Determining the WIMP mass from a single direct detection experiment, a more detailed study
aa r X i v : . [ h e p - ph ] J un Determining the WIMP mass from a single directdetection experiment, a more detailed study
Anne M. Green † † School of Physics and Astronomy, University of Nottingham, University Park,Nottingham, NG7 2RD, UKE-mail: [email protected] , Abstract.
The energy spectrum of nuclear recoils in Weakly Interacting MassiveParticle (WIMP) direct detection experiments depends on the underlying WIMP mass.We study how the accuracy with which the WIMP mass could be determined by a singledirect detection experiment depends on the detector configuration and the WIMPproperties. We investigate the effects of varying the underlying WIMP mass and cross-section, the detector target nucleus, exposure, energy threshold and maximum energy,the local circular speed and the background event rate and spectrum. The numberof events observed is directly proportional to both the exposure and the cross-section,therefore these quantities have the greatest bearing on the accuracy of the WIMPmass determination. The relative capabilities of different detectors to determine theWIMP mass depend not only on the WIMP and target masses, but also on their energythresholds. The WIMP and target mass dependence of the characteristic energy scaleof the recoil spectrum suggests that heavy targets will be able to measure the massof a heavy WIMP more accurately. We find, however, that the rapid decrease of thenuclear form factor with increasing momentum transfer which occurs for heavy nucleimeans that this is in fact not the case. Uncertainty in the local circular speed andnon-negligible background would both lead to systematic errors in the WIMP massdetermination. For deviations of ±
20 km s − in the underlying value of the circularspeed the systematic error is of order 10%, increasing with increasing WIMP mass.This error can be reduced by also fitting for the circular speed. With a single detector itwill be difficult to disentangle a WIMP signal (and the WIMP mass) from background ifthe background spectrum has a similar shape to the WIMP spectrum (i.e. exponentialbackground, or flat background and a heavy WIMP). Keywords : dark matter, dark matter detectors
1. Introduction
Cosmological observations indicate that the majority of the matter in the Universe isdark and non-baryonic (e.g. Ref. [1]). Weakly Interacting Massive Particles (WIMPs)are one of the leading cold dark matter candidates, and supersymmetry provides aconcrete, well-motivated WIMP candidate in the form of the lightest neutralino (e.g.Ref. [2, 3]). The direct detection of WIMPs in the lab [4] (see Ref. [5] for a review ofcurrent and future experiments) would not only directly confirm the existence of darkmatter but would also allow us to probe the WIMP properties, as the shape of thedifferential event rate depends on the WIMP mass [6, 7, 8, 9, 10, 11]. Constraintson, or measurements of, the WIMP mass and elastic scattering cross-section willbe complementary to the information derived from collider and indirect detectionexperiments [12].In paper I [10], see also Refs. [9, 11], we examined the accuracy with which afuture SuperCDMS [13] like direct detection experiment would be able to measurethe WIMP mass, given a positive detection. In this paper we revisit that analysis,studying in more detail the dependence of WIMP mass limits on the detector capabilities(including threshold energy, exposure, maximum energy and target nucleus), the WIMPproperties (mass and cross-section), the local circular speed and the effects of non-zerobackgrounds. We outline the calculation of the differential event rate and the MonteCarlo simulations in Sec. 2 (see paper I [10] for further details), present the results inSec. 3 and conclude with discussion in Sec. 4.
2. Method
The differential event rate, assuming spin-independent coupling, is given by (seee.g. [2, 6]): d R d E ( E ) = σ p ρ χ µ χ m χ A F ( E ) F ( E ) , (1)where ρ χ is the local WIMP density, σ p the WIMP scattering cross section on theproton, µ p χ = ( m p m χ ) / ( m p + m χ ) the WIMP-proton reduced mass, A and F ( E ) themass number and form factor of the target nucleus respectively and E is the recoilenergy of the detector nucleus. We use the Helm form factor [14]. The dependence onthe WIMP velocity distribution is encoded in F ( E ), which is defined as F ( E ) = h Z ∞ v min f E ( v, t ) v d v i , (2)where f E ( v, t ) is the (time dependent) WIMP speed distribution in the rest frame ofthe detector, normalized to unity and h .. i denotes time averaging. The WIMP speeddistribution is calculated from the velocity distribution in the rest frame of the Galaxy, f G ( v ), via Galilean transformation: v → ˜ v = v + v E ( t ) where v E ( t ) is the Earth’svelocity with respect to the Galactic rest frame [6]. The lower limit of the integral, v min ,is the minimum WIMP speed that can cause a recoil of energy E : v min = Em A µ χ ! / , (3)where m A is the atomic mass of the detector nuclei and µ A χ the WIMP-nucleon reducedmass. We use the ‘standard halo model’, an isotropic isothermal sphere, for which thelocal WIMP velocity distribution, in the Galactic rest frame, is Maxwellian f G ( v ) = N (cid:2) exp (cid:0) −| v | /v (cid:1) − exp (cid:0) − v /v (cid:1)(cid:3) | v | < v esc , (4) f G ( v ) = 0 | v | > v esc , (5)where N is a normalization factor and v c = 220 ±
20 km s − [15] and v esc ≈
540 km s − [16]are the local circular and escape speeds respectively. If the ultra-local WIMPdistribution is smooth, then the uncertainties in the detailed shape of the local velocitydistribution lead to relatively small changes in the shape of the differential eventrate [17, 18]. Consequently there is a relatively small, [ O (5%)], systematic uncertaintyin the WIMP mass [10]. We caution that the assumption of a smooth ultra-local WIMPdistribution may, however, not be valid on the sub milli-pc scales probed by directdetection experiments (e.g. Ref. [19] but see also Ref. [20] for arguments that the ultra-local WIMP distribution consists of a large number of streams, and is hence effectivelysmooth).As shown by Lewin and Smith [6], see also Paper I [10], the differential event ratecan, to a reasonable approximation, be written asd R d E ( E ) = (cid:18) d R d E (cid:19) exp (cid:18) − EE R (cid:19) F ( E ) . (6)The event rate in the E → R/ d E ) , and E R , the characteristic energyscale, are given by (cid:18) d R d E (cid:19) = c σ p ρ χ √ πµ χ m χ v c A , (7)and E R = c E R µ χ v c m A , (8)respectively, where c and c E R are constants of order unity which are required when theEarth’s velocity and the Galactic escape speed are taken into account and are determinedby fitting to the energy spectrum calculated using the full expression, eq. (1). The exactvalues of these constants depend on the target nucleus, the energy threshold and theGalactic escape speed. For a Ge detector with energy threshold E th = 0 keV, c ≈ . c E R ≈ .
72, with a weak dependence on the WIMP mass [10]. For the majority ofour calculations we will use the accurate expression, eq. (1), however in Sec. 3.3 where weconsider varying v c we will use the fitting function, eq. (6), as it is not computationallyfeasible to carry out the full calculation in the likelihood analysis in this case. Figure 1.
From top panel to bottom: the expected number of events, N exp , for a E = 3 × kg day exposure (with σ p = 10 − pb), the characteristic energy scale, E R ,and the variation of the characteristic energy scale with mass, dE R / d m χ , as a functionof WIMP mass for a Ge detector (solid line) and a Xe detector (dashed). For N exp thethe energy thresholds of the current CDMS II [21] and Xenon10 [22] experiments havebeen used: E th = 10 keV and 4 . The characteristic energy, E R , depends on the WIMP mass, m χ , and the mass ofthe target nuclei, m A . For WIMPs which are light compared with the target nuclei, m χ ≪ m A , E R ∝ m χ /m A , while for heavy WIMPs, m χ ≫ m A , E R ∼ const. In words,for light WIMPs the energy spectrum is strongly dependent on the WIMP mass whilefor heavy WIMPs the dependence on the WIMP mass is far weaker. Consequentlyit will be easier to measure the mass of light (compared with the target nuclei) thanheavy WIMPs. Since the experiments which currently have the greatest sensitivity arecomposed of Ge and Xe (CDMS II [21] and Xenon10 [22] respectively) we focus on thesetargets. Fig. 1 shows the dependence of the characteristic energy, E R , and d E R / d m χ onthe WIMP mass for Ge and Xe. For m χ < ( > ) ∼
50 GeV E R varies more strongly with m χ for Ge (Xe), reflecting the asymptotic WIMP mass dependences of the expression for E R . This suggests that light (heavy) target nuclei will be better suited to determiningthe mass of light (heavy) WIMPs (see however Sec. 3.2). The detector energy thresholdwill also come into play, in particular for small exposures, as the expected number ofevents depends on the energy threshold. The expected number of events, N exp , for anexposure of E = 3 × kg day is also shown in fig. 1 as a function of WIMP mass for Geand Xe detectors with E th = 10 keV [21, 13] and 4 . N exp is larger (by a factor of order ∼ m χ ≈
50 GeV) for Xe than forGe. This indicates that the relative capabilities of detectors to determine the WIMPmass will depend not only on the WIMP and target masses, but also on their energythresholds.
We use Monte Carlo simulations to examine, for a range of detector configurations andinput WIMP masses, how well the WIMP mass could be determined from the energiesof observed WIMP nuclear recoil events.We estimate the WIMP mass and cross-section by maximizing the extendedlikelihood function (which takes into account the fact that the number of events observedin a given experiment is not fixed), e.g. Ref. [23]: L = λ N expt exp ( − λ ) N expt ! Π N expt i =1 f ( E i ) . (9)Here N expt is the number of events observed, E i ( i = 1 , ..., N expt ) are the energiesof the events observed, f ( E ) is the normalized differential event rate and λ = E R ∞ E th (d R/ d E ) d E is the mean number of events where E is the detector exposure (whichhas dimensions of mass times time) and E th is the threshold energy. We calculate theprobability distribution of the maximum likelihood estimators of the WIMP mass andcross-section, for each detector configuration and input WIMP mass, by simulating 10 experiments. We first calculate the expected number of events, λ in , from the inputenergy spectrum. The actual number of events for a given experiment, N expt , is drawnfrom a Poisson distribution with mean λ in . We Monte Carlo generate N expt eventsfrom the input energy spectrum, from which the maximum likelihood mass and cross-section for that experiment are calculated. Finally we find the (two-sided) 68% and 95%confidence limits on the WIMP mass from the maximum likelihood masses.
3. Results
In Sec. 3.1 we investigate the mass limits for a a SuperCDMS like Ge detector [13] andtheir dependence on the detector energy threshold, maximum energy and exposure, andthe WIMP cross-section. In Sec. 3.2 we compare the Ge mass limits with those for aXe detector, before examining the effects of uncertainties in the local circular speed andnon-negligible background in Secs. 3.3 and 3.4 respectively.
We begin, as in paper I [10], by looking at a SuperCDMS like detector [13], composedof Ge with a nuclear recoil energy threshold of E th = 10 keV. We assume that the Figure 2.
Limits on the WIMP mass, m lim χ , as a function of the input WIMPmass, m in χ , for the benchmark Ge detector ( E th = 10 keV, perfect energy resolution,no upper limit on energy of events detected and zero background) for exposures E = 3 × , × and 3 × kg day and input cross-section σ p = 10 − pb. The dot-dashed line is the input mass and the solid (dotted) lines are the 95% (68%) confidencelimits. background event rate is negligible, as is expected for this experiment located atSNOLab [13], and that the energy resolution is perfect ‡ . For simplicity we assume thatthe nuclear recoil detection efficiency is independent of energy. The energy dependenceof the efficiency of the current CDMS II experiment is relatively small (it increasesfrom ∼ .
22 at E = E th = 10 keV to ∼ .
30 at E = 15 keV and then remains roughlyconstant). For further discussion of these assumptions see Ref. [10].We consider fiducial efficiency weighted exposures § E = 3 × , 3 × and3 × kg day which correspond, roughly, to a detector with mass equal to that ofthe 3 proposed phases of SuperCDMS taking data for a year with a ∼
50% detectionefficiency k . ‡ Gaussian energy resolution, with full width at half maximum of order 1 keV [13], does not affect theWIMP parameters extracted from the energy spectrum [10]. § For brevity we subsequently refer to this as simply the exposure. k The 50% detection efficiency was chosen based on Ref [24]. The more recent CDMS II analysis [21]
Figure 3.
As fig. 2, but with the fractional deviation of the WIMP mass limits fromthe input mass, ( m lim χ − m in χ ) /m in χ , plotted. We use fiducial values for the detector energy threshold and WIMP-proton cross-section of E th = 10 keV and σ p = 10 − pb but later, in Secs. 3.1.1 and 3.1.4 respectively,consider a range of values for these parameters. Note that this fiducial cross-sectionis, given the recent limits from the CDMS II [21] and Xenon10 [22] experiments, anorder of magnitude smaller than that used in Paper I [10]. We use the standard valuesfor the local circular speed and WIMP density, v c = 220 km s − and ρ χ = 0 . − respectively. We examine the effect of uncertainties in the local circular speed in Sec. 3.3.The local WIMP density only affects the amplitude, and not the shape, of the energyspectrum. Therefore it only affects the WIMP mass determination indirectly, throughthe number of events detected. We note that the limits from our idealized simulateddetector are likely to be better than those achievable in reality by a real detector.In fig. 2 we plot the 68% and 95% confidence limits on the WIMP mass, m lim χ , for thefiducial detector configuration as a function of the input WIMP mass, m in χ . Here, andthroughout, the limits terminate when there is a >
5% probability that an experimentwill detect no events. Fig. 3 uses the same data, but shows the fractional limits on has a lower nuclear recoil acceptance, ∼ Table 1.
Dependence of the 95% fractional confidence limits on the WIMP mass,( m lim χ − m χ ) /m χ , on the energy threshold, E th , for the benchmark Ge detector, forinput WIMP masses m in χ = 50 ,
100 and 200 GeV and exposure E = 3 × kg day. E th (keV) m in χ (GeV)50 100 2000 -0.064 , +0.043 -0.12 , +0.14 -0.25 , +0.6310 -0.082 , +0.057 -0.13 , +0.18 -0.27 , +0.8120 -0.11 , +0.073 -0.16 , +0.22 -0.31 , +1.1the WIMP mass, ( m lim χ − m in χ ) /m in χ . With exposures of E = 3 × and 3 × kg dayit would be possible, with this detector configuration, to measure the mass of a light[ m χ ∼ O (50 GeV)] WIMP with an accuracy of roughly 25% and 10% respectively. Thesenumbers, and the upper limits in particular, increase with increasing WIMP mass, andfor heavy WIMPs ( m χ ≫
100 GeV) even with a large exposure it will only be possible toplace a lower limit on the mass. For very light WIMPs, m χ < O (20 GeV), the numberof events above the detector energy threshold would be too small to allow the mass tobe measured accurately. We now examine the effects of varying the energy threshold, E th (for the fiducial detector configuration and WIMP properties described above).Table 1 contains the 95% confidence limits on the fractional deviation of the WIMPmass from the input WIMP mass for input WIMP masses of m in χ = 50 ,
100 and 200 GeV,energy thresholds E th = 0 ,
10 and 20 keV and an exposure of E = 3 × kg day.As the energy threshold is increased the expected number of events decreases. Thesmaller range of recoil energies also reduces the accuracy with which the characteristicscale of the energy spectrum, E R , and hence the WIMP mass can be determined. Theeffect of varying E th is smallest for intermediate WIMP masses; for m χ = 50 GeV (andlight WIMPs in general) the small E R means that expected number of events decreasesrapidly as the energy threshold is increased, while for m χ = 200 GeV (and heavy WIMPsin general) the large E R , and flatter energy spectrum, means that the smaller range ofrecoil energies reduces the accuracy with which E R can be measured. We have previously assumed that recoil events of all energiesabove the threshold energy can be detected. In real experiments there will be amaximum energy, E max , above which recoils are not detected/analysed. For instancefor CDMS II [21] E max = 100 keV. Fig. 4 compares the fractional mass limits for E max = 100 keV with those found previously assuming no upper limit. The differenceis very small for light WIMPs [ m χ < O (50 GeV)] increasing with increasing m χ to O (10%) for m χ ∼ O (200 GeV) and E = 3 × kg day. This reflects the fact that forlight WIMPs the differential event rate above E max is essentially negligible, however this Figure 4.
Fractional mass limits as a function of input mass for the fiducial detectorconfiguration, with no limit on the energy of recoils which can be detected, (solid linesas before) and for a maximum energy E max = 100 GeV (long dashed). For clarity onlythe 95% confidence limits are displayed in this figure. is not the case for heavier WIMPs and finite E max reduces the accuracy with which thecharacteristic energy scale of the spectrum can be measured. Fig. 5 shows the fractional limits on the WIMP mass as a functionof the exposure, E , for input WIMP masses of m in χ = 50 ,
100 and 200 GeV, for thefiducial detector and σ p = 10 − pb. As the exposure is increased the mass limits (andin particular the 95% upper confidence limit) improve, initially rapidly and then moreslowly (reflecting the fact that the expected number of events is directly proportionalto the exposure). An accuracy of ∼ ±
10% in the determination of the WIMP masscan be achieved with an exposure E = 10 (10 ) kg day for m in χ = 50 (100) GeV. For m in χ = 200 GeV even E = 10 kg day would not be sufficient to achieve O (10%) precision. The expected number of events is directly proportional to boththe cross-section and exposure. Varying the cross-section is therefore very similar tovarying the exposure (and hence we do not display a plot of the limits for varying σ p ).0 Figure 5.
Fractional mass limits as a function of exposure, E , for the benchmark Gedetector and σ p = 10 − pb, for, from top to bottom, m in χ = 50 ,
100 and 200 GeV.
Unsurprisingly the accuracy with which it will be possible to determine the WIMPmass depends sensitively on the underlying cross-section. For instance if σ p = 10 − pb,with an exposure of E = 3 × kg day it will be possible to measure the mass withan accuracy of ∼ ±
20% ( +100% − ) for m in χ = 50 (100) GeV. If σ p = 10 − pb, even for m in χ = 50 GeV with an exposure of E = 3 × kg day it will only be possible todetermine the WIMP mass to within a factor of a few and for more massive WIMPs itwill only be possible to place a lower limit on the mass. We now examine the dependence of the mass limits on the detector target material. Infig. 6 we compare the fractional limits for the fiducial (Super-CDMS [13] like) detectorwith E th = 10 keV with those from a Xe detector with E th = 4 . E th = 4 . E th = 10 keV. For heavier WIMPs, m χ >
100 GeV, in contrastto the naive expectation from the m χ dependence of E R (see Sec. 2.1), the Ge detectorproduces slightly better limits than the Xe detector with E th = 4 . E th = 10 keV and a Xe detector with E th = 4 . F ( E ) = 1. Without the form factorthe mass limits are substantially tighter for both Ge and Xe. This is because the eventrate, in particular at large energies, is increased. The improvement in the accuracy ofthe determination of the characteristic energy, E R , and hence m χ , is greater than if theexposure were simply increased so as to increase the expected number of events. Forinstance for Xe, m χ = 200 GeV and E = 3 × kg day without the form factor theone- σ error on the WIMP mass is ∼ σ error is ∼
20 GeV. This is because the greater relative abundance of large energy recoils allows E R to be determined more accurately. With the form factor set to unity the masslimits for large WIMP masses are significantly better for Xe than for Ge, as naivelyexpected from the WIMP mass dependence of E R . However for any real detector therapid decrease of the Xe form factor with increasing energy/momentum transfer meansthat, contrary to naive expectations, the mass of heavy WIMPs can not be measuredmore accurately with Xe than with Ge (assuming similar threshold energies). Thisconclusion, see also recent discussion by Drees and Shan [11], should also hold for anyother heavy target. v c Up until now we have assumed that the local circular speed, v c , is known and equalto its standard value of 220 km s − . There is in fact an uncertainty in v c of order ±
20 km s − [15] and since E R depends on both m χ and v c there is a degeneracy between m χ and v c [10]. Physically, the kinetic energies of the incoming WIMPs depend on theirmass and velocities. For larger (smaller) v c the incoming WIMPs have larger (smaller)mean kinetic energies than assumed, resulting in larger (smaller) maximum likelihoodmass values. This can be made more quantitative by differentiating the expression forthe characteristic energy E R , eq. (8):∆ m χ m χ = − [1 + ( m χ /m A )] ∆ v c v c . (10)For an input WIMP mass of m in χ = 100 GeV and a 20 km s − uncertainty in v c , this givesa ∼
20 GeV shift in the value of the WIMP mass determined.In fig. 8 we plot the fractional mass limits for input circular speeds v inc = 200 , − for E = 3 × kg day. We carry out the likelihood analysis twice, oncewith v c fixed at 220 km s − and once with v c as an additional variable parameter. Asdiscussed in Sec. 2.1 in this section we use the fitting function, eq. (6), for both the2 Figure 6.
Fractional mass limits for the fiducial Ge detector with E th = 10 keV(solid lines for both 68% and 95% confidence limits) and for a Xe detector with E th = 4 . E = 3 × , × and 3 × kg day. input energy spectrum and the likelihood analysis as it is not computationally feasibleto carry out the full calculation of the energy spectrum for each value of v c consideredduring the likelihood analysis.When the underlying value of v inc is different from the (fixed) value used inthe likelihood analysis there is, as expected, a significant systematic error in thedetermination of the WIMP mass. For deviations of ±
20 km s − in the underlyingvalue of v c this systematic error increases with increasing m in χ from ∼
10% for small m in χ to ∼
40% for m χ ≈
200 GeV. The limits are however asymmetric, with the systematicerror in the upper limits being substantially larger for v inc = 200 km s − . Allowing thevalue of v c to vary in the likelihood analysis substantially reduces the error, but therestill appears to be a small systematic shift in the mass limits. While future experiments aim to have negligible backgrounds (e.g. Ref. [13]), non-negligible neutron backgrounds would lead to errors in the determination of the WIMP3
Figure 7.
A comparison of the fractional mass limits obtained, for Ge and Xe withan exposure E = 3 × kg day, with and without the form factor. The solid and longdashed lines are for the fiducial Ge detector with E th = 10 keV and for a Xe detectorwith E th = 4 . F ( E ) = 1. mass. The size of the errors will depend on the amplitude and shape of the backgroundspectrum. In particular if the background spectrum is exponential it can closely mimicthe shape of a WIMP recoil spectrum (see fig. 9).Motivated by simulations of the neutron background in various dark matterdetectors [25] we consider two forms for the background(i) A flat background energy spectrum from E th = 10 keV to E max = 100 keV ¶ ,parametrised by the total background rate, b r , per tonne year.(ii) A exponential energy spectrum from E th = 10 keV to E max = 100 keV, parametrisedby the total background rate, b r , per tonne year, and the characteristic backgroundenergy scale, E b : (cid:18) d R d E (cid:19) back = (cid:18) d R d E (cid:19) E =0 exp [ − ( E/E b )] , (11) ¶ In this section we assume that only recoils up to E max are detected. Figure 8.
Fractional mass limits for the fiducial Ge detector and E = 3 × kg day forvarying circular speed, v c . In the left panel v c is fixed as 220 km s − in the likelihoodanalysis while in the right panel v c is allowed to vary in the likelihood analysis (i.e. it isan additional fitting parameter). Solid, dotted and dashed lines are for v inc = 220 , − respectively. where b r = Z E max E th (cid:18) d R d E (cid:19) back . (12)The limits on the WIMP mass for an exposure of E = 3 × kg day and σ p = 10 − pb in the presence of a flat background with b r = 10 and 100 tonne − year − are displayed in fig. 10. We carry out the likelihood analysis of the WIMP parameterstwice, firstly neglecting the background and then including the background rate as anadditional parameter. Neglecting the background leads to a systematic over-estimate ofthe WIMP mass, since the flat background increases the event rate at large E relativeto that at small E , so that the best fit energy spectrum has larger E R , or equivalentlylarger m χ . Including the background rate in the likelihood analysis avoids the systematicerror but, inevitably, leads to larger statistical error in the WIMP mass limits. Thefractional errors are smallest for m χ ∼ O (50 GeV) and increase for smaller and larger + Note that the backgrounds in real experiments are not expected to be this large. Figure 9.
Solid lines are the differential energy spectra including the form factor,d R/ d E , for (from top to bottom at E = 0 keV) WIMPs with m χ = 50 ,
100 and200 GeV (solid lines). The dotted lines are exponential background spectra with (fromtop to bottom) b r = 1000 tonne − year − & E b = 15 keV, b r = 1000 tonne − year − & E b = 25 keV and b r = 670 tonne − year − & E b = 35 keV (dotted lines). Theparameters of the exponential background spectra have been chosen to demonstratethat, even when the form factor is included, the WIMP recoil spectra are close toexponential and could, in principle, be mimicked by an exponential background. WIMP masses. This is because for small WIMP masses the background event rate islarger compared with the WIMP event rate, while for large WIMP masses the WIMPenergy spectrum (exponential with large characteristic energy scale) is closer in shapeto the flat background spectrum. The systematic (background not included in analysis)and additional statistical (background rate included) errors are both at least 10% for b r = 10 tonne − year − . For b r = 100 tonne − year − the minimum systematic error (for m χ ∼
50 GeV) is ∼ b r = 10 tonne − year − , howeverthe upper limits are increased significantly.The limits on the WIMP mass for an exposure of E = 3 × kg day and σ p = 10 − pb in the presence of a background with an exponential energy spectrumwith E b = 20 keV are displayed in fig. 11. The exponential background spectrum with6 Figure 10.
Fractional mass limits in the presence of a background with a flat energyspectrum and total rate b r = 10 and 100 tonne − year − (top and bottom panelsrespectively). The dashed (dotted) lines are for when the background event rate isnot (is) included in the likelihood analysis. The solid lines are the confidence limitsfor zero background. The fiducial Ge detector configuration with E max = 100 keV, E = 3 × kg day and σ p = 10 − pb is used. E b = 20 keV is similar to the WIMP spectrum with m χ ∼
70 GeV. Therefore if thebackground is neglected in the likelihood analysis, for smaller (larger) input WIMPmasses the WIMP mass is systematically over(under)-estimated. For the exponentialbackground when the background rate, b r , and characteristic energy, E b , are included inthe likelihood analysis there are still large deviations from the zero background limits,this is because the WIMP and background spectra can have extremely similar shapesand can be degenerate. The fluctuations in the limits with varying m in χ reflect the errorsresulting from this degeneracy rather than a real underlying trend.In summary the implications of non-negligible backgrounds for the determinationof the WIMP mass depend strongly on the shape of the background spectra (as wellas, obviously, its amplitude). A flat background spectrum will lead to a systematicerror in the WIMP mass for light WIMPs, which can be avoided, at the cost of largerstatistical error, by fitting for the background event rate. For heavier WIMPs the flat7 Figure 11.
As fig. 10 for an exponential background energy spectrum with E b = 20 keV. background is similar in shape to the WIMP spectrum and it is hence more difficult toseparate the WIMP and background spectra and accurately measure the WIMP mass.An exponential background spectrum is similar in shape to the WIMP spectrum andwould inevitably (even when the background parameters are including in the likelihoodanalysis) lead to increased errors in the determination of the WIMP mass.With a single detector it will be difficult to disentangle a WIMP signal (and theWIMP mass) from background if the background spectrum has a similar shape tothe WIMP spectrum (i.e. exponential background, or flat background with a heavyWIMP). Multiple targets (for instance Ge and Si as used by CDMS II [21]) would helpdue to the dependence of the WIMP spectrum on the mass of the target nuclei. Seeref. [26] for Monte Carlo simulations using CaWO and ZnWO . Detectors composed ofvery different targets (e.g. Ge and Xe) would likely have different background spectrahowever.8
4. Summary
We have studied how the accuracy with which the WIMP mass could be determined by asingle direct detection experiment depends on the detector configuration and the WIMPproperties. Specifically, we investigated the effects of varying the underlying WIMP massand cross-section, the detector target nucleus, exposure, energy threshold and maximumenergy, the local circular speed and the background event rate and spectrum.The accuracy of the mass limits is most strongly dependent on the underlyingWIMP mass and the number of events detected. For light WIMPs (mass significantlyless than that of the target nuclei) small variations in the WIMP mass lead to significantchanges in the energy spectrum. Conversely for heavy WIMPs the energy spectrumdepends only weakly on the WIMP mass. Consequently it will be far easier to measurethe WIMP mass if it is light than if it is heavy. The number of events detected is directlyproportional to both the exposure and the cross-section, therefore these quantities havethe greatest bearing on the accuracy of the WIMP mass determination. For our baseline,SuperCDMS [13] like, Ge detector with negligible background and energy threshold E th = 10 keV for a WIMP-proton cross-section of σ p = 10 − pb, a factor of a few belowthe current exclusion limits from the CDMS II [21] and Xenon10 [22] collaborations,with exposures of E = 3 × and 3 × kg day it would be possible to measurethe mass of a light [ m χ ∼ O (50 GeV)] WIMP with an accuracy of roughly 25% and10% respectively. These numbers, and the upper limits in particular, increase withincreasing WIMP mass, and for heavy WIMPs ( m χ ≫
100 GeV) even with a largeexposure it will only be possible to place a lower limit on the mass. For very lightWIMPs, m χ < O (20 GeV), the number of events above the detector energy thresholdwould be too small to allow the mass to be measured accurately. If σ p = 10 − pb, with anexposure of E = 3 × kg day it will be possible to measure the mass with an accuracyof ∼ ±
20% ( +100% − ) for m in χ = 50 (100) GeV. If σ p = 10 − pb, for m in χ = 50 GeV, evenwith an exposure of E = 3 × kg day it will only be possible to determine the WIMPmass to within a factor of a few and for more massive WIMPs it will only be possibleto place a lower limit on the mass.The energy threshold, E th , and the maximum energy, E max , above which recoilsare not detected/analysed also affect the accuracy with which the WIMP mass can bedetermined. Increasing E th (or decreasing E max ) not only reduces the number of eventsdetected, but also reduces the range of recoil energies and the accuracy with which thecharacteristic energy of the energy spectrum, E R , and hence the WIMP mass, can bemeasured. The effect of increasing E th is smallest for intermediate WIMP masses. Forlight WIMPs the small E R means that the expected number of events decreases rapidlyas the energy threshold is increased, while for heavy WIMPs the large E R , and flatterenergy spectrum, means that the smaller range of recoil energies reduces the accuracywith which E R can be measured. The effect of reducing the maximum energy (frominfinity to 100 keV) is very small for light WIMPs as the differential event rate above E max = 100 keV is negligible, however for heavy WIMPs the fractional mass limits can9change by O (10%).The relative capabilities of different detectors to determine the WIMP mass dependnot only on the WIMP and target masses, but also on their energy thresholds. TheWIMP and target mass dependence of the characteristic energy scale of the recoilspectrum suggests that heavy targets will be able to measure the mass of a heavyWIMP more accurately, however the rapid decrease of the nuclear form factor withincreasing momentum transfer which occurs for heavy nuclei means that this is in factnot the case.If the WIMP distribution on the ultra-local scales probed by direct detectionexperiments is smooth, then the uncertainties in the detailed shape of the local velocitydistribution lead to relatively small changes in the shape of the differential eventrate [17, 18], and hence a relatively small, [ O (5%)], systematic uncertainty in the WIMPmass [10]. There is however an uncertainty in the local circular speed, v c , (and hencethe typical speed of the WIMPs) of order ±
20 km s − [15] and since E R depends onboth m χ and v c this leads to a degeneracy between m χ and v c [10]. For deviations of ±
20 km s − in the underlying value of v c this systematic error increases with increasing m in χ from ∼
10% for small m in χ to ∼
40% for m χ ≈
200 GeV.The assumption of a smooth WIMP distribution may not be valid on the sub milli-pc scales probed by direct detection experiments (see discussion in Paper I [10]). Ifthe ultra-local WIMP distribution consists of a finite number of streams (with a prioriunknown velocities) then the recoil spectrum will consist of a number of (sloping dueto the energy dependence of the form factor) steps. The positions of the steps willdepend on the stream velocities, the target mass and the WIMP mass. In this casemultiple targets would be needed to extract any information on the WIMP mass. Dreesand Shan [11] have recently demonstrated that with multiple targets it is in principlepossible to constrain the WIMP mass without making any assumptions about the WIMPvelocity distribution.Future experiments aim to have negligible backgrounds, however, non-negligibleneutron backgrounds would lead to errors in the determination of the WIMP mass.The size of the errors will depend on the amplitude and shape of the backgroundspectrum. If the background rate is not negligible compared with the WIMP eventrate it will be difficult to disentangle a WIMP signal (and the WIMP mass) from thebackground if the background spectrum has a similar shape to the WIMP spectrum (i.e.exponential background, or flat background with a heavy WIMP). The uncertaintiesfrom backgrounds could be mitigated by using multiple targets (see e.g. Ref. [26]),however detectors composed of very different targets (such as Ge and Xe) would beunlikely to have the same background spectra.
Acknowledgments
AMG is supported by STFC and is grateful to Ben Morgan for useful discussions andMarcela Carena for encouragement to investigate some of the issues considered.0
5. References [1] M. Tegmark et al., Phys. Rev. D astro-ph/0310723 ; S. Cole et al., Mon. Not.Roy. Astron. Soc. , 505 (2005), astro-ph/0501174 ; J. Dunkley et al., arXiv:0803.0586 .[2] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rep. , 195 (1996).[3] L. Bergstr¨om, Rept. Prog. Phys. , 793 (2000), hep-ph/0002126 ; G. Bertone. D. Hooper and J.Silk, Phys. Rep.
279 (2005), hep-ph/0404175 .[4] M. W. Goodman and E. Witten, Phys. Rev. D , 3059 (1985).[5] L. Baudis, arXiv:0711.3788 (2007).[6] J. D. Lewin and P. F. Smith, Astropart. Phys. , 87 (1996).[7] M. J. Lewis and K. Freese, Phys. Rev. D astro-ph/0307190 .[8] J. L. Bourjaily and G. L. Kane, hep-ph/0501262 .[9] http://particleastro.brown.edu/theses/060421 Monte Carlo Simulations Dark Matter Detectors Jackson v3.pdf;http://cosmology.berkeley.edu/inpac/CDMSCE Jun06/Talks/200606CDMSCEmass.pdf[10] A. M. Green, JCAP08(2007)022, hep-ph/0703217 .[11] M. Drees and C-L. Shan, arXiv:0803.4477 (2008).[12] D. Hooper and E. A. Baltz, arXiv.0802.0702 (2008); N. Bernal, A. Goudelis, Y. Mambriniand C. Munoz, arXiv:0804.1976 (2008); B. Altunkaynak, M. Holmes and B. D. Nelson, arXiv:0804.2899 (2008).[13] R. W. Schnee et al., proceedings of DARK 2004, fifth international Heidelberg conference on darkmatter in Astro and Particle Physics, astro-ph/0502435 (2004); P. L. Brink et al., proceedingsof Texas Symposium on Relativistic Astrophysics, astro-ph/0503583 (2004).[14] R. H. Helm, Phys. Rev. , 1023 (1986).[16] M. C. Smith et al., Mon. Not. Roy. Astron. Soc. , 755 (2007), astro-ph/0611671 .[17] M. Kamionkowski and A. Kinkhabwala, Phys. Rev. D , 3256 (1998), hep-ph/9710337 ; F.Donato, N. Fornengo and S. Scopel, Astropart. Phys. , 247 (1998), hep-ph/9803295 .[18] A. M. Green, Phys. Rev. D , 083003 (2002), astro-ph/0207366 .[19] B. Moore et al., Phys. Rev. D astro-ph/0106271 ; S. Stiff and L. Widrow, Phys.Rev. Lett. , 211301 (2003), astro-ph/0301301 .[20] A. Helmi, S. D. M. White and V. Springel, Phys. Rev. D , 0635023 (2002), astro-ph/0201289 ;M. Vogelsberger et al., arXiv:0711.1105 .[21] Z. Ahmed et al., arXiv:0802.3530 (2008).[22] J. Angle et al., Phys. Rev. Lett. , 021303 (2008), arXiv:0706.0039 .[23] G. Cowan, Statistical data analysis , published by Oxford University Press (1998).[24] D. S. Akerib et al., Phys. Rev. D , 052009 (2005), astro-ph/0507190 .[25] M. J. Carson et al., NIMA , 509 (2005); L. Kaufmann and A. Rubbia, hep-ph/0612056 ; H.S.Lee et al. NIMA , 644 (2007).[26] H. Kraus et al., Phys. Lett. B610