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, phi, with a linear effective potential V (phi). We determine the fraction, f, of the total lifetime of the Universe, tu, 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) equivalent to 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) greater than or similar to 2 x 10(-2) (1 X 10(-3)), where omega phi equivalent to p(phi)/rho(phi). These results depend mainly on the linearity of the scalar field potential for -V (phi(0)) less than or similar to V (phi) less than or similar to 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 similar to 1.6(omega(phi) + 1), where 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) less than or similar to -0.6 and t(U) greater than or similar to 2.4t(0) at the 2 sigma level. (c) 2005 Elsevier B.V. All rights reserved.