A Semi-Analytic Approach to Dark Matter Attenuation*

If dark matter interacts too strongly with nuclei, it could be slowed to undetectable speeds in Earth's crust or atmosphere before ever reaching a detector. For sub-GeV dark matter, analytic approximations appropriate for heavier dark matter fail, necessitating the use of computationally expensive simulations. We present a new method of modeling attenuation of light dark matter in the Earth, based on the approximation that the scattering is isotropic in the lab frame. We show that this approach agrees well with Monte Carlo results, and can be much faster when the number of scatterings becomes large, as the runtime for Monte Carlo methods increases exponentially with cross section. We use this method to model attenuation for sub-dominant dark matter--that is, particles that make up a small fraction of the dark matter density--and show that previous papers on sub-dominant dark matter overestimate the sensitivity of direct detection experiments.