Pressure parametrization of dark energy: first and second-order constraints with latest cosmological data

We explore an extension of the ΛCDM 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 ρ is derived from the continuity equation, resulting in a dynamical equation-of-state parameter w ≡ p/ρ 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 Ω1 captures linear deviations, while a second-order parameter, Ω2, 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 ΛCDM and the CPL parameterization. A joint analysis of Planck CMB, DESI, and DESY5 data yields the strongest evidence for DDE, with a 2.7σ deviation in the first-order model and over 4σ 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 wDE(a), including clear phantom-crossing phenomena. Notably, the late-time evolution of wDE(a) remains consistent across datasets and shows strong agreement with the CPL parameterization, underscoring the robustness of the pressure-based approach.