Pressure Parametrization of Dark Energy: First and Second-Order Constraints with Latest Cosmological Data

We explore an extension of the $\Lambda$CDM model in which the pressure $p$ of the dark energy (DE) fluid evolves with the expansion of the Universe, expressed as a function of the scale factor $a$. The corresponding energy density $\rho$ is derived from the continuity equation, resulting in a dynamical equation-of-state parameter $w \equiv p/\rho$ during the late-time expansion of the Universe. The pressure is modeled using a Taylor expansion around the present epoch ($a = 1$), introducing deviations from a cosmological constant within the dynamical dark energy (DDE) framework. At first order, a single new parameter $\Omega_1$ captures linear deviations, while a second-order parameter, $\Omega_2$, accounts for quadratic evolution in the pressure. We constrain the first- and second-order DDE models using multiple observational datasets and compare their performance against $\Lambda$CDM and the CPL parameterization. A joint analysis of Planck CMB, DESI, and DESY5 data yields the strongest evidence for DDE, with a $2.7\sigma$ deviation in the first-order model and over $4\sigma$ in the second-order model, providing strong statistical support for a departure from a cosmological constant. The reconstructed DE evolution in the second-order case reveals a distinctive non-monotonic behavior in both energy density and $w_{\rm DE}(a)$, including clear phantom-crossing phenomena. Notably, the late-time evolution of $w_{\rm DE}(a)$ remains consistent across datasets and shows strong agreement with the CPL parameterization, underscoring the robustness of the pressure-based approach.