What can you learn from what they don't tell you?

So I'm a "phenomenologist." Loosely speaking, this means that I care (practically) about the results of experiments, past, present and future. Of late a great deal of my emphasis has been on dark matter experiments.

One particular experiment I'm interested in is the CRESST experiment. CRESST uses a CaWO4 crystal and measures both the phonon signal as well as a light yield, and the ratio of the two helps determine nuclear recoils from electronic recoils (and thus, hopefully, dark matter scatters from background). It's really a fantastic experiment, capable of better than 1 keV energy resolution.

The only problem is a persistent set of background events at the detector. They may be cracks, and there are plausible suggestions by the collaboration for the source of them, but they remain. It's not impossible that they could be real dark matter events if dark matter is inelastic, for which the tungsten in the target is particularly sensitive. It's speculative, but who knows?

Anyhow, I've been eager to see the updated results from the experiment for a while now. They didn't publish their last run (which was an R&D run) and I've been looking forward to the results from this current run. Unfortunately, those are not yet out.

However, they did show a "selected" set of results at the UCLA DM conference. It sounds like they have 8 or 9 working detectors and showed 2, both which were clean, i.e., had no events in them in the range up to 40 keV.

So what does one make of this?

Well, it occurs to me we can play a bit of a game. We can make a weird sort of prediction, based on psychology, of what they might see. Since clean detectors are necessarily background free (since there are no events) it's good to show them. So, if I were running an experiment, I'd show all the clean detectors I had. We can turn this around, and use the fact that we saw precisely two detectors with no events to extrapolate that these were the only event free detectors.

So, this is a big speculation, but there's no harm in idle speculation. Let's suppose they had 8 or 9 experiments, two of which had no events. Well, if the events come from a Poisson distribution with expectation N, then the probability of seeing 0 events in any given detector is exp[-N].

So, let's say there are 9 operating detectors. Then the likelihood that you have two detectors with nothing in them is 36 exp[-2N](1-exp[-N])^7. What does this look like as a function of N?

This function has a peak at 1.5 and an expectation of 1.8. This is consistent with their commissioning run.

So if they have 9 operating detectors, that's my expectation.

In essence, I'm just taking advantage of the fact that in the limit of many equivalent detectors, the fraction expected to have 0 events is just Exp[-N]. So, if that's 2/9, then N=1.5. Nine may not be many, and they may not be equivalent, but probably not too terrible of assumptions.

But we can still go a bit further. Given that we are assuming that we've seen the clean detectors, the others must have at least one event. If I take the peak expectation value of 1.5, the Poisson expectation for detectors that have at least one event is 1.93. Thus, for the seven remaining detectors I expect 13.5 events in the energy range up to 40 keV.

What does this mean? OK, well, nothing. There are lots of reasons they might have only shown two: maybe they'd only finished analyzing two, maybe they just thought that was the right number. Still, it's fun what one can do when one combines statistics+human psychology+unjustified assumptions. Just like phenomenology.

NW

[Using the expectation value of 1.8 rather than the peak of 1.5, you get an expectation of 15.1 events. If only 8 detectors are working, the peak (expectation value) is at 1.4 (1.7), and the expectation for the remaining 6 is 11 (12.5) events. And then, of course, these are just mean values. Also I should add that I'm very tired at the moment so I'm very likely making stupid mistakes with my Poisson probabilities.]