We use the ACS BViz data from the HUDF and all other deep HST ACS fields ( including the GOODS fields ) to find large samples of star-forming galaxies at z \sim 4 and z \sim 5 and to extend our previous z \sim 6 sample .
These samples contain 4671 , 1416 , and 627 B , V , and i dropouts , respectively , and reach to extremely low luminosities ( 0.01 - 0.04 L _ { z = 3 } ^ { * } or M _ { UV } \sim - 16 to -17 ) , allowing us to determine the rest-frame UV luminosity function ( LF ) and faint-end slope \alpha at z \sim 4 - 6 to high accuracy .
We find faint-end slopes \alpha of -1.73 \pm 0.05 at z \sim 4 , -1.66 \pm 0.09 at z \sim 5 , and -1.74 \pm 0.16 at z \sim 6 – suggesting that the faint-end slope is very steep and shows little evolution with cosmic time .
We find that M _ { UV } ^ { * } brightens considerably in the 0.7 Gyr from z \sim 6 to z \sim 4 ( by \sim 0.7 mag from M _ { UV } ^ { * } = -20.24 \pm 0.19 to M _ { UV } ^ { * } = -20.98 \pm 0.10 ) .
The observed increase in the characteristic luminosity over this range is almost identical to that expected for the halo mass function – suggesting that the observed evolution is likely due to the hierarchical coalescence and merging of galaxies .
The evolution in \phi ^ { * } is not significant .
The UV luminosity density at z \sim 6 is modestly lower ( 0.45 \pm 0.09 \times ) than that at z \sim 4 ( integrated to -17.5 AB mag ) though a larger change is seen in the dust-corrected star-formation rate density .
We thoroughly examine published LF results and assess the reasons for their wide dispersion .
We argue that the results reported here are the most robust available .
The extremely steep faint-end slopes \alpha found here suggest that lower luminosity galaxies play a significant role in reionizing the universe .
Finally , recent search results for galaxies at z \sim 7 - 8 are used to extend our estimates of the evolution of M ^ { * } from z \sim 7 - 8 to z \sim 4 .