Partial Rate Matrix for Dark Matter Scattering

I present a highly efficient integration method for scattering calculations, and a ``partial rate matrix'' that encodes the scattering rate as a function of the $SO(3)$ orientation of the detector. This replaces the original multidimensional rate integral with a simple exercise in vector multiplication, speeding up the rate calculation by a factor of around $10^8$. I include a scheme to fully factorize the dark matter particle model, its astrophysical velocity distribution, and the properties of the target material from each other, enabling efficient calculation of the partial rate matrix even in studies comparing large sets of these input functions. This is now the only sensible way to evaluate the dark matter scattering rate in anisotropic detector materials. It is straightforward to generalize this method to other difficult but linear problems.