Approximate ωϕ ∼ Ωϕ Relations in Quintessence Models

Abstract

Quintessence field is a widely-studied candidate of dark energy. There is “tracker solution” in quintessence models, in which evolution of the field ϕ at present times is not sensitive to its initial conditions. When the energy density of dark energy is neglectable (Ωϕ ≪ 1), evolution of the tracker solution can be well analysed from “tracker equation”. In this paper, we try to study evolution of the quintessence field from “full tracker equation”, which is valid for all spans of Ωϕ. We get stable fixed points of ωϕ and Ωϕ (noted as ϕ and ϕ) from the “full tracker equation”, i.e., ωϕ and Ωϕ will always approach ϕ and ϕ respectively. Since ϕ and ϕ are analytic functions of ϕ, analytic relation of ϕ ∼ ϕ can be obtained, which is a good approximation for the Ωϕ ∼ Ωϕ relation and can be obtained for the most type of quintessence potentials. By using this approximation, we find that inequalities ϕ < ωϕ and ϕ < Ωϕ are statisfied if the ωϕ (or ϕ) decreases with time. In this way, the potential U(ϕ) can be constrained directly from observations, by no need of solving the equations of motion numerically.