Multifield dark energy: Interplay between curved field space and curved spacetime

Exponential quintessence models motivated by string compactifications naturally involve both a dilatonic scalar and its axionic partner evolving on a curved field space, while spatial curvature enlarges the cosmological phase space and may affect late-time dynamics. We perform a systematic analysis of the minimal two-field exponential system in a curved FLRW background including radiation and matter components, combining a complete dynamical systems classification with analytical approximations and numerical integration. In the scalar-dominated limit, non-geodesic trajectories can sustain accelerated expansion on steep potentials, and in curved universes a scaling-curvature fixed point can in principle soften the requirements for acceleration. However, we show that these mechanisms arise in distinct invariant manifolds and cannot be simultaneously realized in the presence of a background fluid: no non-geodesic scaling fixed point exists within an open region of parameter space. As a consequence, in the observationally viable thawing regime the axion does not track the background fluid and spatial curvature becomes dynamically subdominant, leading to an effectively single-field evolution. We further identify a degeneracy between curvature effects and scalar-field dynamics that limits their independent impact on late-time cosmology. Confronting the model with current cosmological background data (Planck 2018 distance priors, Pantheon+, BAO, and cosmic chronometers), we obtain an upper bound $\lambda \lesssim 0.75$ (95 percent CL) on the potential slope. Our results demonstrate that even in the minimal multifield setup with spatial curvature, sustained late-time acceleration requires a sufficiently flat potential, so that the tension between cosmic acceleration and quantum gravity expectations persists within this framework.