Extremely Small Sizes for Faint z~2-8 Galaxies in the Hubble Frontier
  Fields: A Key Input For Establishing their Volume Density and UV Emissivity
  We provide the first observational constraints on the sizes of the faintest
galaxies lensed by the Hubble Frontier Fields (HFF) clusters. Ionizing
radiation from faint galaxies likely drives cosmic reionization, and the HFF
initiative provides a key opportunity to find such galaxies. Yet, we cannot
really assess their ionizing emissivity without a robust measurement of their
sizes, since this is key to quantifying both their prevalence and the faint-end
slope to the UV luminosity function. Here we provide the first such size
constraints with 2 new techniques. The first utilizes the fact that the
detectability of highly-magnified galaxies as a function of shear is very
dependent on a galaxy's size. Only the most compact galaxies will remain
detectable in regions of high shear (vs. a larger detectable size range for low
shear), a phenomenon we carefully quantify using simulations. Remarkably,
however, no correlation is found between the surface density of faint galaxies
and the predicted shear, using 87 faint high-magnification mu>10 z~2-8 galaxies
seen behind the first 4 HFF clusters. This can only be the case if such faint
(~-15 mag) galaxies have significantly smaller sizes than luminous galaxies. We
constrain their half-light radii to be <~30 mas (<160-240 pc). As a 2nd size
probe, we rotate and stack 26 faint high-magnification sources along the major
shear axis. Less elongation is found than even for objects with an intrinsic
half-light radius of 10 mas. Together these results indicate that extremely
faint z~2-8 galaxies have near point-source profiles in the HFF dataset
(half-light radii conservatively <30 mas and likely 5-10 mas). These results
suggest smaller completeness corrections and hence much lower volume densities
for faint z~2-8 galaxies and shallower faint-end slopes than have been derived
in many recent studies (by factors of ~2-3 and by dalpha>~0.1-0.3).
