Preliminaries

DM Flux

The typical velocity of DM particles is of the order of $\sim 10^{3}c$. The local DM density is ${\rho }_0=0.3{\text{ GeV cm}}^{-3}$ and thus, the DM number density is $n=\rho /m$.

$\varphi =\frac{v\rho }{m}\approx \frac{10^7}{m}{\text{ cm}}^{-2}{\text{s}}^{-1}$.

Assuming a typical WIMP cross section $\sigma$, these particles interact very weakly with SM particles.

Kinematics

Direct DM detection is based on the search of scattering between DM particles and nuclei in a detector. This process is observable only through the recoiling nucleus, with an energy $E_R$. Since DM particles move at non-relativistic speeds, the dynamics of their elastic scattering off nuclei can be calculated.

$E_R=\frac{1}{2}{m}_{\chi }{v}^2\frac{4m_{\chi }{m}_N}{(m_{\chi }+m_N{)}^2}\frac{1+\cos \theta }{2}$.

For DM particles with a mass of the order of $100\text{ GeV}$, this leads to recoil energies of approximately $E_R\sim 100\text{ keV}$. In the case of a head-on collision and when the DM and target masses are equal,

$E_R^{\text{max}}=\frac{1}{2}{m}_{\chi }{v}^2=\frac{1}{2}{m}_{chi}\times 10^{-6}=\frac{1}{2}\left(\frac{m_{\chi }}{1\text{ GeV}}\right)\text{ keV}$.