Is phantom barrier crossing inevitable? A cosmographic analysis

Recent findings from the Dark Energy Spectroscopic Instrument (DESI), when combined with supernova observations and CMB measurements, provide indications of a time-evolving dark energy component, with possible hints of phantom-like behavior in the recent past. Motivated by these developments, we investigate late-time cosmic acceleration within a cosmography-inspired framework based on the jerk parameter. Instead of adopting a phenomenological parametrization of the dark energy equation of state, we reformulate the differential equation defining the jerk into a class of integrable anharmonic oscillator equations through a specific closure ansatz. This procedure selects a restricted but analytically solvable subclass of cosmographic evolutions, leading to a closed-form expression for the dark energy equation of state governed effectively by a single parameter. Using DESI-DR2 BAO data, supernova data, and a compressed Planck likelihood, we constrain the cosmological parameters and find deviations from a cosmological constant at late times. Within this integrable model class, we do not find evidence of phantom barrier crossing. Further we show, using a dynamical systems approach, that $$w_{DE} = -1$$ w DE = - 1 corresponds to a generic bifurcation point, which prevents crossing within cosmographic framework.