Fundamental statistical limitations of future dark matter direct detection experiments

We discuss irreducible statistical limitations of future ton-scale dark matter direct detection experiments. We focus in particular on the coverage of confidence intervals, which quantifies the reliability of the statistical method used to reconstruct the dark matter parameters and the bias of the reconstructed parameters. We study 36 benchmark dark matter models within the reach of upcoming ton-scale experiments. We find that approximate confidence intervals from a profile-likelihood analysis exactly cover or overcover the true values of the weakly interacting massive particle (WIMP) parameters, and hence are conservative. We evaluate the probability that unavoidable statistical fluctuations in the data might lead to a biased reconstruction of the dark matter parameters, or large uncertainties on the reconstructed parameter values. We show that this probability can be surprisingly large, even for benchmark models leading to a large event rate of order a hundred counts. We find that combining data sets from two different targets leads to improved coverage properties, as well as a substantial reduction of statistical bias and uncertainty on the dark matter parameters.

Figure 1

The left (right) panels show examples for a good (bad) reconstruction of the WIMP benchmark model with true values $m_{\chi }=50\mathrm{GeV}$, ${\sigma }_p^{\mathrm{SI}}=2.51\times {10}^{-10}$. The difference is exclusively in statistical fluctuations in the simulated data. Top panels: 68.3% and 95.4% confidence levels in the $m_{\chi }-{\sigma }_p^{\mathrm{SI}}$ plane; the red cross shows the true value. Bottom panels: energy spectrum of the mock data (yellow histogram: recall that we use an unbinned likelihood function, the counts are binned for a better visualization), true rate $dR/dE(E)$ (black) and for the “bad” reconstruction an example of a rate (red) with a higher likelihood than the true rate.

Figure 2

Coverage results for the 1D 68.3% (top) and 95.4% (bottom) confidence interval for the WIMP mass in the $m_{\chi }-{\sigma }_p^{\mathrm{SI}}$ plane, for simulated Xe target (left) and for a combination of $\mathrm{Xe}+\mathrm{Ge}$ (right). Green (red) regions show “exact” coverage (overcoverage), as defined in the text. Black regions correspond to a transition from exact coverage to overcoverage. No undercoverage is observed. Isocontours of the expected number of counts in the Xe experiment are given in black. In the upper-left plot, the benchmark points studied are indicated by blue crosses. The “flares” pattern seen in some points are an artefact of the interpolation scheme used to generate the plots.

Figure 3

Difference between the histogram of the profile likelihood test statistic $\lambda (m_{\chi })$ from mock data sets and the value of the chi-square distribution with 1 degree of freedom (as predicted by Wilks’ theorem) at the center of each bin, as a function of $\lambda (m_{\chi })$, for two different WIMP benchmark points. This difference quantifies the deviation from Wilks’ theorem for these two benchmark points. For each benchmark point, ${10}^3$ realizations of mock data sets have been used to construct this histogram. Error bars assume Poisson count statistics.

Figure 4

As in Fig. 2, but for the 1D confidence intervals for ${\sigma }_p^{\mathrm{SI}}$. A significant improvement in the coverage when combining $\mathrm{Xe}+\mathrm{Ge}$ is apparent.

Figure 5

Expected fractional uncertainty for the WIMP mass in the $m_{\chi }-{\sigma }_p^{\mathrm{SI}}$ plane, for a Xe ($\mathrm{Xe}+\mathrm{Ge}$) target in the left (right) plot, quantifying the precision of the mass reconstruction (low e.f.u. corresponding to better precision). Isocontours of the expected number of counts in the Xe experiment are given in black; isocontours of the percentage of “bad” reconstruction (f.u. $>0.75$) are shown in white.

Figure 6

Expected fractional uncertainty on the WIMP mass as a function of exposure for a xenon experiment (bottom axis) and a germanium experiment (top axis) required to achieve this e.f.u. for a WIMP with cross section ${\sigma }_p^{\mathrm{SI}}={10}^{-9}$, for three different benchmark masses $m_{\chi }=25\mathrm{GeV}$ (red), $m_{\chi }=50\mathrm{GeV}$ (black) and $m_{\chi }=250\mathrm{GeV}$ (blue). Solid lines correspond to e.f.u. results for Xe only, dashed lines correspond to e.f.u. results for a $\mathrm{Xe}+\mathrm{Ge}$ target.

Figure 7

As in Fig. 5, but for the fractional bias of the WIMP mass; i.e., the bias of the WIMP mass relative to the benchmark mass (notice that almost no negative bias is observed).