Beyond j=1: Observational Constraints on Almost- CDM Cosmologies

The cosmographic condition j(z)=1 provides the kinematical signature of the spatially flat CDM model independently of any specific dark-energy or modified-gravity theory. We investigate the extent to which current observations permit departures from this condition by considering three phenomenological ``almost- CDM'' cosmographic closures, in which the cosmic jerk differs slightly from unity through a small deformation parameter epsilon. The models are constrained using Markov Chain Monte Carlo analyses of recent DESI baryon acoustic oscillation measurements together with compressed Planck cosmic microwave background likelihoods and the Union3, Pantheon+, and DESY5 Type Ia supernova compilations. Rather than assuming a parameterized dark-energy equation of state, our cosmographic framework reconstructs the expansion history directly from observations, with the effective dark-energy equation of state emerging as a derived quantity. We find that all three closures are tightly constrained to the vicinity of the CDM cosmographic fixed point, with Planck data driving the preferred evolution toward j 0 1 and w DE,0-1. Despite their distinct kinematical constructions, the reconstructed dark-energy evolution consistently exhibits smooth freezing behaviour close to w=-1, without crossing the phantom divide. Model comparison using the Akaike and Bayesian information criteria shows that the almost- CDM models remain statistically competitive with standard dark-energy parameterizations while requiring fewer assumptions about the functional form of w(z). These results demonstrate the power of model-independent cosmography for constraining the cosmic expansion history and provide a natural framework for future studies of cosmological perturbations and structure formation.