The stellar mass function of star-forming galaxies and the
  mass-dependent SFR function since z=2.23 from HiZELS
  We explore a large uniformly selected sample of H${\alpha}$ selected
star-forming galaxies (SFGs) at z=0.40,0.84,1.47,2.23 to unveil the evolution
of the star formation rate (SFR) function and the stellar mass function. We
find strong evolution in the SFR function, with the typical SFR of SFGs
declining exponentially with time in the last 11Gyrs as
SFR$^*$(T[Gyr])=10$^{4.23/T+0.37}$ M$_{\odot}$yr$^{-1}$, but with no evolution
in the faint-end slope, ${\alpha}\approx$-1.6. The stellar mass function of
SFGs, however, reveals little evolution: ${\alpha} \approx$-1.4,
M$^*$$\approx$10$^{11.2\pm0.2}$ M$_{\odot}$ and just a slight increase of
$\approx$2.3x in ${\Phi}^*$ from z=2.23 to z=0.4. The stellar mass density
within SFGs has been roughly constant since z=2.23 at
$\approx$10$^{7.65\pm0.08}$ M$_{\odot}$Mpc$^{-3}$, comprising $\approx$100% of
the stellar mass density in all galaxies at z=2.23, and declining to
$\approx$20% by z=0.4, driven by the rise of the passive population. We find
that SFGs with M$\approx$10$^{10.0\pm0.2}$ M$_{\odot}$ contribute most to the
SFR density (${\rho}_{\rm SFR}$) per dlog$_{10}$M, and that there is no
significant evolution in the fractional contribution from SFGs of different
masses to ${\rho}_{\rm SFR}$ or ${\rho}_{\rm SFR}$(dlog$_{10}$M)$^{-1}$ since
z=2.23. Instead, we show that the decline of SFR$^*$ and of ${\rho}_{\rm SFR}$
are primarily driven by an exponential decline in SFRs at all masses. Our
results have important implications not only on how SFGs need to be quenched
across cosmic time, but also on the driver(s) of the exponential decline in
SFR$^*$ from $\approx$66 M$_{\odot}$yr$^{-1}$ to $\approx$5
M$_{\odot}$yr$^{-1}$ since z=2.23.
