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Direct detection of WIMPs David G.Cerdeno",Anne M.Green 10 a Departamento de Fisica Teorica C-XI,and Instituto de Fisica Teorica UAM-CSIC, Universidad Autonoma de Madrid,Cantoblanco,E-28049 Madrid,Spain bSchool of Physics and Astronomy,University of Nottingham University Park,Nottingham,NG7 2RD,UK October 13,2021 Abstract A generic weakly interacting massive particle(WIMP)is one of the most attractive candidates 'yd-onst to account for the cold dark matter in our Universe,since it would be thermally produced with the correct abundance to account for the observed dark matter density.WIMPs can be searched for directly through their elastic scattering with a target material,and a variety of experiments are currently operating or planned with this aim.In these notes we overview the theoretical calculation of the direct detection rate of WIMPs as well as the different detection signals.We discuss the various ingredients(from particle physics and astrophysics)that enter the calculation and review the theoretical predictions for the direct detection of WIMPs in particle physics models. 6 1 Introduction If the Milky Way's DM halo is composed of WIMPs,then the WIMP flux on the Earth is of order 105(100 GeV/mx)cm-2s-1.This flux is sufficiently large that,even though the WIMPs are weakly interacting,a small but potentially measurable fraction will elastically scatter off nuclei.Direct detection experiments aim to detect WIMPs via the nuclear recoils,caused by WIMP elastic scattering,in dedicated low background detectors [1].More specifically they aim to measure the rate,R,and energies,Er,of the nuclear recoils (and in directional experiments the directions as well). In this chapter we overview the theoretical calculation of the direct detection event rate and the potential direct detection signals.Sec.2 outlines the calculation of the event rate,in- cluding the spin independent and dependent contributions and the hadronic matrix elements. Sec.3 discusses the astrophysical input into the event rate calculation,including the local WIMP velocity distribution and density.In Sec.4 we describe the direction detection signals, specifically the energy,time and direction dependence of the event rate.Finally in Sec.5 we discuss the predicted ranges for the WIMP mass and cross-sections in various particle physics models. "This contribution appeared as chapter 17,pp.347-369,of "Particle Dark Matter:Observations,Models and Searches"edited by Gianfranco Bertone,Copyright 2010 Cambridge University Press.Hardback ISBN 9780521763684,http://cambridge.org/us/catalogue/catalogue.asp?isbn=9780521763684

arXiv:1002.1912v1 [astro-ph.CO] 9 Feb 2010 Direct detection of WIMPs ∗ David G. Cerde˜noa , Anne M. Greenb a Departamento de F´ısica Te´orica C-XI, and Instituto de F´ısica Te´orica UAM-CSIC, Universidad Aut´onoma de Madrid, Cantoblanco, E-28049 Madrid, Spain b School of Physics and Astronomy, University of Nottingham University Park, Nottingham, NG7 2RD, UK October 13, 2021 Abstract A generic weakly interacting massive particle (WIMP) is one of the most attractive candidates to account for the cold dark matter in our Universe, since it would be thermally produced with the correct abundance to account for the observed dark matter density. WIMPs can be searched for directly through their elastic scattering with a target material, and a variety of experiments are currently operating or planned with this aim. In these notes we overview the theoretical calculation of the direct detection rate of WIMPs as well as the different detection signals. We discuss the various ingredients (from particle physics and astrophysics) that enter the calculation and review the theoretical predictions for the direct detection of WIMPs in particle physics models. 1 Introduction If the Milky Way’s DM halo is composed of WIMPs, then the WIMP flux on the Earth is of order 105 (100 GeV/mχ) cm−2 s −1 . This flux is sufficiently large that, even though the WIMPs are weakly interacting, a small but potentially measurable fraction will elastically scatter off nuclei. Direct detection experiments aim to detect WIMPs via the nuclear recoils, caused by WIMP elastic scattering, in dedicated low background detectors [1]. More specifically they aim to measure the rate, R, and energies, ER, of the nuclear recoils (and in directional experiments the directions as well). In this chapter we overview the theoretical calculation of the direct detection event rate and the potential direct detection signals. Sec. 2 outlines the calculation of the event rate, including the spin independent and dependent contributions and the hadronic matrix elements. Sec. 3 discusses the astrophysical input into the event rate calculation, including the local WIMP velocity distribution and density. In Sec. 4 we describe the direction detection signals, specifically the energy, time and direction dependence of the event rate. Finally in Sec. 5 we discuss the predicted ranges for the WIMP mass and cross-sections in various particle physics models. ∗This contribution appeared as chapter 17, pp. 347-369, of “Particle Dark Matter: Observations, Models and Searches” edited by Gianfranco Bertone, Copyright 2010 Cambridge University Press. Hardback ISBN 9780521763684, http://cambridge.org/us/catalogue/catalogue.asp?isbn=9780521763684 1

2 Event rate The differential event rate,usually expressed in terms of counts/kg/day/kev (a quantity referred to as a differential rate unit or dru)for a WIMP with mass mx and a nucleus with mass mN is given by dR =P0 dowN(v,ER)dv, vf(v)dER (1) dER mN mx Jumin where po is the local WIMP density, (v,ER)is the differential cross-section for the ER WIMP-nucleus elastic scattering and f(v)is the WIMP speed distribution in the detector frame normalized to unity. Since the WIMP-nucleon relative speed is of order 100kms-1 the elastic scattering occurs in the extreme non-relativistic limit,and the recoil energy of the nucleon is easily calculated in terms of the scattering angle in the center of mass frame,0* En =2(1-cos0) (2) mN where uN mxmN/(mx+mN)is the WIMP-nucleus reduced mass. The lower limit of the integration over WIMP speeds is given by the minimum WIMP speed which can cause a recoil of energy ER:Umin=V(mN ER)/(2u).The upper limit is formally infinite,however the local escape speed vesc (see Sec.3.2),is the maximum speed in the Galactic rest frame for WIMPs which are gravitationally bound to the Milky Way. The total event rate(per kilogram per day)is found by integrating the differential event rate over all the possible recoil energies: o器e,E (3) mN mx Jvmin where Er is the threshold energy,the smallest recoil energy which the detector is capable of measuring. The WIMP-nucleus differential cross section encodes the particle physics inputs (and as- sociated uncertainties)including the WIMP interaction properties.It depends fundamentally on the WIMP-quark interaction strength,which is calculated from the microscopic description of the model,in terms of an effective Lagrangian describing the interaction of the particular WIMP candidate with quarks and gluons.The resulting cross section is then promoted to a WIMP-nucleon cross section.This entails the use of hadronic matrix elements,which de- scribe the nucleon content in quarks and gluons,and are subject to large uncertainties.In general,the WIMP-nucleus cross section can be separated into a spin-independent(scalar) and a spin-dependent contribution, dowN dowN dowN dER dER (4) SI dER /SD Finally,the total WIMP-nucleus cross section is calculated by adding coherently the above spin and scalar components,using nuclear wave functions.The form factor,F(ER),encodes the dependence on the momentum transfer,q=v2mNER,and accounts for the coherence 2

2 Event rate The differential event rate, usually expressed in terms of counts/kg/day/keV (a quantity referred to as a differential rate unit or dru) for a WIMP with mass mχ and a nucleus with mass mN is given by dR dER = ρ0 mN mχ Z ∞ vmin vf(v) dσW N dER (v, ER) dv , (1) where ρ0 is the local WIMP density, dσWN dER (v, ER) is the differential cross-section for the WIMP-nucleus elastic scattering and f(v) is the WIMP speed distribution in the detector frame normalized to unity. Since the WIMP-nucleon relative speed is of order 100 km−1 s −1 the elastic scattering occurs in the extreme non-relativistic limit, and the recoil energy of the nucleon is easily calculated in terms of the scattering angle in the center of mass frame, θ ∗ ER = µ 2 N v 2 (1 − cos θ ∗ ) mN , (2) where µN = mχmN /(mχ + mN ) is the WIMP-nucleus reduced mass. The lower limit of the integration over WIMP speeds is given by the minimum WIMP speed which can cause a recoil of energy ER: vmin = q (mN ER)/(2µ 2 N ). The upper limit is formally infinite, however the local escape speed vesc (see Sec. 3.2), is the maximum speed in the Galactic rest frame for WIMPs which are gravitationally bound to the Milky Way. The total event rate (per kilogram per day) is found by integrating the differential event rate over all the possible recoil energies: R = Z ∞ ET dER ρ0 mN mχ Z ∞ vmin vf(v) dσW N dER (v, ER) dv , (3) where ET is the threshold energy, the smallest recoil energy which the detector is capable of measuring. The WIMP-nucleus differential cross section encodes the particle physics inputs (and associated uncertainties) including the WIMP interaction properties. It depends fundamentally on the WIMP-quark interaction strength, which is calculated from the microscopic description of the model, in terms of an effective Lagrangian describing the interaction of the particular WIMP candidate with quarks and gluons. The resulting cross section is then promoted to a WIMP-nucleon cross section. This entails the use of hadronic matrix elements, which describe the nucleon content in quarks and gluons, and are subject to large uncertainties. In general, the WIMP-nucleus cross section can be separated into a spin-independent (scalar) and a spin-dependent contribution, dσW N dER = dσW N dER SI + dσW N dER SD . (4) Finally, the total WIMP-nucleus cross section is calculated by adding coherently the above spin and scalar components, using nuclear wave functions. The form factor, F(ER), encodes the dependence on the momentum transfer, q = √ 2mN ER, and accounts for the coherence 2

loss which leads to a suppression in the event rate for heavy WIMPs or nucleons.In general, we can express the differential cross section as dowN dER (oF,(ER)+iDr(En）, mN (5) where oo SI.SD are the spin-independent and -dependent cross sections at zero momentum transfer. The origin of the different contributions is best understood at the microscopic level,by analysing the Lagrangian which describes the WIMP interactions with quarks.The contribu- tions to the spin-independent cross section arise from scalar and vector couplings to quarks, whereas the spin-dependent part of the cross section originates from axial-vector couplings These contributions are characteristic of the particular WIMP candidate (see,e.g.,[2])and can be potentially useful for their discrimination in direct detection experiments. 2.1 Spin-dependent contribution The contributions to the spin-dependent (SD)part of the WIMP-nucleus scattering cross section arise from couplings of the WIMP field to the quark axial current,5q.For example,if the WIMP is a (Dirac or Majorana)fermion,such as the lightest neutralino in supersymmetric models,the Lagrangian can contain the term Ca4(6X)(@nu59). (6) If the WIMP is a spin 1 field,such as in the case of LKP and LTP,the interaction term is slightly different, Ladeupa(Bp8B)(@°sq). (7) In both cases,the nucleus,N,matrix element reads (NIGYYsqIN)=2X (NIJNIN), (8) where the coefficients relate the quark spin matrix elements to the angular momentum of the nucleons.They can be parametrized as y≈AS+△"S (9) where J is the total angular momentum of the nucleus,the quantities Ag"are related to the matrix element of the axial-vector current in a nucleon,(n)=2s,and (Sp.n)=(NISp.nN)is the expectation value of the spin content of the proton or neutron group in the nucleusl.Adding the contributions from the different quarks,it is customary to define ap=∑ △g:an=】 g-△ (10) q=,asV②G IThese quantities can be determined from simple nuclear models.For example,the single-particle shell model assumes the nuclear spin is solely due to the spin of the single unpaired proton or neutron,and therefore vanishes for even nuclei.More accurate results can be obtained by using detailed nuclear calculations. 3

loss which leads to a suppression in the event rate for heavy WIMPs or nucleons. In general, we can express the differential cross section as dσW N dER = mN 2µ 2 N v 2 σ SI 0 F 2 SI (ER) + σ SD 0 F 2 SD(ER) , (5) where σ SI, SD 0 are the spin-independent and -dependent cross sections at zero momentum transfer. The origin of the different contributions is best understood at the microscopic level, by analysing the Lagrangian which describes the WIMP interactions with quarks. The contributions to the spin-independent cross section arise from scalar and vector couplings to quarks, whereas the spin-dependent part of the cross section originates from axial-vector couplings. These contributions are characteristic of the particular WIMP candidate (see, e.g., [2]) and can be potentially useful for their discrimination in direct detection experiments. 2.1 Spin-dependent contribution The contributions to the spin-dependent (SD) part of the WIMP-nucleus scattering cross section arise from couplings of the WIMP field to the quark axial current, ¯qγµγ5q. For example, if the WIMP is a (Dirac or Majorana) fermion, such as the lightest neutralino in supersymmetric models, the Lagrangian can contain the term L ⊃ α A q ( ¯χγµ γ5χ)(¯qγµγ5q). (6) If the WIMP is a spin 1 field, such as in the case of LKP and LTP, the interaction term is slightly different, L ⊃ α A q ǫ µνρσ(Bρ ↔ ∂µ Bν )(¯qγσ γ5q). (7) In both cases, the nucleus, N, matrix element reads hN|qγ¯ µγ5q|Ni = 2λ N q hN|JN |Ni, (8) where the coefficients λ N q relate the quark spin matrix elements to the angular momentum of the nucleons. They can be parametrized as λ N q ≃ ∆ (p) q hSpi + ∆(n) q hSni J , (9) where J is the total angular momentum of the nucleus, the quantities ∆q n are related to the matrix element of the axial-vector current in a nucleon, hn|qγ¯ µγ5q|ni = 2s (n) µ ∆ (n) q , and hSp,ni = hN|Sp,n|Ni is the expectation value of the spin content of the proton or neutron group in the nucleus1 . Adding the contributions from the different quarks, it is customary to define ap = X q=u,d,s α A q √ 2GF ∆ p q ; an = X q=u,d,s α A q √ 2GF ∆ n q , (10) 1These quantities can be determined from simple nuclear models. For example, the single-particle shell model assumes the nuclear spin is solely due to the spin of the single unpaired proton or neutron, and therefore vanishes for even nuclei. More accurate results can be obtained by using detailed nuclear calculations. 3

interaction of the WIMP and the gluon scalar density in the nucleon,with fc =1- They are determined experimentally, fu=0.020±0.004，fa=0.026±0.005，fs=0.118±0.062， (19) with fru=frd,fra=fr,and frs=frs The uncertainties in these quantities,among which the most important is that on frs,mainly stem from the determination of the m-nucleon sigma term. The vector coupling (which is present,for example,in the case of a Dirac fermion but vanishes for Majorana particles)gives rise to an extra contribution.Interestingly,the sea quarks and gluons do not contribute to the vector current.Only valence quarks contribute, leading to the following expression 00= B路 (20) 64π with BN=av(A+Z)+a(2A-Z) (21) Thus,for a general WIMP with both scalar and vector interactions,the spin-independent contribution to the scattering cross section would read dowN 2mN dER sI [Zf+(A-Z)f")2+ B 256 F2(ER) (22) In most cases the WIMP coupling to neutrons and protons is very similar,fp fn,and therefore the scalar contribution can be approximated by dowN 、dEr/sI 2mN A((ER). (23) TU2 The spin-independent contribution basically scales as the square of the number of nucleons (A2),whereas the spin-dependent one is proportional to a function of the nuclear angular momentum,(J+1)/J.Although in general both have to be taken into account,the scalar component dominates for heavy targets (A 20),which is the case for most experiments (usually based on targets with heavy nuclei such as Silicon,Germanium,Iodine or Xenon). Nevertheless,dedicated experiments exist that are also sensitive to the SD WIMP coupling through the choice of targets with a large nuclear angular momentum. As we have seen,the WIMP direct detection rate depends on both astrophysical input (the local DM density and velocity distribution,in the lab frame)and particle physics input (nuclear form factors and interaction cross-sections,which depend on the theoretical frame- work in which the WIMP candidate arises).We will discuss these inputs in more detail in Secs.3 and 5 respectively. 2.3 Hadronic Matrix Elements The effect of uncertainties in the hadronic matrix elements has been studied in detail for the specific case of neutralino dark matter [5-9].Concerning the SI cross section,the quantities f in Eq.(19)can be parametrized in terms of the nucleon sigma term,N,(see in this respect,e.g.,Refs.7;9])which,in terms of the u and d quark masses reads N(m+m)(Nlu+d), (24) 5

interaction of the WIMP and the gluon scalar density in the nucleon, with f p T G = 1 − P q=u,d,s f p T q. They are determined experimentally, f p T u = 0.020 ± 0.004, f p T d = 0.026 ± 0.005, f p T s = 0.118 ± 0.062, (19) with f n T u = f p T d, f n T d = f p T u, and f n T s = f p T s. The uncertainties in these quantities, among which the most important is that on fT s, mainly stem from the determination of the π-nucleon sigma term. The vector coupling (which is present, for example, in the case of a Dirac fermion but vanishes for Majorana particles) gives rise to an extra contribution. Interestingly, the sea quarks and gluons do not contribute to the vector current. Only valence quarks contribute, leading to the following expression σ0 = µ 2 N B2 N 64π , (20) with BN ≡ α V u (A + Z) + α V d (2A − Z). (21) Thus, for a general WIMP with both scalar and vector interactions, the spin-independent contribution to the scattering cross section would read dσW N dER SI = 2 mN πv2 " [Zf p + (A − Z)f n ] 2 + B2 N 256# F 2 (ER). (22) In most cases the WIMP coupling to neutrons and protons is very similar, f p ≈ f n , and therefore the scalar contribution can be approximated by dσW N dER SI = 2 mN A2 (f p ) 2 πv2 F 2 (ER). (23) The spin-independent contribution basically scales as the square of the number of nucleons (A2 ), whereas the spin-dependent one is proportional to a function of the nuclear angular momentum, (J + 1)/J. Although in general both have to be taken into account, the scalar component dominates for heavy targets (A > 20), which is the case for most experiments (usually based on targets with heavy nuclei such as Silicon, Germanium, Iodine or Xenon). Nevertheless, dedicated experiments exist that are also sensitive to the SD WIMP coupling through the choice of targets with a large nuclear angular momentum. As we have seen, the WIMP direct detection rate depends on both astrophysical input (the local DM density and velocity distribution, in the lab frame) and particle physics input (nuclear form factors and interaction cross-sections, which depend on the theoretical framework in which the WIMP candidate arises). We will discuss these inputs in more detail in Secs. 3 and 5 respectively. 2.3 Hadronic Matrix Elements The effect of uncertainties in the hadronic matrix elements has been studied in detail for the specific case of neutralino dark matter [5–9]. Concerning the SI cross section, the quantities f p T q in Eq.(19) can be parametrized in terms of the π nucleon sigma term, ΣπN , (see in this respect, e.g., Refs.[7; 9]) which, in terms of the u and d quark masses reads ΣπN = 1 2 (mu + md)hN|uu¯ + ¯dd|Ni , (24) 5

and is related to the strange quark scalar density in the nucleon.The largest source of uncertainty in fro stems from the determination of this quantity,for which the current data implies EN =(64+8)MeV [10],which translates into a variation of a factor 4 in frs.Notice that in general the WIMP interaction with strange quarks would be the leading contribution to the SI cross section,due to its larger Yukawa coupling.In this case,oo is roughly proportional to frs2 and the above uncertainty in the strange quark content leads to a variation of more than one order of magnitude in the resulting SI cross section [7;9]). Similarly,for the SD cross section the uncertainties in the strange spin contribution As are the dominant contribution to the error in 0o.However,in the case of the neutralino, this can imply a correction of as much as a factor 2 in the resulting cross section [9],being therefore much smaller than the above uncertainty for the SI cross section.It should be emphasized,however,that uncertainties in the determination of the spin form factors S(g) would also affect the theoretical predictions for the dark matter detection rate. 3 Astrophysics input 3.1 Local DM density The differential event rate is directly proportional to the local WIMP density,po =p(r= Ro)where Ro =(8.0+0.5)kpc [11]is the solar radius.Any observational uncertainty in Po therefore translates directly into in an uncertainty in the event rate and the inferred constraints on,or measurements of,the scattering cross-sections. Exclusion limits are traditionally calculated assuming a canonical local WIMP density, Po =0.3GeVcm-3.The local WIMP density is calculated by applying observational con- straints (including measurements of the rotation curve)to models of the Milky Way and the values obtained can vary by a factor or order 2 depending on the models used [12-15]. A recent study [16],using spherical halo models with a cusp (p o r(r)-as r-0)finds p0=(0.30±0.05)GeV cm-3 3.2 Speed distribution The standard halo model,conventionally used in calculations of exclusion limits and signals, has an isotropic,Gaussian velocity distribution (often referred to as Maxwellian) 1 f()= v/2 -exp √2m石 2a2 (25) The speed dispersion is related to the local circular speed by a =v3/2vc and ve=(220+ 20)kms-1 [17](see Sec.3.3)so that o270kms-1.This velocity distribution corresponds to an isotropic singular isothermal sphere with density profile p(r)r-2.The isothermal sphere is simple,and not unreasonable as a first approximation,however it is unlikely to be an accurate model of the actual density and velocity distribution of the Milky Way.Observations and numerical simulations(see chapter 1)indicate that dark matter halos do not have a 1/r2 density profile and are (to some extent at least)triaxial and anisotropic. If the velocity distribution is isotropic there is a one to one relation between f(v)and the the spherically symmetric density profile given by Eddington's formula [18],see Refs.[19;20]. In general the steady state phase-space distribution of a collection of collisionless particles 6

and is related to the strange quark scalar density in the nucleon. The largest source of uncertainty in f p T q stems from the determination of this quantity, for which the current data implies ΣπN = (64 ± 8) MeV [10], which translates into a variation of a factor 4 in fT s. Notice that in general the WIMP interaction with strange quarks would be the leading contribution to the SI cross section, due to its larger Yukawa coupling. In this case, σ0 is roughly proportional to fT s 2 and the above uncertainty in the strange quark content leads to a variation of more than one order of magnitude in the resulting SI cross section [7; 9]). Similarly, for the SD cross section the uncertainties in the strange spin contribution ∆s are the dominant contribution to the error in σ0. However, in the case of the neutralino, this can imply a correction of as much as a factor 2 in the resulting cross section [9], being therefore much smaller than the above uncertainty for the SI cross section. It should be emphasized, however, that uncertainties in the determination of the spin form factors S(q) would also affect the theoretical predictions for the dark matter detection rate. 3 Astrophysics input 3.1 Local DM density The differential event rate is directly proportional to the local WIMP density, ρ0 ≡ ρ(r = R0) where R0 = (8.0 ± 0.5) kpc [11] is the solar radius. Any observational uncertainty in ρ0 therefore translates directly into in an uncertainty in the event rate and the inferred constraints on, or measurements of, the scattering cross-sections. Exclusion limits are traditionally calculated assuming a canonical local WIMP density, ρ0 = 0.3 GeV cm−3 . The local WIMP density is calculated by applying observational constraints (including measurements of the rotation curve) to models of the Milky Way and the values obtained can vary by a factor or order 2 depending on the models used [12–15]. A recent study [16], using spherical halo models with a cusp (ρ ∝ r(r) −α as r → 0) finds ρ0 = (0.30 ± 0.05) GeV cm−3 . 3.2 Speed distribution The standard halo model, conventionally used in calculations of exclusion limits and signals, has an isotropic, Gaussian velocity distribution (often referred to as Maxwellian) f(v) = 1 √ 2πσ exp − |v| 2 2σ 2 ! . (25) The speed dispersion is related to the local circular speed by σ = p 3/2vc and vc = (220 ± 20) km s−1 [17] (see Sec.3.3) so that σ ≈ 270 km s−1 . This velocity distribution corresponds to an isotropic singular isothermal sphere with density profile ρ(r) ∝ r −2 . The isothermal sphere is simple, and not unreasonable as a first approximation, however it is unlikely to be an accurate model of the actual density and velocity distribution of the Milky Way. Observations and numerical simulations (see chapter 1) indicate that dark matter halos do not have a 1/r2 density profile and are (to some extent at least) triaxial and anisotropic. If the velocity distribution is isotropic there is a one to one relation between f(v) and the the spherically symmetric density profile given by Eddington’s formula [18], see Refs. [19; 20]. In general the steady state phase-space distribution of a collection of collisionless particles 6

is given by the collisionless Boltzmann equation and the velocity dispersions of the system are calculated via the Jean's equations (e.g.Ref.[21]).Solving the Jean's equations requires assumptions to be made,and therefore even for a specific density distribution the solution is not unique.Several specific models have been used in the context of WIMP direct detection signals.The logarithmic ellipsoidal model [22]is the simplest triaxial generalisation of the isothermal sphere and has a velocity distribution which is a multi-variate Gaussian.Osipkov- Merritt models [23;24]are spherically symmetric with radially dependent anisotropic velocity distributions.Fitting functions for the speed distributions in these models are available,for a selection of density profiles,in Ref.[25].Velocity distributions have also been extracted from cosmological simulations,with both multi-variate Gaussian [26]and Tsallis [27]distri- butions [28]being advocated as fitting functions.While it is not known whether any of these models provide a good approximation to the real local velocity distribution function,the models are none the less useful for assessing the uncertainties in the direct detection signals. Particles with speed,in the Galactic rest frame,greater than the local escape speed, vesc=√2Φ(Ro)I whereΦ(r）is the potential,are not gravitationally bound.Many of the models used,in particular the standard halo model,formally extend to infinite radii and therefore their speed distribution has to be truncated at vese 'by hand'(see e.g.Ref.[29]). The standard value for the escape speed is vese=650kms-1.A recent analysis,using high velocity stars from the RAVE survey,finds 498kms-<vese 608kms-1 with a median likelihood of 544 kms-1 [30]. In Sec.4 we discuss the impact of uncertainty in the speed distribution on the direct detection signals. 3.3 Earth's motion The WIMP speed distribution in the detector rest frame is calculated by carrying out,a time dependent,Galilean transformation:v=v+ve(t).The Earth's motion relative to the Galactic rest frame,ve(t),is made up of three components:the motion of the Local Standard of Rest (LSR),the Sun's peculiar motion with respect to the LSR,ve,and the Earth's orbit about the Sun,vorb If the Milky Way is axisymmetric then the motion of the LSR is given by the local circular velocity (0,ve,0),where vc =220kms-1 is the standard value.Kerr and Lynden-Bell found, by combining a large number of independent measurements,ve=(222+20)kms-1 [17].A more recent determination,using the proper motions of Cepheids measured by Hipparcos [31], is broadly consistent:vc=(218+7)kms-1(Ro/8kpc). The Sun's peculiar motion,determined using the parallaxes and proper motions of stars in the solar neighbourhood from the Hipparcos catalogue,is ve =(10.0+0.4,5.2+0.6,7.2+ 0.4)kms-1 [32]in Galactic co-ordinates(where x points towards the Galactic center,y is the direction of Galactic rotation and z towards the North Galactic Pole). A relatively simple,and reasonably accurate,expression for the Earth's motion about the Sun can be found by ignoring the ellipticity of the Earth's orbit and the non-uniform motion of the Sun in right ascension [33]:verb =veleisin A(t)-e2 cos A(t)]where ve 29.8kms-1 is the orbital speed of the Earth,A(t)=2(t-0.218)is the Sun's ecliptic longitude(with t in years)and e1(2)are unit vectors in the direction of the Sun at the Spring equinox (Summer solstice).In Galactic co-ordinates e=(-0.0670,0.4927,-0.8676)and e2=(-0.9931,-0.1170,0.01032).More accurate expressions can be found in Ref.[34]. 7

is given by the collisionless Boltzmann equation and the velocity dispersions of the system are calculated via the Jean’s equations (e.g. Ref. [21]). Solving the Jean’s equations requires assumptions to be made, and therefore even for a specific density distribution the solution is not unique. Several specific models have been used in the context of WIMP direct detection signals. The logarithmic ellipsoidal model [22] is the simplest triaxial generalisation of the isothermal sphere and has a velocity distribution which is a multi-variate Gaussian. OsipkovMerritt models [23; 24] are spherically symmetric with radially dependent anisotropic velocity distributions. Fitting functions for the speed distributions in these models are available, for a selection of density profiles, in Ref. [25]. Velocity distributions have also been extracted from cosmological simulations, with both multi-variate Gaussian [26] and Tsallis [27] distributions [28] being advocated as fitting functions. While it is not known whether any of these models provide a good approximation to the real local velocity distribution function, the models are none the less useful for assessing the uncertainties in the direct detection signals. Particles with speed, in the Galactic rest frame, greater than the local escape speed, vesc = p 2|Φ(R0)| where Φ(r) is the potential, are not gravitationally bound. Many of the models used, in particular the standard halo model, formally extend to infinite radii and therefore their speed distribution has to be truncated at vesc ‘by hand’ (see e.g. Ref. [29]). The standard value for the escape speed is vesc = 650 km s−1 . A recent analysis, using high velocity stars from the RAVE survey, finds 498km s−1 < vesc < 608 km s−1 with a median likelihood of 544 km s−1 [30]. In Sec. 4 we discuss the impact of uncertainty in the speed distribution on the direct detection signals. 3.3 Earth’s motion The WIMP speed distribution in the detector rest frame is calculated by carrying out, a time dependent, Galilean transformation: v → v˜ = v + ve(t). The Earth’s motion relative to the Galactic rest frame, ve(t), is made up of three components: the motion of the Local Standard of Rest (LSR), the Sun’s peculiar motion with respect to the LSR, v p ⊙, and the Earth’s orbit about the Sun, v orb e . If the Milky Way is axisymmetric then the motion of the LSR is given by the local circular velocity (0, vc, 0), where vc = 220 km s−1 is the standard value. Kerr and Lynden-Bell found, by combining a large number of independent measurements, vc = (222 ± 20) km s−1 [17]. A more recent determination, using the proper motions of Cepheids measured by Hipparcos [31], is broadly consistent: vc = (218 ± 7) km s−1 (R0/8 kpc). The Sun’s peculiar motion, determined using the parallaxes and proper motions of stars in the solar neighbourhood from the Hipparcos catalogue, is v p ⊙ = (10.0 ± 0.4, 5.2 ± 0.6, 7.2 ± 0.4) km s−1 [32] in Galactic co-ordinates (where x points towards the Galactic center, y is the direction of Galactic rotation and z towards the North Galactic Pole). A relatively simple, and reasonably accurate, expression for the Earth’s motion about the Sun can be found by ignoring the ellipticity of the Earth’s orbit and the non-uniform motion of the Sun in right ascension [33]: v orb e = ve[e1 sin λ(t) − e2 cos λ(t)] where ve = 29.8 km s−1 is the orbital speed of the Earth, λ(t) = 2π(t − 0.218) is the Sun’s ecliptic longitude (with t in years) and e1(2) are unit vectors in the direction of the Sun at the Spring equinox (Summer solstice). In Galactic co-ordinates e1 = (−0.0670, 0.4927, −0.8676) and e2 = (−0.9931, −0.1170, 0.01032). More accurate expressions can be found in Ref. [34]. 7

The main characteristics of the WIMP signals can be found using only the motion of the LSR,and for the time dependence the component of the Earth's orbital velocity in that direction.However accurate calculations,for instance for comparison with data,require all the components described above to be taken into account. 3.4 Ultra-fine structure Most of the WIMP velocity distributions discussed in Sec.3.2 are derived by solving the collisionless Boltzmann equation,which assumes that the phase space distribution has reached a steady state.However this may not be a good assumption for the Milky Way;structure formation in CDM cosmologies occurs hierarchically and the relevant dynamical timescales for the Milky Way are not many orders of magnitude smaller than the age of the Universe. Both astronomical observations and numerical simulations(due to their finite resolution) typically probe the dark matter distribution on kpc scales.Direct detection experiments probe the DM distribution on sub milli-pc scales (the Earth's speed with respect to the Galactic rest frame is 0.2mpcyr-1).It has been argued (see e.g.Refs.[35-37])that on these scales the DM may not have yet reached a steady state and could have a non-smooth phase-space distribution.On the other hand it has been argued that the rapid decrease in density of streams evolving in a realistic,ellipsoidal,Galactic potential means that there will be a large number of overlapping streams in the solar neighbourhood producing an effectively smooth DM distribution 38]. If the local DM distribution consists of a small number of streams,rather than a smooth distribution,then there would be significant changes in the signals which we will discuss in Sec.4.This is currently an open issue;directly calculating the DM distribution on the scales probed by direct detection experiments is a difficult and unresolved problem. It has been suggested that a tidal stream from the Sagittarius (Sgr)dwarf galaxy,which is in the process of being disrupted,passes through the solar neighbourhood with the asso- ciated DM potentially producing distinctive signals in direct detection experiments [39;40]. Subsequent numerical simulations of the disruption of Sgr along with observational searches for local streams of stars suggest that the Sgr stream does not in fact pass through the solar neighbourhood (see e.g.Ref.[41]).None the less the calculations of the resulting WIMP signals in Refs.[39;40]provide a useful illustration of the qualitative effects of streams. 4 Signals We have already seen in Sec.2 that the recoil rate is energy dependent due to the kinematics of elastic scattering,combined with the WIMP speed distribution.Due to the motion of the Earth with respect to the Galactic rest frame the recoil rate is also both time and direction dependent.In this section we examine the energy,time and direction dependence of the recoil rate and the resulting WIMP signals.In each case we first focus on the signal expected for the standard halo model,with a Maxwellian velocity distribution,before discussing the effect on the signal of changes in the WIMP velocity distribution. 8

The main characteristics of the WIMP signals can be found using only the motion of the LSR, and for the time dependence the component of the Earth’s orbital velocity in that direction. However accurate calculations, for instance for comparison with data, require all the components described above to be taken into account. 3.4 Ultra-fine structure Most of the WIMP velocity distributions discussed in Sec. 3.2 are derived by solving the collisionless Boltzmann equation, which assumes that the phase space distribution has reached a steady state. However this may not be a good assumption for the Milky Way; structure formation in CDM cosmologies occurs hierarchically and the relevant dynamical timescales for the Milky Way are not many orders of magnitude smaller than the age of the Universe. Both astronomical observations and numerical simulations (due to their finite resolution) typically probe the dark matter distribution on ∼ kpc scales. Direct detection experiments probe the DM distribution on sub milli-pc scales (the Earth’s speed with respect to the Galactic rest frame is ≈ 0.2 mpc yr−1 ). It has been argued (see e.g. Refs. [35–37]) that on these scales the DM may not have yet reached a steady state and could have a non-smooth phase-space distribution. On the other hand it has been argued that the rapid decrease in density of streams evolving in a realistic, ellipsoidal, Galactic potential means that there will be a large number of overlapping streams in the solar neighbourhood producing an effectively smooth DM distribution [38]. If the local DM distribution consists of a small number of streams, rather than a smooth distribution, then there would be significant changes in the signals which we will discuss in Sec. 4. This is currently an open issue; directly calculating the DM distribution on the scales probed by direct detection experiments is a difficult and unresolved problem. It has been suggested that a tidal stream from the Sagittarius (Sgr) dwarf galaxy, which is in the process of being disrupted, passes through the solar neighbourhood with the associated DM potentially producing distinctive signals in direct detection experiments [39; 40]. Subsequent numerical simulations of the disruption of Sgr along with observational searches for local streams of stars suggest that the Sgr stream does not in fact pass through the solar neighbourhood (see e.g. Ref. [41]). None the less the calculations of the resulting WIMP signals in Refs. [39; 40] provide a useful illustration of the qualitative effects of streams. 4 Signals We have already seen in Sec. 2 that the recoil rate is energy dependent due to the kinematics of elastic scattering, combined with the WIMP speed distribution. Due to the motion of the Earth with respect to the Galactic rest frame the recoil rate is also both time and direction dependent. In this section we examine the energy, time and direction dependence of the recoil rate and the resulting WIMP signals. In each case we first focus on the signal expected for the standard halo model, with a Maxwellian velocity distribution, before discussing the effect on the signal of changes in the WIMP velocity distribution. 8

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. 9

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 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 differential 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