The coincidence problem in linear dark energy models

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, $\phi$, with a linear effective potential $V(\phi)$. 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 $1+\omega_{\phi0} \gapp 2 \times 10^{-2}$ ($1 \times 10^{-3}$), where $\omega_{\phi} \equiv p_\phi/\rho_\phi$. These results depend mainly on the linearity of the scalar field potential for $-V(\phi_0) \lapp V(\phi) \lapp V(\phi_0)$ and are weakly dependent on the specific form of $V(\phi)$ outside this range. We also show that if $\omega_{\phi0}$ is close to -1 then $\omega_{\phi0}+1 \sim 1.6 ({\tilde \omega}_\phi +1)$, where ${\tilde \omega}_\phi$ is the weighted average value of $\omega_{\phi}$ in the time interval $[0,t_0]$. We independently confirm current observational constraints on this class of models which give $\omega_{\phi0} \lapp -0.6$ and $t_U \gapp 2.4 t_0$ at the $2 \sigma$ level.