  We explore the evolution of the specific star formation rate (SSFR) for
3.6um-selected galaxies of different M_* in the COSMOS field. The average SFR
for sub-sets of these galaxies is estimated with stacked 1.4GHz radio continuum
emission. We separately consider the total sample and a subset of galaxies (SF)
that shows evidence for substantive recent star formation in the rest-frame
optical SED. At 0.2<z<3 both populations show a strong and M_*-independent
decrease in their SSFR towards z=0.2, best described by a power- law (1+z)^n,
where n~4.3 for all galaxies and n~3.5 for SF sources. The decrease appears to
have started at z>2, at least above 4x10^10M_Sun where our conclusions are most
robust. We find a tight correlation with power-law dependence, SSFR (M_*)^beta,
between SSFR and M_* at all z. It tends to flatten below ~10^10M_Sun if
quiescent galaxies are included; if they are excluded a shallow index beta_SFG
-0.4 fits the correlation. On average, higher M_* objects always have lower
SSFRs, also among SF galaxies. At z>1.5 there is tentative evidence for an
upper SSFR-limit that an average galaxy cannot exceed. It is suggested by a
flattening of the SSFR-M_* relation (also for SF sources), but affects massive
(>10^10M_Sun) galaxies only at the highest z. Below z=1.5 there thus is no
direct evidence that galaxies of higher M_* experience a more rapid waning of
their SSFR than lower M_* SF systems. In this sense, the data rule out any
strong 'downsizing'. We combine our results with recent measurements of the
galaxy (stellar) mass function in order to determine the characteristic mass of
a SF galaxy (M_*=10^(10.6\pm0.4)M_Sun). In this sense, too, there is no
'downsizing'. Our analysis constitutes the most extensive SFR density
determination with a single technique to z=3. Recent Herschel results are
consistent with our results, but rely on far smaller samples.
