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.