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.

Figures (11)

• Figure 1: Background evolution for the CPL dark-energy profile (effective-fluid benchmark). Top left: normalized dark-energy density $\tilde{\Omega}_{\rm de}(z)=\rho_{\rm de}(z)/\rho_{\rm c0}$. Top right: density parameters $\Omega_{\rm r}(z)$, $\Omega_{\rm m}(z)$, and $\Omega_{\rm de}(z)$. Bottom left: dark-energy pressure $p_{\rm de}(z)/\rho_{\rm c0}$ obtained from the continuity equation. Bottom right: effective equation-of-state parameter $w_{\rm de}(z)=p_{\rm de}(z)/\rho_{\rm de}(z)$ (dotted line marks $w_{\rm de}=-1$). For CPL, $\rho_{\rm de}>0$ by construction, so the $w_{\rm de}=-1$ line coincides with the NEC boundary $\rho_{\rm de}+p_{\rm de}=0$. In this benchmark, $\rho_{\rm de}$ remains positive while the DE history crosses the NEC boundary (i.e. the phantom-divide line) as $z$ decreases toward the present, transitioning from the $p$-phantom regime ($\rho_{\rm de}+p_{\rm de}<0$, $w_{\rm de}<-1$) to the $p$-quintessence regime ($\rho_{\rm de}+p_{\rm de}>0$, $w_{\rm de}>-1$).
• Figure 2: Effective scalar-sector reconstruction associated with the CPL benchmark (treated as an effective-fluid history). Top left: reconstructed (effective) potential $V(\phi)/\rho_{\rm c0}$ obtained by eliminating $z$ between $V(z)$ and $\phi(z)$; the loop/multivalued relation reflects the non-monotonic $\phi(z)$ induced by the NEC-boundary crossing. Top right: effective kinetic and potential contributions, $K(z)/\rho_{\rm c0}$ and $V(z)/\rho_{\rm c0}$, with $K(z)=\tfrac{1}{2}(\rho_{\rm de}+p_{\rm de})$; the sign change of $K(z)$ marks the NEC boundary $\rho_{\rm de}+p_{\rm de}=0$, which for CPL (where $\rho_{\rm de}>0$ by construction) is equivalent to $w_{\rm de}=-1$ (the usual phantom-divide line). Bottom left: reconstructed field trajectory $\phi(z)/M_{\rm Pl}$. Bottom right:$\mathrm{d}(\phi/M_{\rm Pl})/\mathrm{d} z$, showing a turning point ($\mathrm{d}\phi/\mathrm{d} z=0$) coincident with the sign change in $K(z)$. Because $K(z)$ changes sign from negative to positive as $z$ decreases toward the present (i.e. the evolution moves from the $p$-phantom to the $p$-quintessence regime in \ref{['tab:pq-branches']}), a single minimally coupled real scalar field with fixed $\epsilon=\pm1$ cannot realise the full CPL history; the displayed $\phi(z)$ and $V(\phi)$ should therefore be interpreted as an effective one-dimensional mapping of an extended scalar sector (e.g. a quintom-like or non-canonical completion).
• Figure 3: Background evolution for the sign-changing $\tanh$ dark-energy density (a smooth mirror AdS$\rightarrow$dS sign-switching transition). Top left: normalized density $\tilde{\Omega}_{\rm de}(z)=\rho_{\rm de}(z)/\rho_{\rm c0}$, which decreases monotonically and smoothly interpolates from a positive late-time plateau to a negative high-$z$ plateau, crossing $\rho_{\rm de}=0$ at the transition redshift $z_\dagger$. Top right: fractional densities $\Omega_{\rm r}(z)$, $\Omega_{\rm m}(z)$, and $\Omega_{\rm de}(z)$, showing standard radiation/matter domination at early times and the late-time emergence of the DE component; $\Omega_{\rm de}$ crosses zero and becomes negative beyond $z_\dagger$. Bottom left: pressure $p_{\rm de}(z)/\rho_{\rm c0}$ inferred from the continuity equation, which remains finite and exhibits a rapid excursion across the transition. Bottom right: derived equation-of-state parameter $w_{\rm de}(z)=p_{\rm de}/\rho_{\rm de}$; the divergence at $\rho_{\rm de}=0$ is purely kinematic (a ratio singularity). The physically meaningful classifier is instead the NEC combination $\rho_{\rm de}+p_{\rm de}=\tfrac{1}{3}(1+z)\,\mathrm{d}\rho_{\rm de}/\mathrm{d} z$: for $\eta>0$ one has $\mathrm{d}\tilde{\Omega}_{\rm de}/\mathrm{d} z<0$ over the range shown, implying $\rho_{\rm de}+p_{\rm de}<0$ throughout (phantom/NEC-violating). In the notation of \ref{['tab:pq-branches']}, the evolution lies on the $p$-phantom branch for $z<z_{\dagger}$ and on the $n$-phantom branch for $z>z_{\dagger}$.
• Figure 4: Scalar-field reconstruction for the sign-changing $\tanh$ (mirror AdS$\rightarrow$dS) dark-energy profile. Top left: reconstructed effective potential $V(\phi)/\rho_{\rm c0}$ obtained by eliminating $z$ between $V(z)$ and $\phi(z)$; the potential is smooth and single-valued and exhibits a broad maximum around the transition, interpolating between an AdS-like (negative) plateau at high redshift and a dS-like (positive) value at late times. Top right: effective kinetic and potential contributions $K(z)/\rho_{\rm c0}$ and $V(z)/\rho_{\rm c0}$, with $K(z)=\tfrac{1}{2}(\rho_{\rm de}+p_{\rm de})$; although $\rho_{\rm de}$ crosses zero, $K(z)$ remains negative throughout, indicating that the evolution stays on the phantom/NEC-violating branch and does not cross the NEC boundary. Bottom left: reconstructed field trajectory $\phi(z)/M_{\rm Pl}$, showing that the field is nearly frozen at both early and late times and evolves mainly around the transition redshift. Bottom right:$\mathrm{d}(\phi/M_{\rm Pl})/\mathrm{d} z$, sharply localized near the transition. Over the range shown $\mathrm{d}\tilde{\Omega}_{\rm de}/\mathrm{d} z<0$, so the single-field sign-consistency condition selects a phantom realization with $\epsilon=-1$ at the effective background level.
• Figure 5: Background evolution for the shifted-$\tanh$ (emergent, positive-definite) dark-energy profile. Top left: normalized density $\tilde{\Omega}_{\rm de}(z)=\rho_{\rm de}(z)/\rho_{\rm c0}$, which remains strictly positive and smoothly approaches $\tilde{\Omega}_{\rm de}\to 0$ at high redshift, realizing an emergent late-time DE component. Top right: density parameters $\Omega_{\rm r}(z)$, $\Omega_{\rm m}(z)$, and $\Omega_{\rm de}(z)$, showing standard radiation/matter domination at early times and late-time DE domination. Bottom left: dark-energy pressure $p_{\rm de}(z)/\rho_{\rm c0}$ obtained from the continuity equation. Bottom right: equation-of-state parameter $w_{\rm de}(z)=p_{\rm de}/\rho_{\rm de}$, which is finite for all $z$ (no kinematic pole since $\rho_{\rm de}>0$), and becomes strongly phantom at high redshift as $\rho_{\rm de}\to 0^{+}$. For $\eta>0$ one has $\mathrm{d}\tilde{\Omega}_{\rm de}/\mathrm{d} z<0$ over the plotted range, hence $\rho_{\rm de}+p_{\rm de}<0$ throughout; in \ref{['tab:pq-branches']} this corresponds to the $p$-phantom branch.