Joint constraints on $R_h=ct$ cosmology from DESI DR2 BAO, CC, and SN\textit{Ia} Pantheon$^+$ sample

This study compares the standard $\Lambda$CDM model with the alternative $R_h=ct$ cosmology using joint constraints from DESI DR2 BAO, cosmic chronometers, and Pantheon$^+$ supernova data, finding that while both models fit the data, $\Lambda$CDM provides a superior statistical fit and naturally explains the observed transition from deceleration to acceleration, whereas $R_h=ct$ predicts a strictly linear expansion.

The Gist

Imagine the universe as a giant balloon that has been inflating since the Big Bang. For decades, scientists have been trying to figure out exactly how that balloon is inflating. Is it speeding up? Slowing down? Or is it expanding at a perfectly steady, constant pace?

This paper is like a head-to-head race between two different theories about how that balloon behaves, using the most up-to-date "rulers" and "stopwatches" available in the cosmos.

The Two Racers

Racer 1: The Standard Model (ΛCDM)
Think of this as the "Gold Standard" or the "Favorite Team" of modern cosmology. It's the current champion.

• The Story: This model says the universe started out expanding quickly, then slowed down because gravity (like a heavy anchor) pulled everything together. But then, about 5 or 6 billion years ago, a mysterious force called "Dark Energy" kicked in, acting like a rocket booster, and started pushing the universe to expand faster and faster.
• The Prediction: It predicts a bumpy ride: slow down first, then speed up.

Racer 2: The Rh = ct Model
This is the challenger. It's a simpler, more "coasting" theory.

• The Story: This model suggests the universe doesn't need a mysterious rocket booster or a heavy anchor. Instead, it just expands at a perfectly constant speed, like a car on cruise control that never touches the gas or brake pedals.
• The Prediction: It predicts a straight, flat line. No slowing down, no speeding up. Just a steady, linear march through time.

The Race Track (The Data)

To see which racer is actually winning, the authors didn't just guess. They used three massive, high-tech datasets as their race track:

1. Cosmic Chronometers (CC): These are like "cosmic stopwatches." By looking at how old different galaxies are, scientists can measure how fast the universe was expanding at different times in the past.
2. Supernovae (Pantheon+): These are "standard candles." They are exploding stars that always shine with the same brightness. By seeing how dim they look from Earth, we can measure how far away they are and how fast they are moving away.
3. DESI DR2 (BAO): This is the "cosmic ruler." It measures the spacing of galaxies across the universe, like measuring the distance between trees in a forest to see how much the forest has stretched.

The Results: Who Won?

The authors ran a statistical "judge" over the data to see which model fits the observations better. Here is what they found:

• The Fit: Both models could "sort of" explain the data, but the Standard Model (ΛCDM) fit the data much more tightly. It was like the Standard Model hit the bullseye, while the Rh = ct model was a few inches off.
• The "Penalty" Score: In science, simpler models are usually better (like Occam's Razor), but only if they fit the data. The Rh = ct model is simpler, but the data didn't support its simplicity. The Standard Model, even though it's more complex, was rewarded because it actually matched the observations.
• The Age of the Universe:
  • The Standard Model calculated the age of the universe to be about 13.7 billion years. This matches perfectly with other famous measurements (like those from the Planck satellite).
  • The Rh = ct Model calculated the age to be about 16 billion years. Because this model assumes a constant, slower expansion rate, it thinks the universe has had to travel for a longer time to get to where it is now.

The Big Takeaway

The paper concludes that while the "coasting" idea (Rh = ct) is an interesting and simple concept, the universe doesn't actually behave that way. The data clearly shows that the universe did slow down in the past and is speeding up now.

The Standard Model (ΛCDM) remains the best description we have of our cosmic history. However, the authors do note that science is never "done." New observations, like those from the James Webb Space Telescope (JWST) finding very old, mature galaxies that shouldn't exist yet, are making scientists ask: "Is our Standard Model perfect, or just the best one we have right now?"

In short: The universe isn't a car on cruise control; it's a car that hit the brakes and then slammed on the gas. The Standard Model is the map that best describes that journey.

Technical Summary

Technical Summary: Joint Constraints on $R_h = ct$ Cosmology from DESI DR2 BAO, CC, and SNIa Pantheon+ Sample

Problem Statement
The standard $\Lambda$CDM cosmological model, while empirically successful, faces persistent theoretical and observational challenges, including the cosmological constant problem, the nature of dark matter and dark energy, and the Hubble tension. These issues have motivated the exploration of alternative frameworks, such as coasting cosmologies where the Universe expands linearly with time. A prominent candidate is the $R_h = ct$ model, which posits that the gravitational horizon ($R_h$) equals $ct$ at all cosmic times, implying a linear scale factor evolution ($a(t) \propto t$) and a constant expansion rate ($q=0$). While previous studies have tested this model against various datasets, often reporting competitive fits, there remains a need for a rigorous, joint statistical assessment using the latest high-precision observational data to determine its viability relative to $\Lambda$CDM.

Methodology
The authors perform a comparative analysis of the standard flat $\Lambda$CDM model and the $R_h = ct$ framework using a joint dataset comprising:

1. Cosmic Chronometers (CC): 15 differential age measurements of $H(z)$ spanning $0.1791 \leq z \leq 1.965$.
2. Type Ia Supernovae (SNIa): The Pantheon+ compilation (1701 SNe Ia, $0.01 \leq z \leq 2.261$), excluding the SH0ES distance-ladder calibration to maintain independence.
3. Baryon Acoustic Oscillations (BAO): 13 recent measurements from the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2), covering $0.295 \leq z \leq 2.330$.

The analysis employs a unified Bayesian framework using the dynesty nested sampling algorithm to explore the posterior distributions. Key parameters constrained include the Hubble constant ($H_0$), the matter density parameter ($\Omega_{m0}$), and the sound horizon at the drag epoch ($r_d$). Notably, $r_d$ is treated as a free parameter to maintain model independence, acknowledging the degeneracy between $H_0$ and $r_d$ in BAO analyses.

Model performance is evaluated through:

• Likelihood Analysis: Minimizing the total $\chi^2$ function derived from the joint likelihood of all datasets.
• Information Criteria: Calculating the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) to penalize model complexity.
• Bayesian Evidence: Computing the Bayes factor ($\ln B$) to quantify the relative support for one model over the other.
• Cosmographic Probes: Reconstructing the redshift evolution of the Hubble parameter $H(z)$, the deceleration parameter $q(z)$, and the cosmic age $t(z)$ directly from the posterior samples.

Key Results

• Statistical Preference: The $\Lambda$CDM model consistently outperforms the $R_h = ct$ model across all statistical indicators. The $\Lambda$CDM model yields a minimum chi-squared of $\chi^2_{\min} = 1574.88$ ($\chi^2_{\text{red}} = 0.975$), whereas $R_h = ct$ yields $\chi^2_{\min} = 1629.51$ ($\chi^2_{\text{red}} = 1.008$).
• Model Selection Criteria: The differences in information criteria are decisive: $\Delta \text{AIC} = 52.63$ and $\Delta \text{BIC} = 47.24$. According to standard criteria ($\Delta > 10$), this constitutes strong evidence against the $R_h = ct$ model. The Bayes factor is $\ln B = -2.3$, indicating weak to moderate support for $\Lambda$CDM over $R_h = ct$.
• Parameter Constraints:
  • $\Lambda$CDM: $H_0 = 68.3 \pm 4.6$ km s$^{-1}$ Mpc$^{-1}$ and $\Omega_{m0} = 0.309 \pm 0.008$. The derived age is $t_0 = 13.676^{+0.92}_{-0.81}$ Gyr, consistent with Planck 2018 CMB results.
  • $R_h = ct$: $H_0 = 61.1 \pm 4.1$ km s$^{-1}$ Mpc$^{-1}$ and $r_d = 151.4^{+8.9}_{-11}$ Mpc. The derived age is $t_0 = 16.035^{+1.09}_{-0.98}$ Gyr.
• Expansion History: The $H(z)$ evolution for $\Lambda$CDM matches the observational data across the full redshift range. In contrast, the $R_h = ct$ model systematically underestimates $H(z)$ at intermediate and high redshifts ($z \gtrsim 1$).
• Deceleration Parameter: The $\Lambda$CDM model naturally reproduces the transition from early-time deceleration ($q > 0$) to late-time acceleration ($q < 0$) at $z \approx 0.6$. The $R_h = ct$ model predicts a strictly linear expansion with $q(z) = 0$ at all redshifts, failing to capture the observed transition to acceleration.
• BAO Consistency: While both models can reproduce DESI DR2 BAO distance ratios ($D_M/r_d$ and $D_H/r_d$) within observational uncertainties, the $\Lambda$CDM model exhibits smaller, randomly distributed residuals. The $R_h = ct$ model shows a systematic pattern of deviation, particularly in the $D_H(z)/r_d$ indicator.

Significance and Claims
The paper claims to provide the first joint statistical comparison of $R_h = ct$ and $\Lambda$CDM using the combined CC + Pantheon+ + DESI DR2 datasets within a unified Bayesian framework. The authors assert that their results validate the reliability of the DESI DR2, CC, and Pantheon+ constraints, as the $\Lambda$CDM-derived age aligns with Planck 2018 CMB results.

The study concludes that while the $R_h = ct$ model provides an acceptable fit within statistical limits, it is decisively disfavored compared to $\Lambda$CDM due to its inability to reproduce the observed transition from deceleration to acceleration and its poorer performance in information-theoretic metrics. The authors note that while the standard model faces theoretical challenges (e.g., the nature of dark energy and the Hubble tension), the current observational data strongly support the $\Lambda$CDM paradigm as the superior description of the Universe's expansion history. The paper acknowledges that recent JWST observations of mature high-redshift galaxies and DESI hints of evolving dark energy suggest that the concordance paradigm may require future refinements, but these do not currently validate the $R_h = ct$ alternative.