and 4.16.
set.
Seeing as the optimal interval method requires actual events with actual energies to produce a limit, we turn again to Monte Carlo. The procedure is:
1. For each constrained leakage we have the cut positions from the previous section. Apply them.
2. Calculate expected number of misidentified background events in each detec- tor,µi, by summing the remaining background model weight vectors, as well as their expected energy spectrum by taking a simple weighted histogram.
3. Preform 5,000 MC experiments by sampling the correct number of events in each detector from a Poissonion with a mean ofµi.
4. For each sampled event, sample an interaction energy from the appropriate background energy spectrum as calculated in step 2.
5. For each MC experiment calculate the cumulative signal probability for all interactions, and preform the Optimal Interval calculation for the 90% C.L.
upper limit on the expected number of WIMP interactions.
6. Average the resulting upper limits and divide by the SAE to produce a WIMP rate per unit exposure. Because we know the theoretical flux of a proposed wimp with this mass through our detectors, we can calculate the 90% C.L.
upper limit on the WIMP-nucleon cross section. The results of this process can be seen in figure 4.17.
7. Repeat for all∼50 constrained leakage values and all 5 signal model datasets.
Although conceptually fairly straightforward, some care was taken to make this computationally tractable. In particular the standard optimal intervalFortrancode required some refactoring to allow it to be compiled as a shared library. In this way theUpperLim function found in UpperLimNew2.f can be called directly with no context-switching overhead. The build process required to achieve this is extremely fragile, and we were unsuccessful at automating it on all systems. For this reason we have not released this publicly as an improvement to the code.
for higher leakage: although the optimal interval will still potentially allow us a good limit, we would expect more leaked background events. To be conservative, we decided to choose the tightest cut that did not sacrifice sensitivity. To this end the Poisson minimum was selected.
Having chosen a constrained leakage value of 0.14 events, we still have five sets of optimized cuts to choose from, one for each of the five signal model masses. Ideally, we want a final cut position that provides strong limits at a variety of potential WIMP masses, not just the masses that they were optimized for. For this end, we took each of our five optimized cut positions and calculated the 90% C.L. upper limit on the expected WIMP rate for each of our five signal models. The results of this cross check can be seen in figure 4.18. As can be seen each set of cuts preforms the best for the WIMP mass at which it was optimized. This is expected, but is a nice consistency check. Although not the strongest limit at any mass other than 50 GeV c−2 the set of cuts optimized on that particular signal model dataset preforms very well at all WIMP masses. As a result the final cut set defining our signal acceptance region is that which was optimized for a constrained leakage of 0.14 misidentified background events assuming a WIMP mass of 50 GeV c−2.
Figure 4.14: Plots of the position in BDT score of our walkers vs step number for each of our detectors. As can be seen it takes∼200 steps for the BDT score to be fully explored in all detectors (Mass 50 GeV, leakage is 0.6 events).
(a)
Figure 4.14: Marginal distributions of our sampled posterior for a WIMP with a mass of 50 GeV/c2, and a constrained leakage of 0.6 events. Due to the angle of the anisotropy it is not visible in any marginal distribution rendering Gibbs sampling inefficient. The initial point found by the SLSQP optimizer is shown (blue).
(a)
(b) (c)
Figure 4.15: Optimal cut positions are shown for a 50 GeV/c2 WIMP mass. The constrained leakage value is depicted via the color (a) and the resulting optimal detector-dependent BDT-cut position is shown against the SAE and leakage func- tions used to preform the optimization.
Figure 4.16: Total maximized SAE vs allowed constrained leakage for a 50 GeV/c2 WIMP. We expect this to be a curve of diminishing returns, and as can be seen the SLSQP-based optimizer (blue) preforms very poorly at a number of constraint conditions. A few iterations of our affine-invariant basinhopping method (orange) vastly improve this result.
Figure 4.17: The 90% C.L. upper limit on the spin-independent WIMP-nucleon cross section vs allowed leakage calculated using the optimal interval method (fuch- sia) as well as using a simple Poisson upper limit (salmon). We used the Poisson minimum to select the cuts to unblind around (black highlighted) which correspond to an allowed leakage of 0.14 events.
Figure 4.18: Plotted is the expected 90% C.L. upper limit of the spin-independent cross section predicted by Monte Carlo from our signal and background models.
The limit was set using cut definitions optimized for a particular WIMP mass (shown by line color). As can be seen each set of cuts cuts preforms the best for the WIMP mass at which it was optimized, but the 50 GeV c−2
model produced a set of cuts that is well-preforming on a large energy domain.
C h a p t e r 5