# Analytical derivation of the radial distribution function in spherical dark matter halos

## TL;DR

This paper presents an analytical method to derive the velocity distribution function of dark matter in spherical halos, complementing numerical simulations and enhancing understanding of dark matter behavior near Earth.

## Contribution

It introduces a differential equation approach to analytically determine the full velocity distribution function in spherical, equilibrated dark matter halos, extending previous work on tangential components.

## Key findings

- Analytical derivation matches numerical simulations below 0.7 v_esc.
- Complete velocity distribution can be obtained analytically for spherical halos.
- Method enhances understanding of dark matter velocity profiles.

## Abstract

The velocity distribution of dark matter near the Earth is important for an accurate analysis of the signals in terrestrial detectors. This distribution is typically extracted from numerical simulations. Here we address the possibility of deriving the velocity distribution function analytically. We derive a differential equation which is a function of radius and the radial component of the velocity. Under various assumptions this can be solved, and we compare the solution with the results from controlled numerical simulations. Our findings complement the previously derived tangential velocity distribution. We hereby demonstrate that the entire distribution function, below 0.7 v_esc, can be derived analytically for spherical and equilibrated dark matter structures.