Hierarchical Gaussian-process test of DESI's dynamical dark-energy preference

Technical Summary: Hierarchical Gaussian-process test of DESI's dynamical dark-energy preference

Problem Statement
Recent analyses combining Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) Baryon Acoustic Oscillation (BAO) data with Cosmic Microwave Background (CMB) and Type Ia supernovae (SN Ia) have reported a preference for evolving dark energy over the standard $\Lambda$CDM model at a significance of roughly $2.8\text{--}4.2\sigma$ using the Chevallier–Polarski–Linder (CPL) parameterisation. However, these results are sensitive to the specific CMB likelihood treatment (full vs. compressed) and the assumed parametric form of the equation of state $w(z)$. Non-parametric reconstructions, such as Gaussian Processes (GP), offer a way to infer the expansion history without assuming a fixed functional shape for $w(z)$. A common limitation in previous GP studies is the "two-step" procedure where kernel hyperparameters (amplitude $\sigma_f$ and length scale $l$) are fixed at maximum-likelihood values before propagating uncertainties, thereby omitting hyperparameter uncertainty from the final error budget. This paper investigates whether the reported dynamical dark-energy preference persists when using a hierarchical Bayesian framework that co-samples these hyperparameters, specifically under a compressed CMB pipeline.

Methodology
The authors employ a hierarchical Gaussian Process model to reconstruct the late-time expansion history $H(z)$ and the dark energy equation of state $w(z)$. The methodology integrates the following components:

1. Data Combination: The analysis combines four distinct probes:

   • Cosmic Chronometers (CC): 37 differential-age $H(z)$ measurements used to condition the GP directly.
   • DESI DR2 BAO: 13 distance measurements from various tracers (BGS, LRG, ELG, QSO) covering $0.1 < z < 4.2$.
   • Planck 2018 Compressed Distance Priors: Instead of the full Planck likelihood, the study uses compressed priors ($R, \ell_A, \omega_b h^2$) derived from TT, TE, EE, and lowE data.
   • Pantheon+ SN Ia: 1624 supernovae with full statistical and systematic covariance.

2. Hierarchical Architecture:

   • Latent Process: $H(z)$ is treated as a latent function drawn from a zero-mean GP with a squared-exponential (Radial Basis Function) kernel, conditioned on the CC data.
   • Co-sampling: Unlike standard approaches, the kernel hyperparameters ($\sigma_f, l$) are co-sampled with the cosmological parameters ($H_0, \Omega_m, \Omega_k, \omega_b h^2$) within a Markov Chain Monte Carlo (MCMC) framework.
   • Effective Likelihood: Because BAO, CMB, and SN Ia depend non-linearly on $H(z)$, the authors define a Monte-Carlo effective likelihood ($L_{\text{eff}}$). This involves averaging the likelihoods of the probes over multiple independent draws of $H(z)$ from the GP posterior at each MCMC step.

3. Reconstruction: The equation of state $w(z)$ is derived non-parametrically from the reconstructed $H(z)$ via the normalized dark energy density $f_{DE}(z)$, using the relation $w(z) = -1 + \frac{1+z}{3} \frac{f'_{DE}(z)}{f_{DE}(z)}$.

4. Ablation Studies: The framework is tested against several variations: fixing hyperparameters to CC maximum-likelihood values, removing Pantheon+ or specific LRG BAO bins, replacing the RBF kernel with a Matérn-5/2 kernel, and tightening the prior on the length scale $l$.

Key Results

• Equation of State: The baseline hierarchical GP analysis yields a low-redshift equation of state of $w(z \simeq 0) = -0.80^{+0.26}_{-0.23}$ (68% C.L.). This value is approximately $0.8\sigma$ away from the $\Lambda$CDM value of $w = -1$.
• Hyperparameters: The posterior favors a large correlation length scale ($l \simeq 3.79^{+0.81}_{-1.12} \, h^{-1}\text{Gpc}$), which penalizes rapid transitions in $w(z)$ at low redshifts.
• CPL Comparison: A parametric CPL fit performed on the exact same compressed data pipeline improves the fit over nested $\Lambda$CDM by only $\Delta\chi^2 \sim 1$ (approximately $1\sigma$), with $w_0 = -0.94^{+0.07}_{-0.07}$ and $w_a = -0.32^{+0.26}_{-0.31}$.
• Robustness: Ablation studies show that the result is robust. Fixing hyperparameters, removing SN data, removing LRG1/2 BAO bins, or changing the kernel type all leave the median $w(0)$ within $\lesssim 1\sigma$ of $-1$.
• Cosmological Parameters: The derived Hubble constant is $H_0 = 70.1^{+3.6}_{-3.1}$ km s$^{-1}$ Mpc$^{-1}$, lying between Planck and SH0ES values. The matter density $\Omega_m = 0.317^{+0.032}_{-0.029}$ is consistent with Planck.

Significance and Claims
The paper concludes that the mild preference for evolving dark energy reported in this specific analysis is not statistically significant enough to challenge the $\Lambda$CDM model. The authors explicitly state that their findings do not address the stronger $2.8\text{--}4.2\sigma$ preference reported by DESI using the full Planck likelihood. Instead, the results highlight that the significance of dynamical dark energy is highly sensitive to the CMB treatment (compressed vs. full likelihood) and the analysis pipeline.

The primary contribution is the demonstration that when using a hierarchical GP with co-sampled hyperparameters and compressed CMB priors, the data combination does not demand a large evolution in $w(z)$ (i.e., a large $w_a$). The study isolates the impact of analysis choices (kernel type, hyperparameter sampling, data subsets) within a single pipeline, showing that none of these variations restore a strong "quintessence-to-phantom" transition signal. The authors emphasize that their modest results are specific to the compressed-CMB setting and that future work utilizing the full Planck likelihood is required to definitively assess tensions with DESI's headline significance.