Sub-horizon evolution of cold dark matter perturbations through dark matter-dark energy equivalence epoch

We consider a cosmological model of the late universe constituted by standard cold dark matter plus a dark energy component with constant equation of state w and constant effective speed of sound. Neglecting fluctuations in the dark energy component we obtain an equation describing the evolution of sub-horizon cold dark matter perturbations through the epoch of dark matter-dark energy equality. We explore its analytic solutions and calculate an exact w-dependent correction for the dark matter growth function, logarithmic growth function and growth index parameter through the epoch considered. We test our analytic approximation with the numerical solution and find that the discrepancy is less than 1% for k =0 in the epoch of interest.