Fast Multipole Method for Gravitational Lensing. Application to High Magnification Quasar Microlensing

We introduce the use of the Fast Multipole Method (FMM) to speed up gravitational lensing ray tracing calculations. The method allows very fast calculation of ray deflections when a large number of deflectors, $N_*$, is involved, while keeping rigorous control on the errors. In particular, we apply this method, in combination with the Inverse Polygon Mapping technique (IPM), to quasar microlensing to generate microlensing magnification maps with very high workloads (high magnification, large size and/or high resolution) that require a very large number of deflectors. Using, FMM-IPM, the computation time can be reduced by a factor $\sim 10^5$ with respect to standard Inverse Ray Shooting, making the use of this algorithm on a personal computer comparable to the use of standard IRS on GPUs. We also provide a flexible web interface for easy calculation of microlensing magnification maps using FMM-IPM\footnote{http://gloton.ugr.es/microlensing/}. We exemplify the power of this new method by applying it to some challenging interesting astrophysical scenarios, including clustered primordial black holes, or extremely magnified stars close to the giant arcs of galaxy clusters. We also show the performance/use of FMM to calculate ray deflection for a halo resulting from cosmological simulations composed by a large number ($N\gtrsim 10^7$) of elements.