Late time acceleration in Palatini gravity

We investigate the effect of the quadratic correction $\alpha R^2$ and non-minimal coupling $\xi {\phi }^{2}R$ on a quintessence model with an exponential potential $V(\phi )=M^4\exp (-\lambda \phi )$ in the Palatini formulation of gravity. We use dynamical system techniques to analyze the attractor structure of the model and uncover the possible trajectories of the system. We find that the quadratic correction cannot play a role in the late time dynamics, except for unacceptably large values of the parameter $\alpha$; although it can play a role at early times. We find viable evolutions, from a matter-dominated phase to an accelerated expansion phase, with the dynamics driven by the non-minimal coupling. These evolutions correspond to trajectories where the field ends up frozen, thus acting as a cosmological constant.