I'm trying to make sense of how the upper limits of the cross-section of particle dark matter (WIMPs) are computed, like in the figure below. This is taken from an article on dark matter searches.

DM cross section vs DM mass

Quoting from the article:

The number of selected events in the region of interest is used to set a limit on the dark matter cross-section. [p.2]

and then

A limit is set by assuming all events are originating from the dark matter scatters and the maximal possible cross-section is determined by using Yellin's optimal method, which is considered to provide a conservative limit. [p.3]

In skimming this article about the statistical methods used in particle dark matter searches, the optimum limit method is given by eq. 13 on page 15

$$x_i = \int_{E_i}^{E_{i+1}} \frac{dN}{dE}(\sigma_{\text{scatt}} )dE \ .$$

I'm not sure if this is the expression that I'm looking for.

In its treatment, the concept of a confidence interval is used, which still eludes me.

Furthermore, I've read on CLs methods which according to Wikipedia is

a statistical method for setting upper limits (also called exclusion limits) on model parameters, a particular form of interval estimation used for parameters that can take only non-negative values.

Mind you, I have not taken a rigorous course in probability theory, so a brief exposition in how the concept of the confidence interval is involved in calculating the exclusion limits, in the figure above, would be highly appreciated. :)

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    $\begingroup$ Like so many questions about the analysis of significant experiments this can only be condensed into a reasonable size answer by assuming that the reader is already fairly familiar with the field. For instance, any discussion you might find in a paper would expect you to know enough about confidence limits to interpret them even if you aren't familiar with the particular method used to set them in this paper. $\endgroup$ – dmckee --- ex-moderator kitten Jun 5 '18 at 0:30
  • $\begingroup$ Well, I'd tolerate a fair bit of handwaiving in this case. :) $\endgroup$ – Mussé Redi Jun 5 '18 at 7:15

I can talk about how such graphs are drawn in general. The basic idea is to compute a confidence interval based on the observed number of event and Poisson statistics. The upper limit calculated is thus the upper limit on the dark matter-nucleon scattering rate. So all we need to do is find the pair of (mass, cross section) values for which the theoretical rate is within the calculated limits.

Let's say the upper limit is 2.4 and lower is zero, so considering the plot you attached, we will calculate the rate for DM particles with the mass in the range [0.5-30] and cross section in the range [10^-9 - 1]. Now discard all (m, sigma) pairs for which the rate is less than 2.4. So you will end up with a reduced set of sigma and mass pairs. For a given sigma, there can be more than 1 mass point, select the lowest mass corresponding to given sigma, and plot them. That's it.

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  • $\begingroup$ I think I'm way over my head with this question, as I've not done the proper research. I don't know how to compute confidence intervals;how the cross section is inferred; what you mean when you say we should discard all ($m$,$\sigma$) pairs less then a certain upper limit. $\endgroup$ – Mussé Redi Jun 24 '19 at 12:14

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