Hilltop quintessence

We examine hilltop quintessence models, in which the scalar field is rolling near a local maximum in the potential, and w≈-1. We first derive a general equation for the evolution of ϕ in the limit where w≈-1. We solve this equation for the case of hilltop quintessence to derive w as a function of the scale factor; these solutions depend on the curvature of the potential near its maximum. Our general result is in excellent agreement (δw≲0.5%) with all of the particular cases examined. It works particularly well (δw≲0.1%) for the pseudo-Nambu-Goldstone boson potential. Our expression for w(a) reduces to the previously-derived slow-roll result of Sen and Scherrer in the limit where the curvature goes to zero. Except for this limiting case, w(a) is poorly fit by linear evolution in a.