The HUDF09 data are the deepest near-IR observations ever , reaching to 29.5 mag .
Luminosity functions ( LF ) from these new HUDF09 data for 132 z \sim 7 and z \sim 8 galaxies are combined with new LFs for z \sim 5 -6 galaxies and the earlier z \sim 4 LF to reach to very faint limits ( < 0.05 L _ { z = 3 } ^ { * } ) .
The faint-end slopes \alpha are steep : -1.79 \pm 0.12 ( z \sim 5 ) , -1.73 \pm 0.20 ( z \sim 6 ) , -2.01 \pm 0.21 ( z \sim 7 ) , and -1.91 \pm 0.32 ( z \sim 8 ) .
Slopes \alpha \lesssim - 2 lead to formally divergent UV fluxes , though galaxies are not expected to form below \sim - 10 AB mag .
These results have important implications for reionization .
The weighted mean slope at z \sim 6 -8 is - 1.87 \pm 0.13 .
For such steep slopes , and a faint-end limit of -10 AB mag , galaxies provide a very large UV ionizing photon flux .
While current results show that galaxies can reionize the universe by z \sim 6 , matching the Thomson optical depths is more challenging .
Extrapolating the current LF evolution to z > 8 , taking \alpha to be - 1.87 \pm 0.13 ( the mean value at z \sim 6 -8 ) , and adopting typical parameters , we derive Thomson optical depths of 0.061 _ { -0.006 } ^ { +0.009 } .
However , this result will change if the faint-end slope \alpha is not constant with redshift .
We test this hypothesis and find a weak , though uncertain , trend to steeper slopes at earlier times ( d \alpha / dz \sim - 0.05 \pm 0.04 ) , that would increase the Thomson optical depths to 0.079 _ { -0.017 } ^ { +0.063 } , consistent with recent WMAP estimates ( \tau = 0.088 \pm 0.015 ) .
It may thus not be necessary to resort to extreme assumptions about the escape fraction or clumping factor .
Nevertheless , the uncertainties remain large .
Deeper WFC3/IR+ACS observations can further constrain the ionizing flux from galaxies .