Reconstruction of the Quintessence Scalar Field Potential Using Gaussian Processes

Recent cosmological observations, including the latest Dark Energy Spectroscopic Instrument (DESI) data releases DR1 and DR2, have renewed interest in the possibility that dark energy may exhibit dynamical behavior rather than being a strict cosmological constant. In this work, we perform a fully model-independent reconstruction of the quintessence scalar field potential using Gaussian Process regression and current Hubble measurements. Instead of assuming a specific functional form for the scalar field potential, we reconstruct the quintessence potential and the corresponding kinetic energy directly from observational data. Our analysis is based on Hubble parameter measurements obtained from cosmic chronometers and the latest high-precision DESI DR2 baryon acoustic oscillation (BAO) data, together with Type Ia supernova data from the Pantheon+ compilation. Gaussian Processes provide a nonparametric and model-independent framework that allows the data to guide the reconstruction. We employ two covariance functions, namely the squared exponential and the Matern ($\nu = 9/2$) kernels, in order to assess the sensitivity of the reconstruction to the kernel choice. We further explore the impact of background cosmological assumptions by considering different priors on the matter density and spatial curvature. Finally, we compare the reconstructed scalar field potential with two theoretically motivated benchmark models: a power law potential and an exponential potential. We find that both models remain consistent with the reconstructed potential within the inferred confidence intervals.