Dark Matter — Question 9

Question 9

A direct-detection rate depends on both particle mass and cross section

Assume local dark-matter rest-energy density .30 GeV/cm$^3$, particle rest energy 100 GeV and speed $2.0\times10^7$ cm/s. A detector has $10^{28}$ targets and effective scattering cross section $10^{-40}$ cm$^2$ per target; ignore thresholds.

1. Find number density, flux and ideal event rate.

2. Find expected events in one year of $3.156\times10^7$ s, and how the rate changes if particle mass doubles at fixed density, speed and cross section.

Question 9 - Solution

A direct-detection rate depends on both particle mass and cross section

(a) Method and calculation

$$\boxed{n_\chi=.30/100=.0030\ {\mathrm{cm^{-3}}},\quad\Phi=n_\chi v=6.0\times10^4\ {\mathrm{cm^{-2}s^{-1}}}}.$$$$\boxed{R=N_T\sigma\Phi=6.0\times10^{-8}\ {\mathrm{s^{-1}}}}.$$

(b) Interpretation and further calculation

$$\boxed{\mu=Rt=1.8936\ {\mathrm{events/year}}}.$$$$R\propto n_\chi=\rho_\chi/m_\chi\Rightarrow\boxed{R^\prime=R/2}.$$

Physical check

The effective cross section and fixed-speed model are supplied assumptions. Real recoil spectra require velocity distributions, nuclear response and detector acceptance.