Parametrizing the local dark matter speed distribution: A detailed analysis

In a recent paper, a new parametrization for the dark matter (DM) speed distribution $f(v)$ was proposed for use in the analysis of data from direct detection experiments. This parametrization involves expressing the logarithm of the speed distribution as a polynomial in the speed $v$. We present here a more detailed analysis of the properties of this parametrization. We show that the method leads to statistically unbiased mass reconstructions and exact coverage of credible intervals. The method performs well over a wide range of DM masses, even when finite energy resolution and backgrounds are taken into account. We also show how to select the appropriate number of basis functions for the parametrization. Finally, we look at how the speed distribution itself can be reconstructed, and how the method can be used to determine if the data are consistent with some test distribution. In summary, we show that this parametrization performs consistently well over a wide range of input parameters and over large numbers of statistical ensembles and can therefore reliably be used to reconstruct both the DM mass and speed distribution from direct detection data.

Figure 1

Several of the benchmark speed distributions used in this work. They are defined in Eqs. (6) and (7) with parameters from Table 1. These distributions are the SHM (solid blue), $\mathrm{SHM}+\mathrm{DD}$ (dashed green), Lisanti et al. (dot-dashed red) and the stream (dotted magenta).

Figure 2

Benchmark speed distributions used in Sec. 4 to test the performance of the parametrization as a function of the number and type of basis functions.

Figure 3

Bayesian information criterion as a function of the number of basis functions for an underlying bump distribution function, 50 GeV WIMP and using Legendre polynomial basis functions (upper panel). Also shown (lower panel) are the reconstructed WIMP mass (dashed blue line), 68% confidence interval (shaded blue region) and underlying WIMP mass (solid horizontal black line).

Figure 4

As Fig. 3 but for an underlying double-peak distribution function.

Figure 5

Time taken (using four processors in parallel) for the reconstruction of the bump benchmark, as a function of number of basis functions. The time taken using the Chebyshev basis (blue squares) grows more slowly with $N$ than for the Legendre basis (red triangles).

Figure 6

Reconstructed WIMP mass $m_{\text{rec}}$ (central dashed blue line) as a function of input WIMP mass $m_{\chi }$ as well as 68% and 95% intervals (inner and outer blue dashed lines respectively). The line $m_{\text{rec}}=m_{\chi }$ (solid red line) is also plotted for reference.

Figure 7

As Fig. 6 but including the effects of finite energy resolution and nonzero backgrounds, as described in the text.

Figure 8

Distribution of the reconstructed mass $m_{\text{rec}}$ for 250 mock data sets generated using several benchmark speed distributions, defined in Sec. 2. These are the SHM (top), $\mathrm{SHM}+\mathrm{DD}$ (middle) and Lisanti et al. (bottom) distributions. The input WIMP mass of $m_{\chi }=50\mathrm{GeV}$ is shown as a vertical dashed red line.

Figure 9

Reconstructed speed distribution for a single realization of data, generated for a 50 GeV WIMP. The 68% and 95% credible intervals are shown as dark and light shaded regions respectively, while the underlying SHM distribution function is shown as a solid blue line.

Figure 10

Reconstructed speed distribution for the same realization of data as Fig. 9. In this case, we have also normalized $f_1(v)$ to unity above $v_a\approx 171\mathrm{km}{\mathrm{s}}^{-1}$ (vertical dashed line). This is the lowest speed accessible to the experiments for a WIMP of mass 50 GeV. The 68% and 95% credible intervals are shown as dark and light shaded regions respectively, while the underlying SHM distribution function is shown as a solid blue line.

Figure 11

Mean reconstructed values of the rescaled mean inverse speed $\eta (v)/\alpha (v)$ at several values of $v$, calculated over 250 realizations of data using a 50 GeV WIMP and underlying SHM distribution function. Error bars indicate the mean upper and lower limits of the 68% credible intervals. The underlying form of $\eta (v)/\alpha (v)$ obtained from the SHM is shown as a solid blue line.

Figure 12

Rescaled mean inverse speed $\eta (v)/\alpha (v)$, reconstructed from a single realization of data using a 50 GeV WIMP and underlying SHM distribution function. At each value of $v$ we calculate 68%, 95% and 99% credible intervals (shown as shaded intervals). We also show the calculated values of $\eta (v)/\alpha (v)$ for several possible benchmark speed distributions: SHM (solid blue), $\mathrm{SHM}+\mathrm{DD}$ (dashed green), Lisanti et al. (dot-dashed red) and stream (dotted magenta). The benchmark curves are truncated when the underlying distribution function goes to zero.

Figure 13

As Fig. 12, but using as input a Lisanti et al. speed distribution and an exposure time which is 2.5 times longer.

Figure 14

As Fig. 13, but focusing on the region around $v\sim 400\mathrm{km}{\mathrm{s}}^{-1}$. Notice that in the range $400–550\mathrm{km}{\mathrm{s}}^{-1}$, both the SHM and $\mathrm{SHM}+\mathrm{DD}$ curves lie at or below the lower limit of the 95% credible interval.