Unbiased reconstruction of the mass function using microlensing survey data

The large number of microlensing events discovered towards the Galactic bul ge bears the promise of reconstructing the stellar mass function, especiall y at low masses near the brown dwarf regime. However, because of the source confusion, even if the distribution and the kinematics of the lenses are k nown, the estimation of the mass function is extremely biased at low masses . The blending due to source confusion biases the duration of the event, wh ich in turns dramatically biases the estimation of the lens mass. It is pos sible to overcome this problem by using differential photometry of the micr olensing events. Differential photometry is free of any bias due to blendin g, but the baseline flux is unknown. In this paper it is shown that, even w ithout knowledge of the baseline flux, pure differential photometry allows the estimation of the mass function without any bias. The basis of the meth od is that taking the scalar product of the microlensing light curves with a given function and taking its sum over all the microlensing events is equ ivalent to projecting the mass function on to another function. This method demonstrates that there is a direct correspondence between the space of th e observations and the space of the mass function. The optimal functions on to which the light curves are projected are their principal components. Th ere is no additional information about the distribution of the scalar produ cts of the data beyond their sum (first-order moments). Higher-order moment s are only linear combinations of the first-order moments. Thus the sum of the projections on the principal components contains all the information, a nd translates into an equal number of projections of the mass function with functions associated with the principal components. The method is illustra ted with simulated data sets consistent with the microlensing experiments. With 1000 such simulations, we show that for instance the exponent of the m ass function can be reconstructed without any bias.