Are the Newly-Discovered $z \sim 13$ Drop-out Sources Starburst Galaxies
  or Quasars?
  The detection of two $z\sim 13$ galaxy candidates has opened a new window on
galaxy formation at an era only $330$ Myr after the Big Bang. Here, we
investigate the physical nature of these sources: are we witnessing star
forming galaxies or quasars at such early epochs? If powered by star formation,
the observed ultraviolet (UV) luminosities and number densities can be jointly
explained if: (i) these galaxies are extreme star-formers with star formation
rates $5-24\times$ higher than those expected from extrapolations of average
lower-redshift relations; (ii) the star formation efficiency increases with
halo mass and is countered by increasing dust attenuation from $z \sim 10-5$;
(iii) they form stars with an extremely top-heavy initial mass function. The
quasar hypothesis is also plausible, with the UV luminosity produced by black
holes of $\sim 10^8 \, \mathrm{M_{\odot}}$ accreting at or slightly above the
Eddington rate ($f_{\rm Edd}\sim 1.0$). This black hole mass at $z\sim13$ would
require very challenging, but not implausible, growth parameters. If
spectroscopically confirmed, these two sources will represent a remarkable
laboratory to study the Universe at previously inaccessible redshifts.
