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Possible Solution of the long-standing discrepancy in the Microlensing Optical Depth Toward the Galactic Bulge by correcting the stellar number count

We find that significant incompleteness in stellar number counts results in a significant overestimate of the microlensing optical depth $τ$ and event rate per star per year $Γ$ toward the Galactic bulge from the first two years of the MOA-II survey. We find that the completeness in Red Clump Giant (RCG) counts $f_{\rm RC}$ decreases proportional to the galactic latitude $b$, as $f_{\rm RC}=(0.63\pm0.11)-(0.052\pm0.028)\times b$, ranging between 1 and 0.7 at $b=-6^\circ\sim-1.5^\circ$. The previous measurements using all sources by Difference Image Analysis (DIA) by MACHO and MOA-I suffer the same bias. On the other hand, the measurements using a RCG sample by OGLE-II, MACHO and EROS were free from this bias because they selected only the events associated with the resolved stars. Thus, the incompleteness both in the number of events and stellar number count cancel out. We estimate $τ$ and $Γ$ by correcting this incompleteness. In the central fields with $|l|<5^\circ$, we find $Γ=[18.74\pm0.91]\times10^{-6}\exp[(0.53\pm0.05)(3-|b|)]$ star$^{-1}$ yr$^{-1}$ and $τ_{200}=[1.84\pm0.14]\times10^{-6}\exp[(0.44\pm0.07)(3-|b|)]$ for the 427 events with $t_{\rm E}\leq200\,$days using all sources brighter than $I_s\leq20$ mag. Our revised all-source $τ$ measurements are about 2-$σ$ smaller than the other all-source measurements and are consistent with the RCG measurements within 1-$σ$. We conclude that the long-standing problem on discrepancy between the high $τ$ with all-source samples by DIA and low $τ$ with RCG samples can probably be explained by the incompleteness of the stellar number count. A model fit to these measurements predicts $Γ=4.60\pm0.25\times10^{-5}$ star$^{-1}$ yr$^{-1}$ at $|b|\sim-1^\circ.4$ and $-2^\circ.25<l<3^\circ.75$ for sources with $I<20$, where the future space mission WFIRST will observe.