Expected scattering rates

5.1 Expected scattering rates

The scattering rate per unit detector volume $Rt$ depends on the local density of dark matter particles $Nw$ ; their velocity distribution relative to a detector in a terrestrial laboratory $nw (v)$ , and the velocity dependent scattering cross-section $σv$ , via the usual equation

where $nt$ is the number density of nuclei of species $t$ in the detector. The local number density of WIMP particles is $∫∞ Nw = 0 nw (v)dv = ρcdm∕mw$ , where $mw$ is the WIMP mass and $ρcdm$ 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 $m w$ in the context of the lightest stable neutralino within minimal MSSM is $20$ to $2 200 GeV ∕c$ [110 , 111 ], and Han and Hempfling [66 ] quote a lower mass limit from LEP data as $∼ 46 GeV ∕c2$ . In the simplest models the dark matter density distribution in the halo of the Galaxy is taken to be a spherical $1∕r2$ (at least for large $r$ ) distribution with a local density, at the position of the solar system, of $3 ∼ 0.3 GeV ∕cm$ . 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 $N$ neutrons and nuclear spin $J$ requires a scaling as $∼ (N ∕2)2$ for the coherent spin-independent part of the cross-section and $2 λsJ(J + 1)$ for the spin-dependent part. The value of $λs$ depends on the target material [60 ].

Figure 6: Total neutralino elastic scattering cross-section normalised to one nucleon for a range of neutralino models within MSSM and mSUGRA, taken from [79 ]. The pink area corresponds to a neutralino in a dominantly bino state, the green bounded area is dominantly higgsino. The cross-section includes both spin-independent and spin-dependent contributions, and in general the spin-dependent part is likely to be larger.

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.