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 ${\rho }_{\mathrm{de}}(z)$ onto an effective scalar-field description in a spatially flat FLRW universe. Working directly with ${\rho }_{\mathrm{de}}(z)$, we reconstruct the associated field trajectory $\varphi (z)$, and field-space potential $V(\varphi )$, 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$\to$dS sign-switching profile in which ${\rho }_{\mathrm{de}}$ crosses zero at $z_{†}$, interpolating between a positive late-time plateau and a negative high-$z$ plateau (${\Lambda }_{\mathrm{s}}$CDM-like at the background level); and (iii) a shifted-$\tanh$ emergent profile that remains positive definite and approaches ${\rho }_{\mathrm{de}}\to 0^{+}$ at high redshift. Finally, treating the reconstructed potential, $V_{\mathrm{tar}}(\varphi )$, 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.

31 pages, 11 figures, 2 tables; matches the published version on PDU

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