Besan\c{c}on Galactic model analysis of MOA-II microlensing: evidence
  for a mass deficit in the inner bulge
  Galactic bulge microlensing surveys provide a probe of Galactic structure. We
present the first field-by-field comparison between microlensing observations
and the Besan\c{c}on population synthesis Galactic model. Using an updated
version of the model we provide maps of optical depth, average event duration
and event rate for resolved source populations and for difference imaging (DIA)
events. We also compare the predicted event timescale distribution to that
observed. The simulation follows the selection criteria of the MOA-II survey
(Sumi et al. 2013). We modify the Besan\c{c}on model to include M dwarfs and
brown dwarfs. Our best fit model requires a brown dwarf mass function slope of
$-0.4$. The model provides good agreement with the observed average duration,
and respectable consistency with the shape of the timescale distribution
(reduced $\chi^2 \simeq 2.2$). The DIA and resolved source limiting yields
bracket the observed number of events by MOA-II ($2.17\times$ and $0.83\times$
the number observed, respectively). We perform a 2-dimensional fit to the event
spatial distribution to predict the optical depth and event rate across the
Galactic bulge. The most serious difficulty for the model is that it provides
only $\sim 50\%$ of the measured optical depth and event rate per star at low
Galactic latitude around the inner bulge ($|b|<3{^\circ}$). This discrepancy
most likely is associated with known under-estimated extinction and star counts
in the innermost regions and therefore provides additional support for a
missing inner stellar population.
