Dynamical Dark Energy in the Crosshairs: A Joint Analysis with DESI, Pantheon plus, and TDCOSMO Constraints

In this work, we perform a comprehensive joint analysis of three representative dark energy models - $\mathrm{\Lambda }\mathrm{C}\mathrm{D}\mathrm{M}$, the Chevallier-Polarski-Linder (CPL) parametrization, and the Generalised Emergent Dark Energy (GEDE) model - using the latest observational datasets: baryon acoustic oscillation (BAO) measurements from DESI Data Releases 1 and 2 (DR1/DR2), the Pantheon Plus sample of Type Ia supernovae (SNe Ia), and time-delay cosmography from TDCOSMO lensing. The CPL model yields a statistically significant improvement over $\mathrm{\Lambda }\mathrm{C}\mathrm{D}\mathrm{M}$, with $\mathrm{\Delta }{\chi }^2=-3.6$ for DESI DR2+Pantheon Plus+TDCOSMO, favoring a quintessence-like behavior ($w_0=-0.87_{-0.05}^{+0.05}$, $w_a=-0.41_{-0.28}^{+0.28}$ at 1$\sigma$ confidence level). The GEDE model also exhibits slightly favoring than $\mathrm{\Lambda }\mathrm{C}\mathrm{D}\mathrm{M}$ with observations, yielding $\mathrm{\Delta }{\chi }^2=-2.83$ (DR1) and $\mathrm{\Delta }{\chi }^2=-3.82$ (DR2). The captured potential dark energy evolution transition parameter $\mathrm{\Delta }$ in GEDE model is constrained to $-0.52_{-0.20}^{+0.22}$ (DR1) and $-0.50_{-0.17}^{+0.18}$ (DR2), showing a $2.4\sigma$ and $2.9\sigma$ deviation from zero respectively. This statistically significant ($>2\sigma$) non-zero value of $\mathrm{\Delta }$ provides evidence against a pure cosmological constant scenario. The negative sign indicates quintessence-like behavior ($w>-1$) in the late universe. Notably, the GEDE constraints show a $>3\sigma$ tension with the $\mathrm{\Delta }=1$ (Phenomenological Emergent Dark Energy model) predictions. This significant discrepancy implies that while both models belong to the emergent dark energy family, their fundamentally different transition mechanisms lead to distinct cosmological implications. Our results collectively suggest that GEDE provides a phenomenologically viable alternative to $\mathrm{\Lambda }$CDM.