This paper aims at explaining the two phases in the observed specific star formation rate (sSFR), namely the high (>3/Gyr) values at z>2 and the smooth decrease since z=2. In order to do this, we compare to observations the specific star formation rate evolution predicted by well calibrated models of chemical evolution for elliptical and spiral galaxies, using the additional constraints on the mean stellar ages of these galaxies (at a given mass). We can conclude that the two phases of the sSFR evolution across cosmic time are due to different populations of galaxies. At z>2 the contribution comes from spheroids: the progenitors of present-day massive ellipticals (which feature the highest sSFR) as well as halos and bulges in spirals (which contribute with average and lower-than-average sSFR). In each single galaxy the sSFR decreases rapidly and the star formation stops in