The quadratic approximation for quintessence with arbitrary initial conditions

Abstract

We examine models of quintessence in which a minimally-coupled scalar field phi evolves near a local extremum of its potential V ( phi) at phi_*. Assuming that (1/V )(dV/dphi) is small and w ~ -1, we Taylor expand the potential about phi_* and derive a general expression for w(a). The dynamics of this field are determined by the initial and final equation of state parameters w_i and w_0, the quantity V''( phi_*)/V(phi_*), and the direction of \dot\phi_i in relation to \dot\phi _0. This approximation is then tested for six values of V''( phi_*)/V(phi_*) and shown to lie within 2% of the exact solution for five of these cases. However, the model becomes less precise near certain values of V''( phi_*)/V(phi_*) where \dot\phi becomes very large.