Cosmological constraints on curved quintessence

Dynamical dark energy has gained renewed interest due to recent theoretical and observational developments. In the present paper, we focus on a string-motivated dark energy set-up, and perform a detailed cosmological analysis of exponential quintessence with potential $V=V_0 e^{-\lambda\phi}$, allowing for non-zero spatial curvature. We first gain some physical intuition into the full evolution of such a scenario by analysing the corresponding dynamical system. Then, we test the model using a combination of Planck CMB data, DESI BAO data, as well as recent supernovae datasets. For the model parameter $\lambda$, we obtain a preference for nonzero values: $\lambda = 0.48^{+0.28}_{-0.21},\; 0.68^{+0.31}_{-0.20},\; 0.77^{+0.18}_{-0.15}$ at 68% C.L. when combining CMB+DESI with Pantheon+, Union3 and DES-Y5 supernovae datasets respectively. We find no significant hint for spatial curvature. We discuss the implications of current cosmological results for the exponential quintessence model, and more generally for dark energy in string theory.