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 in the last 11 Gyrs 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 ^ { * } \sim 10 ^ { 11.2 \pm 0.2 } M _ { \odot } and just a slight increase of \sim 2.3 \times 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 \sim 10 ^ { 7.65 \pm 0.08 } 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 = 0.40 , driven by the rise of the passive population .
We find that SFGs with \sim 10 ^ { 10.0 \pm 0.2 } M _ { \odot } contribute most to the SFR density ( \rho _ { SFR } ) per d \log _ { 10 } M , and that there is no significant evolution in the fractional contribution from SFGs of different masses to \rho _ { SFR } or \rho _ { SFR } ( d \log _ { 10 } M ) ^ { -1 } since z = 2.23 .
Instead , we show that the decline of SFR ^ { * } and of \rho _ { 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 \sim 66 M _ { \odot } yr ^ { -1 } to 5 M _ { \odot } yr ^ { -1 } since z \sim 2.23 .