Determining the Microlens Mass Function from Quasar Microlensing Statistics

The first investigations of the response of the microlensing magnification pattern (at an optical depth of order unity) to the mass function of the microlenses found that the resulting statistics depend only on the mean microlens mass <m>. In particular the mean microlensing caustic crossing rate was found to be proportional to the squareroot of <m>. We show that while this is true in the limit of mass functions with a narrow range of mass, in general the magnification pattern shows structure that reflects the contribution to the optical depth of microlenses with different masses. We present a better approximation, relating the microlens mass function to light-curve statistics. We show that the variability statistics of quasar microlensing light-curves can (in principle) be inverted to obtain the mass function of the microlenses in the mass range over which the mass density remains comparable, ie. p(m)dm~C/m. A preliminary analysis of the structure function for Q2237+0305 suggests that there is not a significant contribution to the optical depth from very low mass objects (0.001 solar mass). However observations of multiple microlensed quasars for a period of ~20 years may in the future yield a detailed p(m)dm. In the mass range where the number density is comparable, ie. p(m)dm~const., the distribution of flux factors could be inverted to find the microlens mass function. This may be used as a probe of the abundance of planets with orbital radii >100 AU.