Cosmographic Reconstruction of the Quintessence Potential in Scalar-Tensor Gravity

Abstract

We generalise the cosmographic reconstruction programme of Chakraborty, Dunsby & Scherrer to scalar-tensor gravity. Working in the Jordan frame with a general non-minimal coupling $F(\Phi )$, we derive exact closed-form expressions for the potential slope ${\lambda }_U$, the coupling slope ${\lambda }_F$, and their curvature parameters ${\Gamma }_U$ and ${\Gamma }_F$ in terms of the cosmographic parameters $(q,j,s)$, the Planck-mass running ${\alpha }_M=d\ln F/d\ln a$ and its derivatives. We express ${\alpha }_M$ and its derivatives in terms of a set of coefficients $g_n$ describing the time variation of the gravitational constant $G\propto F^{-1}$, casting all four quantities in fully observable form. We then evaluate the reconstruction against current data, including DESI~DR2 baryon acoustic oscillation measurements and Lunar Laser Ranging constraints on $\dot{G}/G$, propagating the cosmographic uncertainties through to the reconstructed potential. All expressions reduce analytically to those of Chakraborty, Dunsby & Scherrer in the minimally-coupled limit, which we verify symbolically. We find that the potential slope ${\lambda }_U$ is recovered at the few-tenths level while the curvature ${\Gamma }_U$ is essentially unconstrained. We further show that the positivity of the scalar kinetic term is equivalent to $w_{\mathrm{eff},0}\ge -1$, so that the region of cosmographic parameter space closed to minimally-coupled quintessence is exactly the phantom region, and that the non-minimal coupling reopens it through a single coefficient $g_2$ of the gravitational tower. Because Lunar Laser Ranging forces $|g_1|\lesssim 2\times 10^{-4}$ while the reconstruction is sensitive to $g_2\sim O(0.1)$, the framework presupposes that ${\alpha }_M$ is passing through zero at the present epoch, as expected on the Damour--Nordtvedt least-coupling attractor.

25 pages, 5 figures. The numerical analysis files used for the reproduction of the figures can be found at: https://github.com/leandros11/st-reconstruction