Paper detail

Approximate $w_ϕ\simΩ_ϕ$ Relations in Quintessence Models

Quintessence field is a widely-studied candidate of dark energy. There is &#34;tracker solution&#34; 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 ($Ω_ϕ\ll1$), evolution of the tracker solution can be well analysed from &#34;tracker equation&#34;. In this paper, we try to study evolution of the quintessence field from &#34;full tracker equation&#34;, which is valid for all spans of $Ω_ϕ$. We get stable fixed points of $w_ϕ$ and $Ω_ϕ$ (noted as $\hat w_ϕ$ and $\hatΩ_ϕ$) from the &#34;full tracker equation&#34;, i.e., $w_ϕ$ and $Ω_ϕ$ will always approach $\hat w_ϕ$ and $\hatΩ_ϕ$ respectively. Since $\hat w_ϕ$ and $\hatΩ_ϕ$ are analytic functions of $ϕ$, analytic relation of $\hat w_ϕ\sim\hatΩ_ϕ$ can be obtained, which is a good approximation for the $w_ϕ\simΩ_ϕ$ relation and can be obtained for the most type of quintessence potentials. By using this approximation, we find that inequalities $\hat w_ϕ<w_ϕ$ and $\hatΩ_ϕ<Ω_ϕ$ are statisfied if the $w_ϕ$ (or $\hat w_ϕ$) is decreasing 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.