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 _ { \phi 0 } \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 % pt \hbox { $ > $ } } 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 } ) \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt% \hbox { $ < $ } } V ( \phi ) \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 % pt \hbox { $ < $ } } V ( \phi _ { 0 } ) and are weakly dependent on the specific form of V ( \phi ) outside this range .
We also show that if \omega _ { \phi 0 } is close to -1 then \omega _ { \phi 0 } +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 _ { \phi 0 } \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt% \hbox { $ < $ } } -0.6 and t _ { U } \mathrel { \hbox to 0.0 pt { \lower 4.0 pt \hbox { $ \sim$ } } \raise 1.0 pt \hbox { $ > $ } } % 2.4 t _ { 0 } at the 2 \sigma level .