Dynamical Dark Energy at Late Time $\Lambda$CDM

We investigate the dynamical properties of dark energy through a detailed analysis of its equation of state parameter $w(z)$ as a function of redshift. We derive a general expression for $w(z)$ from the Friedmann-Lema\^itre-Robertson-Walker (FLRW) equations, establishing a direct relationship between the dark energy equation of state and the observable Hubble parameter $H(z)$ and its derivative. Using the relation $w(z) = -1 + \frac{2(1+z)}{3H(z)} \frac{dH}{dz}$, we develop an approximation method valid for $z \lesssim 1$ that accounts for the changing balance between matter and dark energy contributions to cosmic expansion. We compare our theoretical framework with recent observational data from the Dark Energy Spectroscopic Instrument (DESI) DR2, analysing how well the commonly used Chevallier-Polarski-Linder (CPL) parametrization $w(z) = -1 + w_a \frac{z}{1+z}$ captures the evolution of dark energy. Our results indicate that the dark energy equation of state exhibits a monotonic evolution with redshift, transitioning from deceleration to acceleration around $z \approx 0.7$. Notably, our predicted $w_{\mathrm{DE}}$ remains greater than $-1$ across all redshifts, avoiding phantom energy scenarios that would violate the null energy condition. This work demonstrates how precise measurements of the cosmic expansion history can constrain the nature of dark energy and provides a framework for testing dynamical dark energy models against current and future cosmological observations.