Background-level reconstruction of scalar-field potentials from dark-energy histories and comparison with analytic potential families

We present a unified background-level framework that maps a prescribed late-time dark-energy density history ρ _de(z) onto an effective scalar-field description in a spatially flat Friedmann–Lemaître–Robertson–Walker universe. Working directly with ρ _de(z) and d ρ _de/d z , we reconstruct the associated pressure p _de(z), kinetic contribution K (z), field trajectory ϕ (z), and field-space potential V (ϕ), without relying on the ratio w_de=p_de/ρ_de in regimes where it is ill-defined (e.g. at a density zero crossing). The physically relevant classifier is instead the null-energy-condition (NEC) combination ρ_de+p_de=13(1+z)dρ_de/dz, whose sign controls the kinetic sector and provides a direct single-field consistency check. We apply the method to three benchmark histories: (i) the Chevallier–Polarski–Linder (CPL) form; (ii) a smooth mirror AdS → dS sign-switching profile in which ρ _de crosses zero at z _†, interpolating between a positive late-time plateau and a negative high- z plateau (Λ_sCDM-like at the background level); and (iii) a shifted-tanh emergent profile that remains positive definite and approaches ρ_de→0⁺ at high redshift. For CPL, ρ _de ' 0 by construction but the evolution crosses the NEC boundary ρ_de+p_de=0 (equivalently w_de=−1), so K (z) changes sign and the reconstructed mapping becomes multivalued in field space; consequently, the full CPL history cannot be realized globally by a single minimally coupled real scalar field with fixed kinetic signature and is naturally interpreted as an effective one-dimensional description of an extended sector. By contrast, the tanh -based transition histories satisfy the sign-consistency condition over the reconstruction range, selecting a single-field realization on the phantom branch (ϵ=−1) at the background level. Finally, treating the reconstructed potential, V _tar(ϕ), as a target, we perform Bayesian model comparison directly in potential space and rank representative analytic potential families by their Bayesian evidence. For CPL (restricting to the single-valued phantom branch for the potential-space comparison), the exponential potential has the highest evidence in the baseline analysis, while the shifted-tanh and hilltop quartic forms remain close competitors; for the sign-switching tanh target, the shifted-tanh potential is strongly preferred, and the emergent profile yields the same qualitative ranking. These results provide a practical dictionary between phenomenological expansion histories and the scalar-field potential shapes required to reproduce them at the background level.