Direct cosmographic reconstruction of the quintessence potential

Abstract: We derive expressions for the first and second derivatives of the quintessence potential $V(φ)$, in terms of $λ= -V^{\prime}/V$ and $Γ= (V^{\prime \prime}/V)/(V^\prime/V)^2$, as functions of the quintessence density fraction $Ω_φ$ and the cosmographic parameters $q$, $j$, and $s$. Our mapping is not explicitly a function of the equation of state parameter $w$. We use these results, along with recent observational data, to derive expansions of $V(φ)$ about the present-day value of the scalar field, $φ_0$.