Early against Late: A contrast on dark energy in the light of DESI DR2

Recent findings from Dark Energy Spectroscopic Instrument (DESI) Baryon Acoustic Oscillation (BAO) measurements, combined with Type Ia supernovae and Cosmic Microwave Background (CMB) data, suggest potential parameter-level deviations from $Λ\text{CDM}$. However, parameter exclusions omit prior-volume penalties, whereas Bayesian evidence uncovers a stark dichotomy between early- and late-time dynamics. To quantify this effect, we perform a Bayesian model comparison that explicitly accounts for the cosmological epoch of dark energy dynamics, contrasting two schemes acting in opposite epochs of cosmic history: pre-recombination Early Dark Energy (EDE) and late-time Chevallier--Polarski--Linder (CPL), both measured against a baseline $Λ$CDM. We validate the learned harmonic-mean estimator against \texttt{UltraNest} (at background level) before applying it to the CMB analysis. Background-only data already disfavor both extensions, yielding $\ln B_{Λ\text{CDM},\text{EDE}} = 2.45 \pm 0.25$ and $\ln B_{Λ\text{CDM},\text{CPL}} = 3.45 \pm 0.24$. Including CMB sharpens this result: EDE is decisively rejected ($\ln B_{Λ\text{CDM},\text{EDE}} = 12.44 \pm 0.05$) with its early-time fraction constrained to $10^3\,Ω_e^{\rm EDE} = 2.0^{+0.6}_{-1.0}$, whereas CPL remains moderately disfavored ($\ln B_{Λ\text{CDM},\text{CPL}} = 2.33 \pm 0.06$), even though 2D parameter posteriors for both models display a multi-$σ$ deviation from $Λ\text{CDM}$. Thus, Bayesian evidence weakens reported preferences for dark energy dynamics, with the impact depending on the cosmic epoch involved, confirmed by our functional reconstructions of $w_{\rm de}(z)$ and $Ω_{\rm de}(z)$ from posterior samples.