that goes a steady 60 mph.

The Old Study looked at the blurry photo and said, "The truck fits the photo better!"
The New Study says: "Wait. If we look at the photo without knowing what kind of car it usually is, it's a tie. But if we know for a fact it's a Ferrari (based on other photos), the blurry photo makes the Ferrari look weird. The truck wins only because the Ferrari looks out of place in this specific photo, not because the truck is definitely the right answer."

Conclusion: The universe is still a mystery. The HII galaxies are an interesting clue, but they aren't the smoking gun yet. We need more evidence to know if the universe is a Ferrari or a truck.

1. Problem Statement

The paper addresses the ongoing debate regarding the validity of the $R_h = ct$ cosmological model (a linear expansion model where the Hubble radius equals $ct$) compared to the standard $\Lambda$ CDM (Lambda Cold Dark Matter) and $w$ CDM (dark energy with a variable equation of state) models.

Context: A recent study by Wei and Melia (WM25, 2025) used a dataset of 231 HII galaxies (HIIGx) and Giant Extragalactic HII Regions (GEHR) as standard candles. Using the Bayesian Information Criterion (BIC), WM25 concluded that $R_h = ct$ is strongly favored over $\Lambda$ CDM and $w$ CDM.
The Conflict: The authors note that BIC relies on best-fit parameters and assumes independent, identically distributed data, which may not hold for all cosmological datasets. Furthermore, previous work by the authors using other probes (Cosmic Chronometers, BAO) has shown $\Lambda$ CDM to be favored.
Objective: To re-evaluate the WM25 analysis using Bayesian Model Comparison (BMC), a more robust statistical framework that integrates over the entire parameter space rather than relying on single best-fit points, to determine if $R_h = ct$ is truly favored.

2. Methodology

Data and Likelihood Construction

Dataset: The authors utilized the same dataset as WM25: 231 sources comprising 36 GEHRs, 181 HIIGx (joint local and high-z), and 14 newly discovered HIIGx samples, covering redshifts $z \approx 0$ to $z \approx 7.5$ .
Standard Candle Relation: The analysis relies on the correlation between H $\beta$ luminosity ( $L$ ) and ionized gas velocity dispersion ( $\sigma$ ). The observed distance modulus ( $\mu_{obs}$ ) is derived from this relation.
Likelihood Function: The total likelihood ( $L_{tot}$ $L_{t o t}$ ) is constructed as the sum of likelihoods for the HIIGx sample and an "anchor" sample (sources with independent distance moduli).
- The authors considered two scenarios:
  1. Without Intrinsic Scatter: Assuming the $L-\sigma$ relation is perfect.
  2. With Intrinsic Scatter ( $\sigma_{int}$ ): Adding an unknown intrinsic scatter parameter to the error budget, as suggested by WM25 to account for potential evolution in the relation.

Cosmological Models

The study compares three models:

Flat $\Lambda$ CDM: Standard model with $\Omega_k=0$ , $\Omega_\Lambda = 1 - \Omega_m$ .
Flat $w$ CDM: Generalized dark energy with a constant equation of state $w$ .
$R_h = ct$ : A specific model where $D_L(z) = \frac{c}{H_0}(1+z)\ln(1+z)$ .

Bayesian Inference Setup

Priors: Two distinct sets of priors were tested to ensure robustness:
1. Uniform Priors: Broad, non-informative ranges ( $H_0 \in [60, 110]$ , $\Omega_m \in [0, 1]$ , $w \in [-3, 0]$ ).
2. Normal (Gaussian) Priors: Centered on Planck 2020 cosmology best-fit values ( $H_0 \sim \mathcal{N}(67.4, 0.5)$ , $\Omega_m \sim \mathcal{N}(0.315, 0.007)$ ).
Evidence Calculation: The Bayesian Evidence ( $Z$ ) was calculated using the Dynesty nested sampler with dynamic nested sampling.
Metric: The Bayes Factor ( $B_{21}$ ) was computed to compare $R_h = ct$ (Model 2) against $\Lambda$ CDM/ $w$ CDM (Model 1). Significance was assessed using Jeffreys' scale.

3. Key Results

The results are presented in two main tables (Table I and Table II) distinguishing between analyses with and without intrinsic scatter.

A. Uniform Priors (Model-Neutral)

Without Intrinsic Scatter: The Bayes factors for $R_h = ct$ vs. $\Lambda$ CDM and $w$ CDM were close to 1 (0.27 and 0.32 respectively).
With Intrinsic Scatter: The Bayes factors remained close to 1 (0.48 and 0.58 respectively).
Interpretation: Under uniform priors, there is no preference for $R_h = ct$ over the standard models; they are statistically equivalent. This contradicts the WM25 BIC results, which claimed strong evidence for $R_h = ct$ .

B. Normal Priors (Planck-Constrained)

Without Intrinsic Scatter:
- $R_h = ct$ vs. $\Lambda$ CDM: Bayes Factor $\approx 50$ ("Very Strong" evidence).
- $R_h = ct$ vs. $w$ CDM: Bayes Factor $\approx 6$ ("Substantial" evidence).
With Intrinsic Scatter:
- $R_h = ct$ vs. $\Lambda$ CDM: Bayes Factor $\approx 18.6$ ("Strong" evidence).
- $R_h = ct$ vs. $w$ CDM: Bayes Factor $\approx 2.5$ ("Barely worth mentioning" evidence).
Interpretation: When priors are constrained by Planck data, $R_h = ct$ appears favored. However, the authors argue this is an artifact of the tension between the HIIGx-derived constraints (which prefer higher $H_0$ and different $\Omega_m$ ) and the Planck priors, rather than a genuine superiority of the $R_h = ct$ model itself.

C. BIC Reproduction (Appendix A)

The authors independently reproduced the BIC analysis.
Finding: Unlike WM25, they found that Flat $\Lambda$ CDM had the lowest BIC (best fit), with a $\Delta$ BIC of $\sim 1$ between $\Lambda$ CDM and $R_h = ct$ .
Discrepancy: The difference in best-fit parameters (e.g., $H_0 \approx 93$ in this analysis vs. lower in WM25) likely caused the divergence in BIC rankings, highlighting the sensitivity of BIC to specific best-fit values.

4. Key Contributions

Robust Model Comparison: The paper demonstrates that Bayesian Model Comparison (integrating over parameters) yields different conclusions than BIC (based on best-fit parameters) for the same dataset.
Prior Sensitivity Analysis: It highlights that the "strong evidence" for $R_h = ct$ reported in previous literature is highly sensitive to the choice of priors. Under model-neutral (uniform) priors, the models are indistinguishable.
Reproduction of BIC: The authors successfully reproduced the BIC analysis but found the opposite ranking to WM25, attributing the difference to slight variations in best-fit parameter estimation.
Identification of Tension: The study identifies that the apparent preference for $R_h = ct$ under Planck priors is driven by the discrepancy between HIIGx constraints and Planck cosmology, not by the Hubble diagram data alone favoring $R_h = ct$ .

5. Significance

Challenging $R_h = ct$ : The paper provides strong evidence against the claim that HII galaxies decisively favor the $R_h = ct$ model over the standard $\Lambda$ CDM.
Methodological Warning: It serves as a cautionary tale for cosmological model selection, emphasizing that information criteria like BIC can be misleading if the dataset is not strictly independent and identically distributed, or if the best-fit parameters are unstable.
Future Directions: The authors suggest that while current BAO data (DESI) favors $\Lambda$ CDM, the specific Lyman- $\alpha$ point driving that conclusion is a single data point. Therefore, independent probes like HII galaxies remain crucial, but must be analyzed with rigorous Bayesian methods and careful consideration of priors.

Conclusion: The paper concludes that $R_h = ct$ is not decisively favored over $\Lambda$ CDM or $w$ CDM by HII galaxy data when using a robust Bayesian framework with uniform priors. The strong evidence cited in previous works appears to be an artifact of the specific information criterion used (BIC) and the tension between the specific dataset and external cosmological priors.