1. Ben Morgan & Anne Green University of Sheffield
Directional Statistics for
WIMP Direct Detection
Ben Morgan & Anne Green
University of Sheffield

2. 1:Directional Sensitivity
Directional detectors aim to confirm/refute galactic nature of an anomalous nuclear recoil signal via measurement of recoil arrival direction distribution.
Strong (20:1) peak expected in distribution in direction of solar motion, cf few% change in rate for annual modulation.
Difficult, if not impossible, for backgrounds to mimic this anisotropy.
Critical question: How many recoil events are required to reject isotropy of recoil directions at a given confidence level?
This number determines the exposure required at a given WIMP-nucleon cross section – a `Directional Sensitivity’.
Copi & Krauss (Phys. Rev. D 63: ), Lehner(DARK2000, Heidelberg): 5-30 events needed, but…
To answer this question must consider:
Accuracy of recoil track reconstruction in detector.
Realistic models for local WIMP velocity distribution f(v).
Statistical analysis independent of assumptions about f(v).

3. 2:Recoil Direction Reconstruction
Initial Recoil Direction
Reconstructed Direction
Simulated LPNI-TPC directional detector:
0.05bar CS2 (negative ion drift).
1kVcm-1 drift field.
MICROMEGAS avalanche stage, 200mm pixel readout (Bellazini et al, NIM A 513:231).
10cm drift length.
5-200keV 32S recoils generated with SRIM2003.
Recoil directions estimated as principal axis ±r of moment analysis of pixel signals.
Lack of charge asymmetry along principal axes => Recoil sense (+r or –r) probably not measurable, but needs experimental test.
Charge diffusion limits accurate axis reconstruction to recoils with E>20keV .
RMS deviation between primary and reconstructed axes is 10-20o, but decreases with increasing recoil energy.
Primary Recoil Direction
Reconstructed Recoil Direction
Primary limitations to accurate direction reconstruction are multiple scattering of recoil and diffusion.
40keV S recoil in 0.05bar CS2

4. 3:Calculation of Recoil Spectra
WIMP Arrival Directions
Recoil Arrival Directions E>20keV
Triaxial Halo
v0=220kms-1, p=0.9,q=0.8,b=0.4,axis=Intermediate
100GeV WIMPs on 32S
WIMP Arrival Directions
Recoil Arrival Directions, E>20keV
Maxwell-Boltzmann Halo
v0=220kms-1
100GeV WIMPs on 32S
WIMP Arrival Directions
Recoil Arrival Directions E>20keV
Maxwell-Boltzmann Halo plus Maxwellian Sagittarius WIMP stream
v0=220kms-1, rsgr=0.25rloc, ssgr=30kms-1
100GeV WIMPs on 32S
Primary recoil direction distribution given by radon transform of WIMP velocity distribution f(v).
Models for f(v) selected to account for features of simulated dark haloes and observed luminous tracers: Flattening, Radial Orbit Bias, Sagittarius Dwarf Tidal Stream.
Year-averaged distributions of >20keV 32S recoil directions in galactic coordinates generated through Monte Carlo, taking angular resolution of simulated detector into account.

5. 4:Statistical Tests of Isotropy
Vectorial Statistics
Axial Statistics
Rayleigh-Watson:
Bingham:
Beran:
Gine:
Dipole:
|Dipole|:
Recoil direction vectors (+r or –r) or, if sense isn’t known, axes (+r and –r) are equivalently points on the unit sphere.
Halo model independent tests for the isotropy of N recoil vectors/axes ri can therefore be made using inference methods developed for spherically distributed data.
Distributions of statistics under null hypothesis of N isotropic vectors/axes f0(T;N) are generally cn2 or gaussian.
Null distributions of Beran and Gine statistics determined from Monte Carlo.

6. 5:Hypothesis Testing of Isotropy
From a given HM recoil distribution, Monte Carlo generate 105 experiments each observing N random recoils (assume zero electron/neutron background).
Calculate statistic T for each experiment – build f1(T;N) for alternative (halo) hypothesis.
For each N, Calculate Rejection and Acceptance factors at all T from f0(T;N) and f1(T;N).
Determine N such that 95% of experiments observing the halo hypothesis have T such that the null (isotropic) hypothesis is rejected at 95%.
‘Require N to reject isotropy at 95% confidence in 95% of experiments’.

7. 6:Directional Sensitivity Results 1
If sense (i.e +r or –r) of recoil is known, only need ~20 events to reject isotropy at 95% confidence in 95% of experiments over each halo model.
If sense cannot be measured, events are required.
Effect of recoil reconstruction more pronounced on axial data.
For axial tests, events in `backward’ hemisphere are added to `forward’ one – reduces anisotropy.

8. 7:Directional Sensitivity Results 2
Recoil Sense Known
Recoil Sense Unknown
Convert numbers to equivalent exposure in kgdays required to observe N recoils for given WIMP-nucleon cross-section and local WIMP density.
Recoil sense known: Isotropy can be rejected for s0>3x10-9pb with 105kgdays.
Recoil sense unknown: Isotropy can be rejected for s0>3x10-8pb with 105kgdays.
105kgdays = 100kgx3yr = 0.05bar).
Now looking at analysis with 2D readout.

9. 8:Tests for a Triaxial Halo
Recoil distributions of triaxial halo models are flattened.
Suggests suitable test is rotational symmetry of recoil azimuthal angles around solar motion direction vsun.
Test statistic here is Kuiper V statistic (see astro-ph/ for details) – valid for vectors and axes.
Select two most flattened triaxial models:
5: p=0.9, q=0.8, b=0.4, axis=long.
7: p=0.72, q=0.7, b=0.4, axis=inter.
Repeat hypothesis testing procedure described earlier. f
Model 5: N= required to reject symmetry around vsun at 90% confidence in 90% of experiments.
Model 7: N=1710 required to reject symmetry around vsun at 95% confidence in 95% of experiments.
For 105kgday and (arguably extreme) HM7, rotational symmetry can be rejected at 95% confidence down to s0~3x10-7pb for r0~0.3GeVcm-3.

10. 9:Tests for Sgr Tidal Stream
Presence of Sgr tidal stream leads to peak in recoil direction distribution deviating from direction of solar motion vsun.
Two possible tests:
Compatibility of median recoil direction with vsun.
Rotational symmetry around vsun.
Former based on ‘X2’ statistic (see astro-ph/ for more details), latter as for triaxial halo.
Choose rsgr=0.25r0, ssgr=30kms-1 to give optimum conditions for detection.
Repeat hypothesis testing process.
‘X2’ Statistic requires N=294 for 95% rejection of vsun as median direction in 95% of experiments (valid only for vectorial data).
Kuiper V Statistic requires N=574 for 95% rejection of symmetry around vsun in 95% of experiments (valid for vectorial and axial data).
For 105kgday, X2 could identify Sgr stream at 95%cl down to s0=6x10-8pb.
For same exposure, V could identify Sgr stream at 95%cl down to s0=1x10-7pb.

11. See astro-ph/0408047 for more details
10:Conclusions
Directional WIMP sensitivity of a generic gas TPC has been studied.
Accuracy of recoil direction measurement limited by multiple scattering and diffusion.
Recoil sense may not be measurable, but experimental tests needed.
Realistic set of WIMP halo velocity distributions considered.
Directional sensitivity determined from hypothesis testing based on halo model independent spherical statistics – techniques can be applied to real data.
Recoil sense known(unknown): 10-20( ) events needed to reject isotropy at 95% confidence in 95% of experiments.
Even extreme triaxial haloes difficult to detect – >2000 events required.
High density (0.25r0) Sgr stream requires events to detect at 95% confidence.
More powerful directional tests incorporating recoil energies/momenta may be possible.
See astro-ph/ for more details