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The Quadratic Approximation for Quintessence with Arbitrary Initial Conditions

We examine quintessence models for dark energy in which the scalar field, $ϕ$, evolves near the vicinity of a local maximum or minimum in the potential $V(ϕ)$, so that $V(ϕ)$ be approximated by a quadratic function of $ϕ$ with no linear term. We generalize previous studies of this type by allowing the initial value of $d ϕ/dt$ to be nonzero. We derive an analytic approximation for $w(a)$ and show that it is in excellent agreement with numerical simulations for a variety of scalar field potentials having local minima or maxima. We derive an upper bound on the present-day value of $w$ as a function of the other model parameters and present representative limits on these models from observational data. This work represents a final generalization of previous studies using linear or quadratic approximations for $V(ϕ)$.