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The idea of direct detection of dark matter is that a dark matter particle striking some underground target will cause the target nucleus to recoil. From the recoil, one can determine the scattering cross-section. In literature, there exists a plot of the WIMP-nucleon cross-section $\sigma$ versus WIMP mass ($M$). There exist upper bounds from several experiments such a LUX. Which formula is used to obtain this plot? Do we know a precise relation between WIMP-nucleon cross-section $\sigma$ and the WIMP mass $M$?

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I'll be a bit rough, so if any experimentalist friend wants to correct me I'll be happy.

The number of scattering events that you expect to see in your experiment is given related to the cross section by definition:

$$\sigma = \frac{\text{N events}}{\text{time}} \frac{1}{\text{N target particles}}\frac{1}{\text{incoming flux}}$$

The number of target particles is a constant of those experiments and you can find it by knowing the total mass and kind of liquid in your giant detector tank.

The flux contains informations about the incoming Dark Matter on your experiment and it is defined something like

$$\text{flux} = \text{number density}\times \text{velocity}\times\text{area}$$

The area is some effective area of the detector that is exposed to the flux. You know the number density by knowing the energy density of dark matter from cosmological measurements and it is related to the particle mass by

$$n = \frac{\rho}{m}$$

While the velocity is estimated using the speed of the Earth in our galaxy and with some models of the velocity of the dark matter halo relative to the galaxy that I am not sure.

There are probably some more factors that I'm missing and everything is obviously more complicated in the real world, but it does not matter.

Now in this formula you know everything except the cross section and the particle mass. By setting the number of observed events (0) in a certain period of time (several years) you can get an upper bound on the cross section by using statistics, asking yourself what is the probability of not seeing anything for a given cross section and mass.

If you want to know why the bound has that Nike-logo shape, it is really easy. The number of events is proportional to the Dark Matter number density. If the mass of the particle is bigger, for a fixed known mass density, the number density is smaller and thus the number of events is smaller. Less statistics $\implies$ weaker bound. For smaller masses it is different. You are expecting a scattering between a Dark Matter particle and a nucleus (with mass of several GeV). For smaller masses the nucleus recoil becomes much smaller and thus you start to lose sensitivity on your detector to measure it.

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    $\begingroup$ Thanks for this explanation. I'm confused about how exactly the sensitivity comes into the exclusion plot. Could you elaborate further? $\endgroup$ – SabrinaChoice Jan 5 at 2:03

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