Model Independent Accelerating Universe and the Cosmological Coinc idence Problem

We show that the evolution of the quintessence energy density ${\mathrm{\Omega }}_Q$ is model independent in an accelerating universe. The accelerating behaviour has lasted at most 0.5 e-folds of expansion assuming a present day value of ${\mathrm{\Omega }}_Q=2\mathrm{/}3$. The generic evolution differs from the exact solution by a percentage given by $r=2{\text{gm}}_Q$, with ${\text{gm}}_Q=({\rho }_Q+p_Q)\mathrm{/}{\rho }_Q$. For a small ${\text{gm}}_Q$ the evolution remains a good approximation for a long period. Nucleosynthesis bounds on ${\mathrm{\Omega }}_Q$ suggest that the model independent solution is valid for at least 12 e-folds of expansion which includes then the scale of radiation and matter equality. We can, therefore, establish model independent conditions on the cosmological parameters at some factor scale $a_i \ll a_o$. Finally, we discuss the relevance of this result to the cosmological coincidence problem.