Colloquium: Annual modulation of dark matter

Direct detection experiments, which are designed to detect the scattering of dark matter off nuclei in detectors, are a critical component in the search for the Universe’s missing matter. This Colloquium begins with a review of the physics of direct detection of dark matter, discussing the roles of both the particle physics and astrophysics in the expected signals. The count rate in these experiments should experience an annual modulation due to the relative motion of the Earth around the Sun. This modulation, not present for most known background sources, is critical for solidifying the origin of a potential signal as dark matter. The focus is on the physics of annual modulation, discussing the practical formulas needed to interpret a modulating signal. The dependence of the modulation spectrum on the particle and astrophysics models for the dark matter is illustrated. For standard assumptions, the count rate has a cosine dependence with time, with a maximum in June and a minimum in December. Well-motivated generalizations of these models, however, can affect both the phase and amplitude of the modulation. Shown is how a measurement of an annually modulating signal could teach us about the presence of substructure in the galactic halo or about the interactions between dark and baryonic matter. Although primarily a theoretical review, the current experimental situation for annual modulation and future experimental directions is briefly discussed.

Figure 1

A simplified view of the WIMP velocities as seen from the Sun and Earth. Because of the rotation of the galactic disk (containing the Sun) through the essentially nonrotating dark matter halo, the Solar System experiences an effective “WIMP wind.” From the perspective of the Earth, the wind changes throughout the year due to the Earth’s orbital motion: the wind is at maximum speed around the beginning of June, when the Earth is moving fastest in the direction of the disk rotation, and at a minimum speed around the beginning of December, when the Earth is moving fastest in the direction opposite to the disk rotation. The Earth’s orbit is inclined at $\sim 60°$ relative to the plane of the disk.

Figure 2

Comparison of the standard halo model (SHM) and an example stream, representative of the smooth background halo and a cold flow, respectively. The stream, modeled after the Sagittarius (Sgr) stream, is roughly orthogonal to the galactic plane with speed $\sim 350\mathrm{km}/\mathrm{s}$ relative to the Sun. Upper panel: The speed distribution [one dimensional $f(v)$ in the frame of the Earth] for both components. Lower panel: The differential signal in a detector is directly proportional to the mean inverse speed $\eta (v_{\min })$. Here the $x$ axis is $v_{\min }$, the lower limit of the integration in Eq. (4). The approximately exponential SHM and steplike stream $\eta$’s are each shown at two periods of the year, corresponding to the times of year at which $\eta$ is minimized and maximized; note these times are different for the two components.

Figure 3

Comparison of the shapes of the total rate shown at two periods of the year, corresponding to the times of year at which the rate is minimized and maximized, as well as the modulation amplitude, for three different halo components: SHM (left), debris flow (middle), and stream (right). The normalization between panels is arbitrary.

Figure 4

The residual rate for the SHM (dashed) and an example stream (dotted) is plotted at several values of $v_{\min }$. The stream is modeled after the Sgr stream. Also shown is the total $\mathrm{SHM}+\mathrm{\text{stream}}$ modulation, assuming the local density of the stream is 10% that of the SHM. The residual rates are given relative to the average rate in each case, i.e., curves show the fractional modulation, except for the stream, where the relative rate has been divided by 10 for visual clarity as its relative modulation is much larger. The corresponding recoil energy for each $v_{\min }$, given by Eq. (5), depends on the WIMP and nuclear target masses; for a WIMP mass of 60 GeV and a germanium target, the $v_{\min }$ values of 150, 350, and $550\mathrm{km}/\mathrm{s}$ correspond to recoil energies of 7, 40, and 100 keV, respectively. The phase reversal of the SHM component, which occurs at smaller $v_{\min }$, can be seen by comparing the dashed curves in the top two panels. The recoil energy at which this phase reversal takes place can be used to determine the WIMP mass. From 146.

Figure 5

The residual rate measured by DAMA/NaI (circles, 0.29 ton yr exposure over 1995–2002) and DAMA/LIBRA (triangles, 0.87 ton yr exposure over 2003–2010) in the 2–6 keVee energy interval, as a function of time. Data are from 44, 46. The solid line is the best-fit sinusoidal modulation $A\cos [(2\pi /T)(t-t_0)]$ with an amplitude $A=0.0116\pm 0.0013\mathrm{cpd}/(\mathrm{kg}\mathrm{keV})$, a phase $t_0=0.400\pm 0.019\mathrm{yr}$ (May $26\pm 7$ days), and a period $T=0.999\pm 0.002\mathrm{yr}$ (46). The data are consistent with the SHM expected phase of 1 June.

Figure 6

The bin-averaged modulation amplitude observed by DAMA (top), CoGeNT (bottom, narrower bins), and CDMS (bottom, wider bins) as a function of energy. Boxes indicate the $1\sigma$ uncertainty for each bin. The DAMA results are fit at the SHM expected phase with a peak on 1 June, while the CoGeNT and CDMS bins are given at the CoGeNT best-fit phase with a peak on 16 April. The DAMA data are from 46 while the CoGeNT and CDMS binning and results are taken from 10. To allow for direct comparison between the two germanium experiments, we present both CoGeNT and CDMS data in keV (nuclear-recoil energy); on the other hand, DAMA data are presented in keVee (electron-equivalent energy). Also shown for DAMA are the best-fit spectra to the data for spin-independent (SI) scattering, corresponding to a WIMP with mass 11 GeV (76 GeV) and SI cross section $2\times {10}^{-4}\mathrm{pb}$ ($1.5\times {10}^{-5}\mathrm{pb}$).

Figure 7

WIMP mass and SI cross sections consistent with the anomalies seen by DAMA, CoGeNT, and CRESST, as well as constraints placed by the null results of CDMS and XENON (as of summer 2012). The halo model is assumed to be the SHM with the given parameters. The lack of overlap between the regions of the three anomalous results and their locations above the exclusion curves of CDMS and XENON indicate a conflict between the experimental results in this case. Alternative couplings, modified halo models, and systematic issues have been proposed to reconcile this apparent incompatibility. Figure courtesy of J. Kopp (138).