Theory of Pixel Lensing

Abstract

Pixel lensing, gravitational microlensing of unresolved stars, is potentially much more sensitive and much more widely applicable than is generally recognized. I give explicit expressions for the pixel noise induced by a time-variable PSF, by photometric and geometric misalignment, and by discrete pixelization, and I show that these can all be reduced below the photon noise. Pixel lensing can be divided into two regimes. In the ``semi-classical'' regime, it is similar to classical lensing in that it measures the time scale of individual events. In the ``spike'' regime, it measures the total optical depth but not individual time scales. I present simple expressions for the boundary between the two regimes and for the event rate in the latter one. These expressions can be used to quickly classify all potential pixel lensing experiments. Pixel lensing can measure the luminosity function as well as the mass function of stars in target galaxies to a distance of a few Mpc. Future space-based pixel lensing could be $\sim 5$ times more effective in the infrared than in the optical, depending on developments in detector technology. Pixel techniques can also be applied to non-pixel-lensing problems including the search for unresolved variable stars and follow up observations of lensing events found in classical lensing searches. To benefit fully from pixel-lensing techniques, follow-up observations should have resolutions of at least 5 pixels per FWHM.