Fast numerical evaluation of dark matter direct detection event rates

TL;DR

The paper tackles the computational bottleneck in predicting dark matter direct-detection event rates for general velocity distributions and anisotropic detector responses. It introduces a fast algorithm based on a closed-form Radon transform of three-dimensional Zernike functions to convert a high-dimensional velocity integral into a tractable expansion, enabling rapid evaluation of angle-integrated rates. Convergence studies and benchmarks show 2–4 order-of-magnitude speedups over optimized numerical integration, with exponential convergence for smooth distributions and manageable memory requirements for practical truncations. The approach is broadly applicable to three-dimensional Radon-transform problems in DM physics and beyond, with practical implications for parameter scans, modulation analyses, and potential extensions to DM–electron, Migdal, and phonon/coupling phenomena; reference implementations ZebraDM and zest are provided.

Abstract

The mathematical expression for the dark matter nuclear recoil event rate in a detector consists of a six dimensional integral over the velocity distribution of dark matter in the detector frame, and over the recoil momentum of the nucleus. While a significant part of this integral can be solved in closed form under traditional assumptions of the standard halo model and/or isotropic detector response, analysis of alternate assumptions is conventionally impeded by the inefficiency of multidimensional numerical integration methods. In this work we introduce a novel, fast and efficient algorithm for direct detection event rate computations. This algorithm takes advantage of a closed form expression for the Radon transform of three dimensional Zernike functions, which are used as basis functions for the velocity distribution. We demonstrate an implementation of this algorithm ZebraDM (https://github.com/sebsassi/zebradm), which is typically faster than an optimized numerical integration approach by a factor between $10^2$ and $10^4$.

Figures (5)

• Figure 1: Test responses (normalized to maximum value) as a function of the direction $\hat{n}$ at $w=1$.
• Figure 2: Error in the truncated Zernike expansion on the angle integrated test function $F_2(x)=\int f_2(\vec{x})\,d\Omega$ for truncation degrees $L=20,40,80,100,140,180$. From bottom to top: the values $F_2^{(L)}(x)=\sum_{k=0}^{L/2}f^{(2)}_{2k,0,0}R_{2k}^0(x)$, the absolute errors $|F_2(x)-F_2^{(L)}(x)|$, and relative errors $|F_2(x)-F_2^{(L)}(x)|/|F_2(x)|$.
• Figure 3: Median relative errors of Zernike expansion of the test distributions as a function of the truncation degrees $L$, demonstrating exponential convergence of the Zernike expansion.
• Figure 4: Evaluation times of different steps of the Zernike-based Radon transform as a function of the truncation degrees $L$. The time $T_R$ is for an isotropic response with $N_t=10$ and $N_E=100$.
• Figure 5: Comparison of the performance of the Zernike-based method (this work, solid lines) and direct numerical integration (dashed lines) for the distributions $f_1$, $f_2$, and $f_3$. From top to bottom: isotropic response, response function $S_1$, and response function $S_2$. The left column corresponds to the conventional Radon transform, while the right column corresponds to the transverse Radon transform. The $y$-axis corresponds to the median relative error, while the $x$-axis corresponds to the effective integration time $T_I^\text{eff}$ defined in Eq. (\ref{['Teff']}).