Statistics of Microlensing Optical Depth

Abstract

We show analytically that the statistical error, στ, in estimating the optical depth, τ, due to microlensing is substantially higher than the naive Poisson estimate: = στ = (η/N)½τ, where N is the number of detections and 1.65 ≤ η ≲ 3. We construct a "microlensing map" using an observationally well-constrained barred bulge model and compare this with a map based on an axisymmetric bulge model. The result indicates a significant difference in the distribution of optical depths. If the optical depths are determined over a wide range of positions around the Galactic center, one could make a rudimentary determination of the geometry of the mass distribution of the Galactic bulge by observing ˜700-800 events. About 1800 events are required to determine the relative contributions to lensing events by bulge and disk massive compact objects.