Two main modes of star formation are know to control the growth of galaxies : a relatively steady one in disk-like galaxies , defining a tight star formation rate ( SFR ) -stellar mass sequence , and a starburst mode in outliers to such a sequence which is generally interpreted as driven by merging . Such starburst galaxies are rare but have much higher SFRs , and it is of interest to establish the relative importance of these two modes . PACS/Herschel observations over the whole COSMOS and GOODS-South fields , in conjunction with previous optical/near-IR data , have allowed us to accurately quantify for the first time the relative contribution of the two modes to the global SFR density in the redshift interval 1.5 < z < 2.5 , i.e. , at the cosmic peak of the star formation activity . The logarithmic distributions of galaxy SFRs at fixed stellar mass are well described by Gaussians , with starburst galaxies representing only a relatively minor deviation that becomes apparent for SFRs more than 4 times higher than on the main sequence . Such starburst galaxies represent only 2 % of mass-selected star forming galaxies and account for only 10 % of the cosmic SFR density at z \sim 2 . Only when limited to SFR > 1000 M _ { \odot } / yr , off-sequence sources significantly contribute to the SFR density ( 46 \pm 20 % ) . We conclude that merger-driven starbursts play a relatively minor role for the formation of stars in galaxies , whereas they may represent a critical phase towards the quenching of star formation and morphological transformation in galaxies .