Energy loss during Dark Matter propagation in an overburden

As experimental constraints on DM interactions become ever more sensitive and push into new regimes of DM mass, it becomes more and more challenging to accurately model the process by which Dark Matter particles lose energy through scattering in the Earth's surface or other overburdens. We show that a commonly-used approximation due to Starkman, Gould, Esmailzadeh and Dimopoulos (SGED) can fail badly in computing the attenuation, even while being useful for an order-of-magnitude estimate of the maximum cross section reach. We introduce a method of importance sampling which makes Monte-Carlo simulation of energy loss feasible, in spite of factor-$10^7$ or greater attenuation. We demonstrate the validity of our new method and expose multiple problems with the SGED approximation, this reveals interesting features of the energy loss process. We spot-check the recent Emken, Kouvaris and Shoemaker "$5\,\Delta v$" prescription to place limits on cross sections based on a limited-statistics analysis, and find that an accurate simulation yields a factor of $\rm{4.4\times10^6}$ and $\rm{2.4\times10^4}$ larger number of events, for 50 MeV and 1 GeV DM mass respectively, than if the EKS $5\,\Delta v$ prescription were valid.