Improved determination of the WIMP mass from direct detection data

Direct detection experiments searching for weakly interacting massive particle (WIMP) dark matter typically use a simplified model of the Galactic halo to derive parameter constraints. However, there is strong evidence that this Standard Halo Model is not a good approximation to our Galaxy. We discuss previous attempts to extract the WIMP mass, cross section and speed distribution from direct detection data and show that these lead to significant biases in the reconstructed parameter values. We develop and test an alternative model-independent method based on parametrizing the momentum distribution of the WIMPs. This allows us to limit the analysis only to those regions of momentum space to which the experiments are sensitive. The method can be applied to a single experiment to extract information about the distribution function, as well as information on the degenerate WIMP mass and interaction cross section combined in a single parameter. This degeneracy can be broken by including data from additional experiments, meaning that the WIMP mass and speed distribution can be recovered. We test the momentum parametrization method using mock data sets from proposed ton-scale direct detection experiments, showing that it exhibits improved coverage properties over previous methods, as well as significantly reduced bias. We are also able to accurately reconstruct the shape of the WIMP speed distribution but distinguishing between different underlying distributions remains difficult.

Figure 1

Range of accessible WIMP speeds for each of the three mock experiments: XENON1T-like (solid blue), SuperCDMS-like (dashed green) and WArP-like (dot-dashed red). Each pair of lines corresponds to the maximum and minimum accessible WIMP speeds for a given experiment. The outermost dotted lines (red) show the accessible speeds for the adjusted parametrization range described in Sec. 6.

Figure 2

1-D and 3-D speed distributions, $f(v)$ and $f_3(v)$, and mean inverse speed, $\eta (v)$, for the 3 benchmark speed models with parameter values as given in Sec. 3a: Standard Halo Model (SHM-solid blue), $\mathrm{\text{Standard}}\mathrm{\text{Halo}}\mathrm{\text{Model}}+\mathrm{\text{Dark}}\mathrm{\text{Disk}}$ (DD-dashed green) and Via Lactea 2 (VL-2-dotted red).

Figure 3

WIMP masses reconstructed using the speed parametrization method from 250 realizations. The benchmark speed distribution is the SHM. The true mass of 50 GeV is shown as a dashed vertical line.

Figure 4

Distribution of the $\Delta$ statistic, defined in Sec. 3b, for 250 realizations using the speed parametrization method. The benchmark speed distribution is the SHM, with a 50 GeV WIMP.

Figure 5

Reconstructed speed distribution, $f(v)$, and mean inverse speed, $\eta (v)$, using the speed parametrization method. The benchmark model used was a 50 GeV WIMP with an SHM speed distribution. The upper pane shows the underlying SHM speed distribution (solid blue) and the fitted values of the speed bin parameters (red points). The errors on the bin values are within-chain standard deviations as described in Sec. 3b. The lower pane shows the mean inverse speed corresponding to these fitted values (dashed red line) and the true mean inverse speed (solid blue).

Figure 6

The rescaled mean inverse speed, $\eta /m_{\chi }$, measured in the SuperCDMS-like experiment as a function of recoil energy, $E_{\mathrm{R}}$. The same mock data set was used as in Fig. 5. The underlying SHM distribution (solid blue) uses the true WIMP mass of 50 GeV, as does the binned approximation to the SHM (dashed red). The reconstructed mean inverse speed (dot-dashed black) uses the reconstructed value of 30 GeV.

Figure 7

Reconstructed values for the scale parameter, $D_{\mathrm{rec}}$, for the Argon experiment using the momentum parametrization method from 250 realizations. The benchmark speed distribution is the SHM. The value of $D_{\mathrm{true}}=75.6\times {10}^7{\mathrm{cm}}^2{\mathrm{kg}}^{-2}$ is shown as a dashed vertical line.

Figure 8

Reconstructed momentum distribution for a single Argon experiment using a benchmark of a 50 GeV WIMP and the SHM. The upper pane shows the SHM momentum distribution (solid blue) and reconstructed bin values (red points). Because the posterior is unimodal, we also display vertical error bars showing the extent of the 68% confidence region for each bin. Note that these errors are strongly correlated. The lower pane shows the corresponding reconstructed mean inverse momentum (dashed red) and the mean inverse momentum in the SHM (solid blue). The underlying distribution has been rescaled by $1/{\alpha }_{\mathrm{Ar}}$ for comparison to the reconstructed values.

Figure 9

WIMP masses reconstructed using the speed and momentum parametrization methods from 250 realization. The benchmark speed distribution is the SHM. The true mass of 50 GeV is shown as a dashed vertical line.

Figure 10

Reconstructed momentum distribution from all three mock experiments using a benchmark of a 50 GeV WIMP and the SHM. The upper pane shows the SHM momentum distribution (solid blue) and reconstructed bin values (red points). The errors on the bin values are within-chain standard deviations as described in Sec. 3b. The lower pane shows the corresponding reconstructed mean inverse momentum (dashed red) and the mean inverse momentum in the SHM (solid blue). The reconstructed values have been rescaled by $\alpha$ for comparison to the true distribution.

Figure 11

Distribution of reconstructed masses, $m_{\mathrm{rec}}$, using the momentum method for 250 reconstructions. The true mass of 50 GeV is shown as a dashed vertical line.

Figure 12

As Fig. 11 for $m_{\chi }=100\mathrm{GeV}$.

Figure 13

Reconstructed speed distribution from all three mock experiments using the momentum parametrization method. The benchmark is a 50 GeV WIMP and the SHM distribution function. The upper pane shows the underlying SHM speed distribution (solid blue) and the fitted values of the speed bin parameters (red points). The errors on the bin values are within-chain standard deviations as described in Sec. 3b. The lower pane shows the mean inverse speed corresponding to these fitted values (dashed red line) and the true mean inverse speed (solid blue). The underlying distributions have been rescaled by $\alpha$ for comparison to the reconstructions.

Figure 14

As Fig. 13 for a 100 GeV WIMP with DD distribution function using 5 momentum bins (left-hand panes) and 7 momentum bins (right-hand panes).

Figure 15

Distribution of reconstructed masses using the 7-bin momentum method for 250 reconstructions for a DD benchmark distribution. The true mass of 100 GeV is shown as a dashed vertical line.

Figure 16

Reconstructed speed distribution from all three mock experiments using a benchmark of a 50 GeV WIMP with SHM distribution. The reconstructed values have been rescaled by $\alpha$ for comparison to the true distribution. The three different benchmark speed distributions defined in Sec. 3a have been overlaid: SHM (solid blue), DD (dashed green) and VL-2 (dotted red).