4.1 Energy dependence The shape of the differential event rate depends on the WIMP and target masses,the WIMP velocity distribution and the form factor.For the standard halo model the expression for the differential event rate,eq.1,can be rewritten approximately (c.f.Ref.[42])as dR dR dER F2(ER)exp ΓE (26) where (dR/dER)o is the event rate in the E>0 keV limit.The characteristic energy scale is given by E=(c12u2)/mN where c is a parameter of order unity which depends on the target nuclei.If the WIMP is much lighter than the target nuclei,mx<mN,then Ec o m/mN while if the WIMP is much heavier than the target nuclei Ecx my.The total recoil rate is directly proportional to the WIMP number density,which varies as 1/mx. In fig.1 we plot the differential event rate for Ge and Xe targets and a range of WIMP masses.As expected,for a fixed target the differential event rate decreases more rapidly with increasing recoil energy for light WIMPs.For a fixed WIMP mass the decline of the differen- tial event rate is steepest for heavy target nuclei.The dependence of the energy spectrum on the WIMP mass allows the WIMP mass to be estimated from the energies of detected events (e.g.Ref.[43]).Furthermore the consistency of energy spectra measured by experiments using different target nuclei would confirm that the events were due to WIMP scattering (rather than,for instance,neutron backgrounds)[42].In particular,for spin independent interactions,the total event rate scales as A2.The is sometimes referred to as the 'materials signal'. The WIMP and target mass dependence of the differential event rate also have some general consequences for experiments.The dependence of the total event rate on my means that,for fixed cross-section,a larger target mass will be required to detect heavy WIMPs than lighter WIMPs.For very light WIMPs the rapid decrease of the energy spectrum with increasing recoil energy means that the event rate above the detector threshold energy,Er, may be small.If the WIMP is light,<(10 GeV),a detector with a low,<O(kev), threshold energy will be required. The most significant astrophysical uncertainties in the differential event rate come from the uncertainties in the local WIMP density and circular velocity.As discussed in Sec.3.1 the uncertainty in the local DM density translates directly into an uncertainty in constraints on (or in the future measurements of)the scattering cross-section.The time averaged differential event rate is found by integrating the WIMP velocity distribution,therefore it is only weakly sensitive to changes in the shape of the WIMP velocity distribution.For the smooth halo models discussed in Sec.3.2 the time averaged differential event rates are fairly similar to that produced by the standard halo model [44;45].Consequently exclusion limits vary only weakly [45;46]and there would be a small (of order a few per-cent)systematic uncertainty in the WIMP mass deduced from a measured energy spectrum [47].With multiple detectors it would in principle be possible to measure the WIMP mass without any assumptions about the WIMP velocity distribution [48. In the extreme case of the WIMP distribution being composed of a small number of streams the differential event rate would consists of a series of (sloping due to the form factor)steps.The positions of the steps would depend on the stream velocities and the target and WIMP masses,while the relative heights of the steps would depend on the stream densities. 94.1 Energy dependence The shape of the differential event rate depends on the WIMP and target masses, the WIMP velocity distribution and the form factor. For the standard halo model the expression for the differential event rate, eq. 1, can be rewritten approximately (c.f. Ref.[42]) as dR dER ≈  dR dER  0 F 2 (ER) exp  − ER Ec  , (26) where (dR/dER)0 is the event rate in the E → 0 keV limit. The characteristic energy scale is given by Ec = (c12µ 2 N v 2 c )/mN where c1 is a parameter of order unity which depends on the target nuclei. If the WIMP is much lighter than the target nuclei, mχ ≪ mN , then Ec ∝ m2 χ/mN while if the WIMP is much heavier than the target nuclei Ec ∝ mN . The total recoil rate is directly proportional to the WIMP number density, which varies as 1/mχ. In fig. 1 we plot the differential event rate for Ge and Xe targets and a range of WIMP masses. As expected, for a fixed target the differential event rate decreases more rapidly with increasing recoil energy for light WIMPs. For a fixed WIMP mass the decline of the differen￾tial event rate is steepest for heavy target nuclei. The dependence of the energy spectrum on the WIMP mass allows the WIMP mass to be estimated from the energies of detected events (e.g. Ref. [43]). Furthermore the consistency of energy spectra measured by experiments using different target nuclei would confirm that the events were due to WIMP scattering (rather than, for instance, neutron backgrounds) [42]. In particular, for spin independent interactions, the total event rate scales as A2 . The is sometimes referred to as the ‘materials signal’. The WIMP and target mass dependence of the differential event rate also have some general consequences for experiments. The dependence of the total event rate on mχ means that, for fixed cross-section, a larger target mass will be required to detect heavy WIMPs than lighter WIMPs. For very light WIMPs the rapid decrease of the energy spectrum with increasing recoil energy means that the event rate above the detector threshold energy, ET , may be small. If the WIMP is light, < O(10 GeV), a detector with a low, < O( keV), threshold energy will be required. The most significant astrophysical uncertainties in the differential event rate come from the uncertainties in the local WIMP density and circular velocity. As discussed in Sec. 3.1 the uncertainty in the local DM density translates directly into an uncertainty in constraints on (or in the future measurements of) the scattering cross-section. The time averaged differential event rate is found by integrating the WIMP velocity distribution, therefore it is only weakly sensitive to changes in the shape of the WIMP velocity distribution. For the smooth halo models discussed in Sec. 3.2 the time averaged differential event rates are fairly similar to that produced by the standard halo model [44; 45]. Consequently exclusion limits vary only weakly [45; 46] and there would be a small (of order a few per-cent) systematic uncertainty in the WIMP mass deduced from a measured energy spectrum [47]. With multiple detectors it would in principle be possible to measure the WIMP mass without any assumptions about the WIMP velocity distribution [48]. In the extreme case of the WIMP distribution being composed of a small number of streams the differential event rate would consists of a series of (sloping due to the form factor) steps. The positions of the steps would depend on the stream velocities and the target and WIMP masses, while the relative heights of the steps would depend on the stream densities. 9