Dark Matter — Question 8

Question 8

A compact-object null search limits only the tested mass population

A microlensing survey would detect an expected 20 events if all halo mass were compact objects of one specified mass. It observes zero events with negligible background. Detection efficiency is already included in 20.

1. Find the one-sided 95 percent upper bound on the compact-object halo fraction f using $P(0)=e^{-20f}$.

2. If the expected count for another object mass is only 2, calculate the formal bound and explain its physical meaning.

Question 8 - Solution

A compact-object null search limits only the tested mass population

(a) Method and calculation

$$e^{-20f_{95}}=.05\Rightarrow f_{95}=-\ln(.05)/20.$$$$\boxed{f_{95}=.149787}.$$

(b) Interpretation and further calculation

$$\boxed{f_{95,\mathrm{other}}=-\ln(.05)/2=1.49787}.$$$$0\le f\le1\Rightarrow\boxed{\text{no exclusion of }f=1\text{ at 95 percent in the second case}}.$$

Physical check

A null survey does not rule out every compact-object mass or all dark matter. Sensitivity depends on event timescale, exposure and efficiency.