Challenging the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Λ</mml:mi></mml:math>CDM model: 5<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi></mml:math> evidence for a dynamical dark energy late-time transition

This paper addresses a central question in modern cosmology: is dark energy (DE) consistent with a cosmological constant ( $w=-1$) or does it undergo a late-time transition between “quintessence-like” behavior ( $w>-1$) and “phantom-like” behavior ( $w<-1$)? The motivation is twofold. First, the standard $\Lambda$CDM model is extremely successful, yet several tensions in late- vs early-universe measurements (notably the Hubble constant $H_0$ tension and related growth/amplitude discrepancies) suggest that the simplest DE picture may be incomplete. Second, new high-precision large-scale structure data from DESI (including BAO) provide an opportunity to detect subtle changes in the expansion history and, by extension, in the DE equation of state (EoS).

The authors propose an “agnostic” observational test designed to let the data determine both the timing and the direction of a DE EoS transition, without imposing a priori whether the universe transitions from quintessence to phantom or vice versa. Concretely, they introduce two extra parameters: a deviation amplitude $\Delta$ controlling the magnitude of departure from $w=-1$, and a critical redshift $z^{†}$ (equivalently a critical scale factor $a^{†}$) marking when the transition occurs. In the baseline phenomenological framework (called $w^{†}$CDM), the EoS is modeled as an abrupt piecewise switch around $a^{†}$: $$w(a)=-1+\Delta \mathrm{sgn}[a^{†}-a].$$ Thus, for $a<a^{†}$ the EoS is $w=-1+\Delta$, while for $a>a^{†}$ it becomes $w=-1-\Delta$. Importantly, the sign of $\Delta$ is not fixed by theory; the MCMC sampling determines whether the best fit corresponds to a phantom-to-quintessence or quintessence-to-phantom evolution. Because abrupt switches can be theoretically problematic for perturbations in a perfect-fluid picture, the authors also embed the idea into a minimally modified gravity scenario known as VCDM (their $w^{†}$VCDM), where the transition is smoothed using a $\tanh$ form: $$w(N)=-1+\Delta \tanh [\zeta (N^{†}-N)],$$ with $\zeta$ fixed to $10^{1.5}$ to control rapidity. This embedding is used to place the phenomenology into a theoretically stable framework without introducing extra propagating degrees of freedom.

Methodologically, the study performs global parameter inference using CLASS (Boltzmann solver) and MontePython (MCMC sampler). They assume a flat universe and use standard $\Lambda$CDM parameters $\{{\Omega }_b{h}^2,{\Omega }_c{h}^2,{\tau }_{\mathrm{reio}},100{\theta }_s,\ln (10^{10}{A}_s),n_s\}$ plus the two DE-transition parameters $\{\Delta ,z^{†}\}$. Priors are taken as flat over broad ranges, including $\Delta \in [-5,5]$ and $z^{†}\in [0,10]$. Convergence is checked with the Gelman–Rubin criterion requiring $R-1\le 10^{-2}$. For model comparison, they use (i) a chi-square difference test relative to $\Lambda$CDM, accounting for the two extra degrees of freedom ($\Delta N=2$), (ii) the Akaike Information Criterion (AIC) to penalize extra parameters, and (iii) Bayesian evidence via Bayes factors $B_{ij}$ interpreted with the Jeffreys–Kass–Raftery scale.

The data combinations are comprehensive: CMB constraints from Planck 2018 (high-$\ell$ Plik TT/TE/EE, low-$\ell$ TT-only and EE-only SimAll, plus CMB lensing), BAO constraints from DESI DR2 (including galaxy/quasar and Lyman-$\alpha$ tracers, with isotropic and anisotropic BAO in nine redshift bins over $0.295\le z\le 2.330$), and multiple SN Ia compilations: PantheonPlus (PP), PantheonPlus calibrated with SH0ES Cepheid anchors (PPS), Union3, and DESY5 (DES supernova sample). The key results are reported across both $w^{†}$CDM and $w^{†}$VCDM, with the authors emphasizing that conclusions are broadly consistent between them.

The central finding is a statistically significant preference for a late-time transition in the DE EoS from phantom-like behavior at higher redshift to quintessence-like behavior at lower redshift (i.e., a sign pattern corresponding to $\Delta <0$ in their convention). The strongest evidence arises when combining Planck + DESI DR2 + DESY5. In the $w^{†}$CDM case, they obtain $\Delta =-0.162\pm 0.031$ and $z^{†}=0.490_{-0.079}^{+0.063}$ (68% CL). The improvement over $\Lambda$CDM is quantified as $\Delta {\chi }_{\min }^2=-27.34$, corresponding to a $4.9\sigma$ significance; using the VCDM embedding gives $\Delta {\chi }_{\min }^2=-29.24$ and $5.1\sigma$. AIC differences are also strongly negative ($\Delta \mathrm{AIC}=-23.34$ for $w^{†}$CDM and $-25.24$ for $w^{†}$VCDM), indicating that the better fit outweighs the penalty for two additional parameters. Bayesian evidence is reported as $\ln B_{ij}\sim -9$ for the best combination, which the authors interpret as decisive support for the extended models over $\Lambda$CDM.

They also find that the transition redshift clusters around $z^{†}\sim 0.3\text{–}0.5$ across dataset choices. For example, Planck + DESI DR2 alone yields $\Delta =-0.226\pm 0.070$ and $z^{†}=0.533\pm 0.064$ with $3.2\sigma$ preference over $\Lambda$CDM ($\Delta {\chi }_{\min }^2=-10.18$). Adding PantheonPlus (Planck + DESI + PP) gives $\Delta =-0.102_{-0.035}^{+0.031}$ and $z^{†}=0.553_{-0.140}^{+0.047}$ with $2.9\sigma$ significance ($\Delta {\chi }_{\min }^2=-11.44$). Planck + DESI + Union3 yields $\Delta =-0.174\pm 0.044$ and $z^{†}=0.524\pm 0.069$ with $3.6\sigma$ significance ($\Delta {\chi }_{\min }^2=-14.86$). The DESY5 SN sample is singled out as providing the strongest statistical leverage; the authors note a reported $\sim 0.04$ mag discrepancy between low- and high-redshift parts of the PP vs DESY5 samples, which could either reflect systematics or genuine new physics.

Regarding cosmological tensions, the transition does not fully resolve the Hubble tension. In the best-fit Planck + DESI + DESY5 combination, the inferred $H_0$ is $66.40\pm 0.49$ km/s/Mpc ($w^{†}$VCDM) or $66.40\pm 0.49$ km/s/Mpc in the table for $w^{†}$CDM as well (the table shows $H_0=66.40\pm 0.49$ for the $w^{†}$CDM column and $66.41\pm 0.50$ for VCDM). This remains in tension with local distance-ladder values, though it is close to DESI’s own $H_0$ expectations in related parameterizations. The authors also state that the model does not substantially mitigate the $S_8$ discrepancy.

The paper’s limitations are primarily those inherent to the modeling and inference strategy. First, the EoS transition is parameterized phenomenologically (abrupt in $w^{†}$CDM, smoothed in $w^{†}$VCDM), so the result is not a unique microphysical mechanism; it is an effective description of the data. Second, the authors acknowledge that CMB-only constraints are weak due to degeneracies introduced by the extra parameters (Planck alone yields only an upper bound $z^{†}<0.626$ at 68% CL and weak constraints on $\Delta$). Third, the strongest evidence relies on specific SN datasets (especially DESY5), and the authors explicitly discuss the possibility that observed differences between SN samples could be due to systematics rather than new physics. Finally, they attribute some features in the reconstructed $w(z)$ confidence regions near $z^{†}$ to numerical error propagation amplified by the rapid transition, which cautions against over-interpreting the detailed shape of reconstructed uncertainties.

Practically, the results matter for anyone using late-time expansion history to infer the nature of DE. If confirmed, the evidence implies that future analyses of BAO and SN Ia should allow for EoS transitions rather than assuming a constant $w$ or $\Lambda$CDM. Observational teams working with DESI-like BAO and next-generation SN surveys should pay particular attention to the redshift range $z\sim 0.3\text{–}0.5$, where the transition is inferred and where model predictions diverge most from $\Lambda$CDM. Theoretical model builders should also note that the authors provide a stable embedding (VCDM) but still require a microphysical explanation for why such a phantom-to-quintessence transition would occur. Overall, the paper claims discovery-level statistical preference (near $5\sigma$) for a late-time DE dynamical transition, while emphasizing that it does not fully solve the $H_0$ tension or the $S_8$ discrepancy.