The standard model of cosmology, known as Lambda cold dark matter or LambdaCDM, treats dark energy as a cosmological constant, a fixed energy density with an equation of state parameter w locked at exactly −1. This simple picture has been remarkably successful at explaining the cosmic microwave background, the relic radiation from the early universe. Yet tensions persist. The Hubble constant inferred from Planck CMB measurements sits in roughly 5-sigma tension with local determinations by the SH0ES collaboration, and weak-lensing surveys prefer lower values of the clustering amplitude S8 than the CMB predicts. Most provocatively, DESI’s baryon acoustic oscillation data, when combined with Type Ia supernovae and CMB observations, favour dynamical dark energy over LambdaCDM at the 2.6 to 3.1 sigma level within the popular Chevallier–Polarski–Linder, or CPL, framework. Crucially, however, if dark energy is allowed only a constant equation of state, LambdaCDM still lies well within the 1-sigma confidence region even with recent data. The noticeable deviation emerges only when more flexible low-redshift parametrizations are permitted, with LambdaCDM lying more than 2 sigma from the best-fit model.

That observation lies at the heart of the problem the study set out to probe. The CPL parametrization, w(a) = w0 + wa(1−a), where a is the cosmic scale factor, is essentially a first-order Taylor expansion of the equation of state around the present epoch. It is convenient and well constrained at low redshifts, roughly z < 1, but may be inadequate beyond that regime, since the true dynamics of dark energy need not follow a linear extrapolation. In the absence of strong theoretical guidance, the choice of parametrization injects an element of arbitrariness into the inference. If the data appear to prefer a phantom-like phase, in which w dips below −1 and the dark energy density grows, the question is whether that reflects a genuine physical signal or an artifact of the functional form the analyst happened to adopt.

To answer that question, the analysis compared five parametrizations with equal numbers of free parameters. Alongside CPL, the study included the Barboza–Alcaniz model, in which the evolution term is divided by a² + (1−a)², and an exponential model, w(a) = w0 + wa(e^(1−a) − 1). It also introduced two new phenomenological forms designed to allow greater flexibility around z ∼ 1: a power-law parametrization, w(a) = w0 a^(−α) = w0(1+z)^α, and a modified power-law, w(a) = 2w0/(1 + a^α). The power-law form diverges as a approaches zero for α > 1, making it an effective description for low and intermediate redshifts rather than a complete early-universe model, while the modified version regularizes that behaviour, asymptotically approaching 2w0 at high redshift. Both reduce to CPL at very low redshift, with a derived slope of αw0 or αw0/2 respectively, but they rapidly deviate from CPL at higher redshifts, exhibiting significantly enhanced phantom behaviour in matched comparisons.

The observational arsenal behind the analysis is formidable. The study used compressed distance priors reconstructed from Planck 2018 TT, TE, EE and lowE measurements to capture the CMB information relevant to late-time cosmology. Baryon acoustic oscillation constraints came from DESI Data Release 2, comprising over 14 million extragalactic objects, including emission line galaxies, luminous red galaxies, quasars and Lyman-alpha forest tracers, providing distance ratios over the redshift range 0.4 < z < 4.2. Three separate Type Ia supernova compilations were tested in turn: PantheonPlus, trimmed to 1590 luminosity distance measurements after removing calibrator supernovae and objects below z = 0.01; Union3, providing 22 binned data points from 2087 cosmologically useful supernovae; and DESY5, totaling 1829 data points after the latest DES-Dovekie recalibration. Thirty-one cosmic chronometer measurements of the Hubble parameter, spanning 0.07 ≤ z ≤ 1.965, completed the combinations, labeled +PantheonPlus, +Union3 and +DESY5. Markov chain Monte Carlo sampling with the EMCEE sampler, checked for convergence with the Gelman–Rubin statistic, constrained the parameters of LambdaCDM and all five dark energy models, with the sound horizon computed from standard early-universe relations rather than sampled independently.

The headline result is a study in statistical modesty. All five dynamical dark energy parametrizations improved the fit relative to LambdaCDM, but the likelihood-ratio significance of that preference was 2.2 to 2.5 sigma for +PantheonPlus, 3.1 to 3.2 sigma for +Union3 and 3.8 to 3.9 sigma for +DESY5. For any given dataset, the five parametrizations differed by at most 0.25 sigma, meaning the strength of the preference depended far more on which supernova sample was used than on which functional form was assumed. Because current data provide limited direct constraints beyond z = 2, the signal was primarily driven by the equation of state in the range 1 ≤ z ≤ 2, precisely where parametrizations with enhanced flexibility, such as the power-law forms, mildly improved the fit. The phantom-like excursions below w = −1 at intermediate redshifts, a feature also seen in recent DESI analyses, were consistent across all five models but never decisively established.

A qualitative hierarchy nonetheless emerged in the reconstructed equations of state. For the +PantheonPlus combination, the power-law model exhibited the most enhanced phantom behaviour beyond z ∼ 1, followed by the modified power-law, exponential, Barboza–Alcaniz and CPL forms, and the same ordering was reflected in the statistical results, with the power-law delivering the largest improvement in chi-squared. Similar hierarchies appeared for +Union3 and +DESY5, with the power-law again ranking first and Barboza–Alcaniz last. Yet this ordering is largely driven by the intrinsic extrapolation properties of the parametrizations rather than by strong constraints from the data. The minimum chi-squared of the power-law and CPL models, which carry the same number of free parameters, differed by at most 1.3 across all three data combinations. The 1-sigma credible regions of the different models overlapped substantially, so the hierarchy should not be read as evidence for a preferred model or a definitive detection of enhanced phantom dynamics.

Model selection criteria told a similarly nuanced story. The Akaike Information Criterion, which penalizes each free parameter by a fixed amount, favoured the dynamical dark energy models over LambdaCDM for all three supernova samples. The Bayesian Information Criterion, whose penalty grows with the logarithm of the number of data points, was harsher: it mildly favoured the dynamical models for +Union3 and +DESY5, with differences of −4.7 to −2.5, but disfavoured them for +PantheonPlus, with differences of +6.3 to +7.6. Since the preference appeared in all five functional forms, it is not tied to any specific parametrization, but its dependence on the supernova sample underscores how fragile the case for evolving dark energy remains.

The broader lesson concerns what present observations can and cannot constrain. Current data primarily constrain the low-redshift behaviour of dark energy, while its evolution at higher redshifts remains weakly constrained and parametrization-dependent. Models with restricted asymptotic behaviour, such as CPL and Barboza–Alcaniz, limit the extent of phantom evolution, whereas less constrained forms like the power-law allow more pronounced departures from w = −1 beyond z ∼ 1, changing the fit only slightly. All five parametrizations show a modest, consistent preference for dynamical dark energy with phantom crossing in the recent past, in agreement with recent DESI analyses, but the strength of that phantom evolution at intermediate redshifts is not robustly pinned down. Future observations with higher precision will be needed to determine whether these features reflect genuine dark energy dynamics or arise from parametrization-dependent effects. Until then, the mathematics chosen to describe the universe’s most mysterious ingredient may matter as much as the data themselves.

Subject of Research: Sensitivity of dynamical dark energy inference to equation of state parametrization choice

Article Title: Probing the sensitivity of dark energy dynamics to equation of state parametrization flexibility

Article References: Hossain, M. W. (2026). Probing the sensitivity of dark energy dynamics to equation of state parametrization flexibility. The European Physical Journal C, 86(10), Article 1219. https://doi.org/10.1140/epjc/s10052-026-16526-x

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16526-x

Keywords: dark energy, LambdaCDM, DESI, equation of state, CPL parametrization, phantom regime, Type Ia supernovae, baryon acoustic oscillations, cosmic microwave background, model selection, cosmology, Probing