Exact analytic solution of sub - horizon scales linear perturbation for general dark energy models

Three decades ago Heath found the integral form of the exact analytic growing mode solution of linear density perturbation δ in subhorizon scales including the cosmological constant or the curvature term. Interestingly, we are able to obtain the analytic solution for general dark energy models with the constant equation of state ωde. We compare the correct analytic growing mode solution δ with the text book solution δ. Indeed, both solutions are equal to each other when ωde = −1. We also able to extend this solution for the specific form of time varying ωde. The background evolution equations in a flat Friedmann-RobertsonWalker universe (ρm + ρde = ρcr) are H ≡ ( ȧ a )2 = 8πG 3 (ρm + ρde) = 8πG 3 ρcr , (1) 2 ä a + ( ȧ a )2 = −8πGωdeρde , (2) where ωde is the equation of state (eos) of dark energy, ρm and ρde are the energy densities of the matter and the dark energy, respectively. We consider the constant ωde. The sub-horizon scale linear perturbation equation with respect to the scale factor a are given in the reference [1], dδ da + (