The coincidence problem in linear dark energy models

We show that a solution to 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, tU, which lies within the interval t0−ΔtA,t0+ΔtA, where t0 is the age of the Universe at the present time, ΔtA≡t0−tA and tA 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+ωϕ0≳2×10−2 (1×10−3), where ωϕ≡pϕ/ρϕ. These results depend mainly on the linearity of the scalar field potential for −V(ϕ0)≲V(ϕ)≲V(ϕ0) and are weakly dependent on the specific form of V(ϕ) outside this range. We also show that if ωϕ0 is close to −1 then ωϕ0+1∼1.6(ω˜ϕ+1), where ω˜ϕ is the weighted average value of ωϕ in the time interval 0,t0. We independently confirm current observational constraints on this class of models which give ωϕ0≲−0.6 and tU≳2.4t0 at the 2σ level.