  A microlensing survey by Sumi et al. (2011) exhibits an overabundance of
short-timescale events (STEs; t_E<2 days) relative to what is expected from
known stellar populations and a smooth power-law extrapolation down to the
brown dwarf regime. This excess has been interpreted as a population of
~Jupiter-mass objects that outnumber main-sequence stars by nearly twofold;
however the microlensing data alone cannot distinguish between events due to
wide-separation (a>~10 AU) and free-floating planets. Assuming these STEs are
indeed due to planetary-mass objects, we aim to constrain the fraction of these
events that can be explained by bound but wide-separation planets. We fit the
observed timescale distribution with a lens mass function comprised of brown
dwarfs, main-sequence stars, and stellar remnants, finding and thus
corroborating the initial identification of an excess of STEs. We then include
a population of bound planets that are expected not to show signatures of the
primary lens (host) in their microlensing light curves and that are also
consistent with results from representative microlensing, radial velocity, and
direct imaging surveys. We find that bound planets alone cannot explain the
entire STE excess without violating the constraints from the surveys we
consider and thus some fraction of these events must be due to free-floating
planets, if our model for bound planets holds. We estimate a median fraction of
STEs due to free-floating planets to be f=0.67 (0.23-0.85 at 95% confidence)
when assuming "hot-start" planet evolutionary models and f=0.58 (0.14-0.83 at
95% confidence) for "cold-start" models. Assuming a delta-function distribution
of free-floating planets of mass m_p=2 M_Jup yields a number of free-floating
planets per main-sequence star of N=1.4 (0.48-1.8 at 95% confidence) in the
"hot-start" case and N=1.2 (0.29-1.8 at 95% confidence) in the "cold-start"
case.
