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 $\delta$ in sub-horizon 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 $\omega_{de}$. We compare the correct analytic growing mode solution $\delta$ with the text book solution $\delta^{D}$. Indeed, both solutions are equal to each other when $\omega_{de} = -1$. We also able to extend this solution for the specific form of time varying $\omega_{de}$.