### 5.1 Expected scattering rates

The scattering rate per unit detector volume depends on the local density of dark matter particles
; their velocity distribution relative to a detector in a terrestrial laboratory , and the velocity
dependent scattering cross-section , via the usual equation
where is the number density of nuclei of species in the detector. The local number density of
WIMP particles is , where is the WIMP mass and is the
assumed local cold dark matter density. The WIMP velocity distribution and the cross-section both have a
wide range of uncertainty, which makes accurate predictions impossible. The preferred range for in
the context of the lightest stable neutralino within minimal MSSM is to [110, 111], and
Han and Hempfling [66] quote a lower mass limit from LEP data as . In the
simplest models the dark matter density distribution in the halo of the Galaxy is taken to be a
spherical (at least for large ) distribution with a local density, at the position of
the solar system, of . The velocity distribution is taken to be a Maxwellian,
consistent with a virialised system but truncated above the Galactic escape velocity. Models
involving non-spherical density distributions [73], rotating halos [44, 73], and/or non-virial
velocity distributions, such as Galactic in-fall components with cusps [116] or bound Solar System
Earth-crossing components [40], can individually give factor-of-two differences in the predicted scattering
rates. The WIMP velocity distribution as seen by a terrestrial detector has a bias imposed
by the Earth’s velocity through the halo and its spin. This produces a temporal modulation
of the apparent WIMP velocity distribution, which results in an annual modulation of the
WIMP scattering rate and recoil spectrum, and daily and annual modulations in the directional
distributions.
The scattering cross-section itself has a very wide range of possible values [45]. Different neutralino
models, within MSSM or SUGRA (supergravity), exhibit an enormous range of interaction strengths that
can be pure axial in nature (coupling only to nuclei with non-zero spin), pure coherent (coupling to
all nucleons), or any combination of the two. Figure 6 shows the allowed range of parameter
space for the scattering cross-sections. The plot [79] has been produced using output from the
DarkSusy [58] code, using up to 65 free parameters. Even in this plot some ‘reasonable’ assumptions
have been made in allowing the parameters to vary; Ellis [45] relaxes some of these and, not
surprisingly, finds a wider range of resulting cross-sections. The cross-sections are normalised to
one nucleon; to calculate the total cross-section for a target nucleus with neutrons and
nuclear spin requires a scaling as for the coherent spin-independent part of the
cross-section and for the spin-dependent part. The value of depends on the target
material [60].

Form factor effects, which arise due to the finite size of the nucleus, are significant for the heavier target
nuclei, are different for axial and coherent scattering, and again have uncertainties [47, 106].
Predicted event rates typically range from 10^{–4} to 10 events/day/kg. To achieve sensitivity to
such rare events requires low-background instruments operating in well shielded underground
environments.