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, $\varphi$, with a linear effective potential $V(\varphi )$. We determine the fraction, $f$, of the total lifetime of the universe, $t_U$, which lies within the interval $[t_0-\Delta t_A,t_0+\Delta t_A]$, where $t_0$ is the age of the universe at the present time, $\Delta t_A\equiv t_0-t_A$ 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 ${\omega }_{\varphi }\equiv p_{\varphi }/{\rho }_{\varphi }$. 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 $V(\varphi )$ outside this range. We also show that if ${\omega }_{\varphi 0}$ is close to -1 then ${\omega }_{\varphi 0}+1\sim 1.6({\tilde{\omega }}_{\varphi }+1)$, where ${\tilde{\omega }}_{\varphi }$ is the weighted average value of ${\omega }_{\varphi }$ 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 $2\sigma$ level.

7 pages, 3 figures, typos corrected, references added