Optimizing WIMP directional detectors

We study the dependence of the exposure required to directly detect a WIMP directional recoil signal on the capabilities of a directional detector. Specifically we consider variations in the nuclear recoil energy threshold, the background rate, whether the detector measures the recoil momentum vector in two or three dimensions and whether or not the sense of the momentum vector can be determined. We find that the property with the biggest effect on the required exposure is the measurement of the momentum vector sense. If the detector cannot determine the recoil sense, the exposure required is increased by an order of magnitude for 3-d read-out and two orders of magnitude for 2-d read-out. For 2-d read-out the required exposure, in particular if the senses cannot be measured, can be significantly reduced by analyzing the reduced angles with the, time dependent, projected direction of solar motion subtracted. The background rate effectively places a lower limit on the WIMP cross-section to which the detector is sensitive; it will be very difficult to detect WIMPs with a signal rate more than an order of magnitude below the background rate. Lowering the energy threshold also reduces the required exposure, but only for thresholds above 20 keV.