1. Directional Statistics for WIMP direct detection Anne Green Astro-Particle Theory and Cosmology Group University of Sheffield Ben Morgan, AMG and Neil Spooner, astro- ph/0408047

2. WIMP directional detection Our motion with respect to the Galactic rest frame produces a large front-back asymmetry in the WIMP flux. Peak in direction of Sun’s motion: l=87.5 b=1.3. Difficult, if not impossible, for a background to mimic this signal.

3. Earlier analysis by Copi & Krauss and Lehner et al. found 5-30 events required, however: Need to include the uncertainties in the reconstruction of the recoil direction (due to multiple scattering and diffusion). WIMP velocity distribution is not known a priori. How many events would be required for a positive detection? Need to used non-parametric statistics (can not use likelihood analysis).

4. Recoil direction reconstruction Simulated a TPC detector with 0.05bar CS 2 (negative ion drift), 1 kV/m drift field MICROMEGAS 200 micro-m pixel readout, 10cm drift length S recoils generated with SRIM2003. Recoil direction reconstructed using moment analysis. r.m.s. deviation between primary and reconstructed dirs ~15 degrees (decreases with increasing E). Reconstruction impossible for E<20keV. May not be possible to measure sense of recoil. Assume background is negligible. (see J. Davies’ talk)

5. Halo modelling Steady state phase space distribution function of a collection of collisionless particles is given by the solution of the collisionless Boltzmann equation: For spherical, isotropic systems there is a unique (one-to-one) relationship between the radial density distribution and f. (if the density varies as 1/r 2 then f is gaussian). BUT observations and simulations indicate that real galaxy halos are not perfectly spherical and isotropic. Triaxial and anisotropic systems are far more complicated. No unique solution: assumptions have to be made about the form of the anisotropy or velocity distribution.

6. Logarithmic ellipsoidal model (Evans, Carollo & De Zeeuw) Simplest triaxial generalisation of the isothermal sphere (f(v) is a multi-variate gaussian). Triaxiality and anisotropy are independent of radius. Osipkov-Merritt (Osipkov & Merritt) Spherically symmetric, anisotropy increases with radius. No guarantee that either provides a good description of the Milky Way halo, but provide us with reasonable benchmarks to asses the effect of uncertainty in the structure of the MW halo on WIMP signals. The MW halo has not completely reached a steady state. In CDM cosmologies structure forms hierarchically, and some substructure survives: tidal debris from the Sagittarius dwarf galaxy, stream of high velocity halo stars in the solar neighbourhood.

7. WIMP fluxes and Directional recoil rates Standard halo Triaxial halo Standard halo+stream for m=100 GeV, cross-section=10 -5 pb, density=0.3 GeV/cm 3

8. Directional Statistics Recoil directions constitute unit vectors on sphere (or axes if the senses of the recoils can not be measured)-can therefore use directional statistics (used extensively in geophysics). Tests for a WIMP signal (anisotropy of recoils): Rayleigh (total length of vectors), Bingham (shape of distribution of axes) Beran/Gene (most general tests for asymmetry), dipole (mean of cosine of angle between recoil direction and solar motion). Test for non-spherical halo (rotational symmetry about direction of solar motion): Kuiper (KS test on azimuthal angles, relative to solar motion). Tests for presence of stream (deviation of mean direction from direction of solar motion): Kuiper and X 2

9. Results If the sense of the recoils can be measured of order 10 events will be sufficient to identify a WIMP signal for any non-contrived halo model (the dipole test is most powerful). With a 10 5 kg day exposure could reject isotropy for cross sections down to 3 x 10 -9 pb. If the senses can not be measured between ~50 and ~300 events will be required, depending on the halo model. Thousands of events would be needed to detect even extreme flattening of the WIMP flux/MW halo (more powerful statistics may exist however). With ~300 events could detect a tidal stream of WIMPs (provided the stream density is sufficiently high and the stream direction is sufficiently different from the direction of solar motion).

10. Conclusions Multiple scatterings and diffusion limit the accuracy with which the nuclear recoil direction can be reconstructed. The local WIMP distribution is not known. Therefore need to use non-parametric statistics. If the sense of the recoils can (can not) be measured then ~10 (~50-300) events will be required to reject isotropy (and detect a WIMP signal). A moderately high density WIMP stream could be detected with several hundred events: WIMP ASTRONOMY ??