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

Abstract

We present a unified \emph{background-level} framework that maps a prescribed late-time dark-energy density history $ρ_{\rm de}(z)$ onto an effective scalar-field description in a spatially flat FLRW universe. Working directly with $ρ_{\rm de}(z)$, we reconstruct the associated field trajectory $ϕ(z)$, and field-space potential $V(ϕ)$, together with a null energy condition (NEC) consistency check. We apply the method to three benchmark histories: (i) the Chevallier--Polarski--Linder (CPL) form; (ii) a smooth mirror AdS$\rightarrow$dS sign-switching profile in which $ρ_{\rm de}$ crosses zero at $z_\dagger$, interpolating between a positive late-time plateau and a negative high-$z$ plateau ($Λ_{\rm s}$CDM-like at the background level); and (iii) a shifted-$\tanh$ emergent profile that remains positive definite and approaches $ρ_{\rm de}\to 0^{+}$ at high redshift. Finally, treating the reconstructed potential, $V_{\rm tar}(ϕ)$, as a target, we perform Bayesian model comparison directly in \emph{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.