🔭 astrophysics

Beyond CPL: Evidence for dynamical dark energy in three-parameter models

By introducing two new three-parameter dark energy models (MmAH1 and MmAH2) and analyzing them alongside standard parametrizations using a comprehensive joint dataset, this study finds consistent statistical evidence (up to 4–5 $\sigma$ with DESY5 data) for departures from the $\Lambda$ CDM model in favor of dynamical dark energy.

Original authors: Sonej Alam, Md. Wali Hossain

Published 2026-04-17

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Sonej Alam, Md. Wali Hossain

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 the universe is a giant, expanding balloon. For decades, scientists have believed that the air inside this balloon (Dark Energy) is pushing it out at a perfectly constant, unchanging rate. This is the standard model, called $\Lambda$ CDM (Lambda Cold Dark Matter). It's the "flat tire" theory: the pressure is steady, predictable, and boring.

But recently, when scientists looked at the balloon with super-precise telescopes, they noticed something weird. The balloon isn't just expanding; it seems to be changing how fast it's expanding. The pressure inside might be shifting. This is the "Hubble Tension" and other cosmic mysteries.

This paper is like a group of mechanics proposing new, more flexible engines to explain why the balloon is acting up.

The Old Engines (The Problem)

The scientists first looked at the existing engines:

The Constant Engine ( $\Lambda$ CDM): The pressure never changes.
The Simple Variable Engine ( $w$ CDM): The pressure is constant, but we don't know exactly what that constant is.
The "CPL" Engine: This is the current favorite. It assumes the pressure changes in a simple, straight line over time (like a car accelerating at a steady rate).

The Issue: The CPL engine is too rigid. It's like trying to describe a rollercoaster with a straight line. It works okay for a short ride, but if the track twists, loops, or dips unexpectedly (which the new data suggests it might), the straight-line engine fails to capture the full picture.

The New Engines (The Solution)

The authors of this paper, Sonej Alam and Md. Wali Hossain, built two new, more flexible engines called MmAH1 and MmAH2.

Think of the CPL engine as a ruler. It can only measure straight lines.
The new MmAH engines are like playdough. They can be squished, stretched, and molded into smooth curves. They allow the "pressure" of the universe to change in a more complex, natural way, while still being able to act like the old constant engine if the universe decides to be boring again.

The Big Test Drive

To see if these new engines work, the authors took them for a spin with the most up-to-date cosmic data available:

The Cosmic Microwave Background (CMB): The "baby photo" of the universe.
DESI (Dark Energy Spectroscopic Instrument): A massive survey mapping millions of galaxies (like a GPS for the cosmos).
Supernovae (SNeIa): Exploding stars used as "standard candles" to measure distance. They used three different lists of these stars (PantheonPlus, Union3, and the new, high-precision DESY5).

The Results: What Did They Find?

When they ran the numbers, here is what happened:

The Old Model is Stuck: The standard "Constant Pressure" model ( $\Lambda$ CDM) fits the data okay, but it's starting to look a bit strained. It's like trying to force a square peg into a round hole.
The New Engines Fit Better: Both MmAH1 and MmAH2 fit the data significantly better than the old models.
- With the older data (PantheonPlus), the new engines were slightly better.
- With the newest, most precise data (DESY5), the new engines were dramatically better. The fit improved so much that the probability of the old "Constant" model being right dropped to less than 1 in a million (a "5-sigma" result).
The "Tension" Meter: The authors used a statistical tool called the Mahalanobis distance to measure how "stressed" the old model is compared to the new ones.
- With the new data, the old model is stressed out by about 4 to 5 standard deviations. In the world of physics, that's a screaming alarm bell. It means the universe is almost certainly not expanding at a constant rate.

The Takeaway

Imagine you are watching a car drive away.

The Old View: You think the car is driving at a steady 60 mph.
The New Data: You look closer and see the car is actually speeding up and slowing down in a complex pattern.
The Paper's Conclusion: The "steady 60 mph" theory is likely wrong. The universe's Dark Energy is dynamic—it's changing, evolving, and behaving like a living thing rather than a static force.

The authors' new "MmAH" models are the best tools we have right now to describe this complex behavior. They are smooth, flexible, and they match the universe's new, noisy data much better than the old, rigid theories.

In short: The universe is not doing what we thought it was doing. It's more dynamic, more complex, and these new models are the first to successfully capture that complexity without falling apart.

1. Problem Statement

The standard $\Lambda$ CDM cosmological model, while successful, faces significant tensions with observational data, most notably the Hubble tension ( $H_0$ discrepancy) and the $S_8$ tension (matter fluctuation amplitude). Recent data from the Dark Energy Spectroscopic Instrument (DESI) and various Type Ia supernova (SNeIa) compilations suggest hints of Dynamical Dark Energy (DDE), where the equation of state (EoS) of dark energy, $w(z)$ , evolves with redshift rather than remaining a constant cosmological constant ( $w = -1$ ).

Existing frameworks for probing DDE have limitations:

CPL Parametrization: The standard Chevallier-Polarski-Linder form, $w(a) = w_0 + w_a(1-a)$ , is a first-order Taylor expansion. It is analytically tractable but may be too restrictive to capture complex transitions at higher redshifts.
mAH Parametrization: The Akhtar-Hossain (mAH) model allows for thawing and phantom behaviors but has rigid asymptotic behaviors (e.g., $w \to -1$ at high redshift) and forbids phantom crossing, limiting its physical flexibility.
Higher-order extensions: Adding quadratic terms (e.g., CPL-wb) increases flexibility but introduces free parameters that are often poorly constrained, leading to degeneracies.

The authors aim to investigate observational evidence for DDE using three-parameter models that offer greater flexibility than CPL or mAH while maintaining physical coherence and bounded evolution.

2. Methodology

Proposed Models: Modified Minimal AH (MmAH)

The authors introduce two new three-parameter extensions of the mAH model, termed MmAH1 and MmAH2, designed to provide smooth, bounded evolution of $w(z)$ and reduce to $\Lambda$ CDM in the appropriate limit.

MmAH1: Defined by $w(z) = \beta + \frac{\alpha}{1 + \frac{1+z}{\Omega_\delta}}$ $w (z) = β + \frac{α}{1 + \frac{1 + z}{Ω _{δ}}}$ .
- Introduces a free parameter $\beta$ (asymptotic value at $z \to \infty$ ) and $\alpha$ (amplitude).
- Uses the matter-dark energy equality redshift ( $z_{eq}$ ) as a free parameter instead of the transition scale $\Omega_\delta$ for better observational constraints.
MmAH2: Defined by $w(z) = \beta + \frac{\alpha z}{1 + \frac{1+z}{\Omega_\delta}}$ $w (z) = β + \frac{α z}{1 + \frac{1 + z}{Ω _{δ}}}$ .
- Similar structure but with a different redshift dependence in the numerator, allowing for different asymptotic behaviors ( $w \to \beta + \alpha\Omega_\delta$ as $z \to \infty$ ).
- Uses the transition redshift $z_t$ as a free parameter.

Both models allow for positive and negative values of the effective CPL slope parameter $w_a$ , enabling a wider range of dynamical behaviors (including phantom crossing) compared to the original mAH.

Data Analysis

The study performs a joint analysis using Markov Chain Monte Carlo (MCMC) techniques (via the EMCEE sampler) on a comprehensive dataset:

CMB: Planck 2018 distance priors (TT, TE, EE + lowE).
BAO: DESI Data Release 2 (DR2) covering $0.1 < z < 4.2$ .
H(z): 31 cosmic chronometer measurements.
RSD: Redshift-space distortion measurements of the growth rate $f\sigma_8$ .
SNeIa: Three distinct compilations to test robustness:
1. PantheonPlus (1550 SNe).
2. Union3 (2087 SNe, binned).
3. DESY5 (Dark Energy Survey Year 5, 1829 SNe).

Statistical Tools

Model Comparison: Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) to penalize model complexity.
Tension Quantification: The Mahalanobis distance is used to quantify the statistical tension between dynamical models and $\Lambda$ CDM in 1D, 2D, and 3D parameter subspaces. This involves computing the squared distance between mean parameter vectors normalized by the combined covariance matrix.

3. Key Contributions

Novel Parametrizations: The formulation of MmAH1 and MmAH2, which generalize the mAH model by promoting the high-redshift asymptote to a free parameter, allowing for more diverse dark energy dynamics (thawing, freezing, phantom) without the restrictive asymptotes of the original mAH.
Comprehensive Comparative Study: A systematic comparison of seven models ( $\Lambda$ CDM, wCDM, CPL, CPL-wb, mAH, MmAH1, MmAH2) across three distinct SNeIa datasets combined with CMB, BAO, H(z), and RSD.
Global vs. Local Tension Analysis: The paper distinguishes between "local" tensions (comparing effective CPL parameters $w_0, w_a, w_b$ derived from the models) and "global" tensions (comparing the fundamental parameters of the MmAH models directly against $\Lambda$ CDM), providing a more robust assessment of model consistency.

4. Key Results

Parameter Constraints and Fit Quality

Standard Parameters: The six standard cosmological parameters ( $\Omega_{m0}, h, \omega_b, r_d, \sigma_8, M$ ) remain stable across all models and datasets, varying within $1\sigma$ .
$\Lambda$ CDM vs. wCDM: The constant EoS parameter $w_0$ in wCDM remains consistent with $-1$ across all datasets, showing no significant preference for a constant deviation from the cosmological constant.
CPL Performance: CPL shows modest to strong evidence for dynamical dark energy, particularly with the DESY5 dataset ( $\Delta\chi^2 \approx -36.1$ , $\Delta AIC \approx -32.1$ ). However, the BIC penalizes the extra parameter for lower-precision datasets (PantheonPlus).
MmAH Performance:
- MmAH1 and MmAH2 consistently outperform $\Lambda$ CDM and the original mAH model.
- For PantheonPlus: $\Delta\chi^2 \approx -7.1$ (comparable to CPL's $-5.98$).
- For Union3: $\Delta\chi^2 \approx -11.5$ (matching CPL).
- For DESY5: $\Delta\chi^2 \approx -37.6$ (slightly outperforming CPL's $-36.1$).
- The MmAH models achieve these improvements while maintaining well-behaved parameter correlations and physical boundedness.

Statistical Tensions

Local Tensions (Low-z):
- PantheonPlus: MmAH1 and CPL show mild-to-moderate tensions with $\Lambda$ CDM ( $\sim 2.3\sigma - 2.5\sigma$ ).
- Union3: Tensions increase to $\sim 2.8\sigma - 3.2\sigma$ .
- DESY5: Significant deviations are observed, with tensions reaching $4.0\sigma - 5.0\sigma$ for CPL and MmAH models.
Global Tensions (Full Parameter Space):
- For PantheonPlus and Union3, MmAH1 shows moderate global deviations ( $\sim 2.4\sigma - 3.1\sigma$ ), while MmAH2 remains largely consistent ( $< 1.3\sigma$ ).
- For DESY5, the global tension becomes highly significant for both models, reaching $\sim 5.5\sigma$ . This indicates that the DESY5 data drives a strong preference for dynamical dark energy over the cosmological constant, even when considering the full parameter space of the MmAH models.

Model Selection Criteria

AIC: Favors CPL and MmAH models significantly for Union3 and DESY5 ( $\Delta AIC < -5$ ), indicating the improved fit outweighs the penalty for extra parameters.
BIC: Penalizes the extra parameters more heavily. For PantheonPlus, BIC disfavors the complex models. However, for DESY5, the BIC also favors the dynamical models ( $\Delta BIC \approx -15$ for MmAH, $-21$ for CPL), suggesting the evidence for DDE is robust in high-precision data.

5. Significance and Conclusion

The paper provides consistent evidence for departures from the $\Lambda$ CDM model, specifically favoring dynamical dark energy when high-precision data (DESY5) is included.

Superiority of MmAH: The MmAH1 and MmAH2 models are presented as physically coherent alternatives to CPL. They offer similar statistical improvements in fit quality but with the advantage of being defined globally over the entire redshift range, avoiding the Taylor expansion limitations of CPL at high $z$ .
DESY5 Impact: The inclusion of the DESY5 SNeIa sample is the critical factor that elevates the statistical significance of DDE from "mild hints" ( $\sim 2\sigma$ ) to "strong evidence" ( $\sim 4-5\sigma$ ).
Implications: The results suggest that the dark energy equation of state is likely not a constant. The smooth, bounded evolution provided by the MmAH parametrizations offers a viable theoretical framework to describe this dynamics, potentially resolving tensions or pointing toward new physics beyond the cosmological constant.

The authors conclude that while $\Lambda$ CDM remains a viable baseline, the data increasingly supports a richer, evolving dark energy sector, and future high-precision surveys will be essential to distinguish between statistical fluctuations and genuine new physics.