The coincidence problem in linear dark energy models

Abstract

We show that a solution to the the coincidence problem can be found in the context of a generic class of dark energy models with a scalar field, $ϕ$, with a linear effective potential $V(ϕ)$. We determine the fraction, $f$, of the total lifetime of the universe, $t_U$, which lies within the interval $[t_0-Δt_A,t_0+ Δt_A]$, where $t_0$ is the age of the universe at the present time, $Δt_A \equiv t_0-t_A$ and $t_A$ is the age of the universe when it starts to accelerate. We find that if we require $f$ to be larger than 0.1 (0.01) then $1+ω_{\phi0} \gapp 2 \times 10^{-2}$ ($1 \times 10^{-3}$), where $ω_ϕ \equiv p_ϕ/ρ_ϕ$. These results depend mainly on the linearity of the scalar field potential for $-V(ϕ_0) \lapp V(ϕ) \lapp V(ϕ_0)$ and are weakly dependent on the specific form of $V(ϕ)$ outside this range. We also show that if $ω_{\phi0}$ is close to -1 then $ω_{\phi0}+1 \sim 1.6 ({\tilde ω}_ϕ+1)$, where ${\tilde ω}_ϕ$ is the weighted average value of $ω_ϕ$ in the time interval $[0,t_0]$. We independently confirm current observational constraints on this class of models which give $ω_{\phi0} \lapp -0.6$ and $t_U \gapp 2.4 t_0$ at the $2 σ$ level.