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Model selection with the Pantheon+ Type Ia SN sample

Using the Pantheon+ Type Ia supernova catalog, this study demonstrates that the $R_{\rm h}=ct$ universe model is strongly favored over the standard $\Lambda$ CDM model because it satisfies general relativistic energy conditions while $\Lambda$ CDM violates the strong energy condition, resulting in an approximately 89.5% likelihood for the former.

Original authors: Namit Chandak, Fulvio Melia, Junjie Wei

Published 2026-02-18

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CC BY 4.0

Original authors: Namit Chandak, Fulvio Melia, Junjie Wei

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 Picture: A Cosmic Rivalry

Imagine the universe as a giant, expanding balloon. For the last few decades, scientists have been using a specific set of rules to explain how this balloon inflates. This set of rules is called $\Lambda$ CDM (Lambda-CDM). It's the "Standard Model" of cosmology. It says the universe started with a Big Bang, expanded, and then, about 5 billion years ago, started speeding up its expansion because of a mysterious force called "Dark Energy."

However, a new challenger has entered the ring: the $R_h = ct$ model. This model suggests the universe has been expanding at a steady, constant speed since the beginning, without any sudden "speed-ups" or mysterious dark energy pushing it.

This paper is a referee's report. The authors took a massive collection of data (1,701 exploding stars, known as Type Ia Supernovae) and asked: Which model actually fits the data better, and which one follows the fundamental laws of physics?

The "Energy Rules" (The Traffic Laws of the Universe)

To understand the paper's main finding, you need to know about the "Energy Conditions." Think of these as the traffic laws of General Relativity.

In our everyday world, gravity is attractive. If you drop a rock, it falls down. If you have two massive objects, they pull toward each other. The "Strong Energy Condition" is a rule that says: "Gravity must always be attractive; it should never push things apart."

The Problem with the Standard Model ( $\Lambda$ CDM): To make the universe speed up (accelerate), the Standard Model relies on "Dark Energy," which acts like anti-gravity. It pushes things apart. The authors argue that this is like a car driving on a highway while ignoring the "No Pushing" sign. It breaks the traffic laws of physics.
The $R_h = ct$ Model: This model suggests the universe expands at a constant speed, like a car cruising on cruise control. It doesn't need anti-gravity to explain the data. Therefore, it obeys all the traffic laws.

The Experiment: The Supernova Race

The authors used the Pantheon+ Sample, which is a massive database of 1,701 Type Ia Supernovae.

What are they? These are "Standard Candles." Imagine them as lightbulbs of a known brightness scattered across the universe. By measuring how dim they look from Earth, we can calculate how far away they are and how fast the universe was expanding when the light left them.

The team ran a simulation with two different maps:

Map A ( $\Lambda$ CDM): The standard map that says the universe is accelerating.
Map B ( $R_h = ct$ ): The alternative map that says the universe is expanding steadily.

The Results: Who Won?

When they compared the maps to the actual data, here is what happened:

The Fit: Both maps actually fit the data okay. If you just looked at the dots on a graph, both lines went through the middle of the data points.
The Violation: However, when they checked the "traffic laws" (the Strong Energy Condition), Map A ( $\Lambda$ CDM) broke the rules. In the redshift range of 0 to 2 (which covers a huge chunk of the universe's history), the Standard Model's prediction requires gravity to be repulsive (anti-gravity), which the authors argue is physically impossible.
The Compliance: Map B ( $R_h = ct$ ) followed all the rules. It fits the data and respects the laws of gravity.

The Verdict:
Using a statistical tool called the "Bayes Information Criterion" (think of it as a judge weighing the evidence), the data favored the $R_h = ct$ model about 90% of the time, while the Standard Model only got about 10%.

The Takeaway

The authors are saying:

"We don't need to invent a mysterious 'Dark Energy' that breaks the laws of physics to explain the universe. The data actually supports a simpler model where the universe expands steadily, obeying all the known laws of gravity."

In a nutshell:
Imagine you are trying to solve a mystery.

Theory A says: "The suspect ran away because a ghost pushed them." (This requires believing in ghosts/anti-gravity).
Theory B says: "The suspect ran away because they were already running fast." (This requires no ghosts).

The evidence (the supernovae) shows that the suspect was indeed running fast, and there is no need to invent a ghost. The "Ghost" theory (Standard Model) is statistically less likely and breaks the rules of the game. The "Running Fast" theory ( $R_h = ct$ ) is the winner.

1. Problem Statement

The paper addresses two critical issues in modern cosmology:

The Tension in $\Lambda$ CDM: The standard inflationary $\Lambda$ CDM model faces growing inconsistencies, including the "Hubble Tension" (discrepancies in $H_0$ measurements between high-redshift CMB data and low-redshift supernovae) and the discovery of early supermassive black holes and galaxies that contradict the model's timeline.
Violation of General Relativity (GR) Energy Conditions: The paper posits that the $\Lambda$ CDM model, particularly its reliance on a cosmological constant ( $\Lambda$ ) and an inflationary epoch, violates fundamental energy conditions derived from General Relativity. Specifically, the Strong Energy Condition (SEC), which requires gravity to be attractive ( $\rho + 3p \geq 0$ ), is violated by dark energy (acting as a cosmological constant) and the inflaton field.
The Competitor: The authors propose the $R_h = ct$ universe (where the Hubble radius $R_h$ equals $ct$) as a theoretically motivated alternative that adheres to all GR energy conditions and requires fewer free parameters (only $H_0$ ) compared to the 11–12 parameters of $\Lambda$ CDM.

2. Methodology

The authors utilize the Pantheon+ sample, consisting of 1,701 spectroscopically confirmed Type Ia Supernovae (SNe Ia), to perform a rigorous model selection between $\Lambda$ CDM and $R_h = ct$ .

Model-Independent Constraints: A novel aspect of this study is the derivation of model-independent bounds on the distance modulus ( $\mu$ ) based on the Strong Energy Condition (SEC). The SEC implies that the expansion of the universe cannot be accelerating ( $\ddot{a} \leq 0$ ). Mathematically, this imposes an upper limit on the distance modulus:
$\mu_{SEC}(z) \leq 5 \log_{10}\left[ \frac{c}{H_0} (1+z) \ln(1+z) \right] + 25$
(Note: The paper notes that the $R_h=ct$ prediction coincides exactly with this SEC limit).
Data Fitting:
- Distance Modulus: The observed distance modulus is calculated using the modified Tripp relation: $\mu_{obs} = -2.5 \log_{10}(x_0) + \alpha x_1 - \beta c + M_B$ .
- Theoretical Models:
  - $\Lambda$ CDM: Uses the standard luminosity distance integral involving matter density $\Omega_m$ .
  - $R_h = ct$ : Uses the analytic form $D_L(z) = \frac{c}{H_0}(1+z)\ln(1+z)$ .
- Optimization: The authors employ Maximum Likelihood Estimation (MLE) using the Python EMCEE (MCMC) module. They optimize nuisance parameters ( $\alpha, \beta, M_B$ ) and the intrinsic dispersion ( $\sigma_{int}$ ) simultaneously with the cosmological parameters.
- Fixed Parameters: $H_0$ is fixed at $70 \text{ km s}^{-1} \text{ Mpc}^{-1}$ because it is degenerate with $M_B$ .
Model Selection Criteria: The authors use the Bayes Information Criterion (BIC) to compare the goodness of fit while penalizing model complexity. The difference in BIC ( $\Delta$ BIC) is used to calculate the relative likelihoods of the two models.

3. Key Contributions

SEC Violation Quantification: The paper provides a direct visualization and quantification of how the $\Lambda$ CDM model violates the Strong Energy Condition in the local universe ( $z \in (0, 2)$ ). It demonstrates that the predicted Hubble diagram for $\Lambda$ CDM exceeds the theoretical upper bound imposed by the SEC.
Pantheon+ Analysis: It applies the latest, most robust Type Ia SN dataset (Pantheon+) to test these energy conditions, moving beyond previous studies that used smaller or less precise samples.
Theoretical Consistency Check: It establishes that while $\Lambda$ CDM fits the data reasonably well, it does so only by violating fundamental GR constraints (specifically the SEC), whereas the $R_h = ct$ model fits the data and remains fully compliant with all energy conditions (Null, Weak, Dominant, and Strong).

4. Results

Fit Quality: Both models provide a good fit to the Pantheon+ data, with residuals that are visually indistinguishable. The optimized nuisance parameters ( $\alpha, \beta, M_B$ ) are very similar in both models.
SEC Compliance:
- $R_h = ct$ : The model's predicted distance modulus curve lies exactly on the SEC upper bound, satisfying the condition $\rho + 3p \geq 0$ across the entire redshift range $z \in (0, 2)$ .
- $\Lambda$ CDM: The model's prediction violates the SEC limit in the redshift range $z \in (0.125, 2)$ . This confirms that the accelerated expansion required by $\Lambda$ CDM is physically inconsistent with the SEC.
Model Selection (BIC):
- The $R_h = ct$ model yields a significantly lower BIC ($1439.27$) compared to $\Lambda$ CDM ($1443.62$).
- The $\Delta$ BIC $\approx 4.35$ translates to a likelihood of $\sim 89.8\%$ for $R_h = ct$ versus $\sim 10.2\%$ for $\Lambda$ CDM.
- This indicates that the linear expansion of the $R_h = ct$ universe is strongly favored by the data over the accelerated expansion of $\Lambda$ CDM.

5. Significance

Challenging the Standard Model: The results suggest that the "acceleration" of the universe, a cornerstone of the $\Lambda$ CDM paradigm, may be an artifact of the model's assumptions rather than a physical reality. The data actually favor a linear expansion ( $R_h = ct$ ) which requires no dark energy in the form of a cosmological constant.
Validation of Energy Conditions: The study reinforces the importance of General Relativity's energy conditions as a filter for viable cosmological models. It argues that a model relying on "speculative antigravity" (inflation and $\Lambda$ ) is disfavored by observational data compared to a model that strictly adheres to GR constraints.
Resolution of Tensions: By favoring $R_h = ct$ , the paper suggests a potential resolution to the Hubble Tension and the "early universe" anomalies (like early galaxies), as the $R_h = ct$ model does not require an inflationary epoch and naturally accommodates the observed timeline of structure formation.
Methodological Shift: The paper advocates for a shift in cosmological model selection criteria, prioritizing adherence to fundamental physical laws (like the SEC) alongside statistical fit quality.

In conclusion, the authors argue that the Pantheon+ Type Ia SN data, when analyzed through the lens of General Relativistic energy conditions, strongly disfavor the standard inflationary $\Lambda$ CDM model in favor of the $R_h = ct$ universe.