Integrating In Dark Matter Astrophysics at Direct Detection Experiments
LLA-UR-12-26557
Integrating In Dark Matter Astrophysics at Direct Detection Experiments
Alexander Friedland ∗ and Ian M. Shoemaker † Theoretical Division T-2, MS B285, Los Alamos National Laboratory, Los Alamos, NM 87545, USA (Dated: May 29, 2013)We study the capabilities of the M
AJORANA D EMONSTRATOR , a neutrinoless double-beta decay experimentcurrently under construction at the Sanford Underground Laboratory, as a light WIMP detector. For a crosssection near the current experimental bound, the M
AJORANA D EMONSTRATOR should collect hundreds or eventhousands of recoil events. This opens up the possibility of simultaneously determining the physical propertiesof the dark matter and its local velocity distribution, directly from the data. We analyze this possibility and findthat allowing the dark matter velocity distribution to float considerably worsens the WIMP mass determination.This result is traced to a previously unexplored degeneracy between the WIMP mass and the velocity dispersion.We simulate spectra using both isothermal and Via Lactea II velocity distributions and comment on the possibleimpact of streams. We conclude that knowledge of the dark matter velocity distribution will greatly facilitatethe mass and cross section determination for a light WIMP.
PACS numbers: 95.35.+d
I. INTRODUCTION
Direct detection experiments offer the possibility of a non-gravitational detection of dark matter (DM), the most com-mon form of matter in the Universe [1]. Swift progress isbeing made in this field, with the LUX and XENON1T ex-periments aiming to push the bound on the spin-independentcross section of a 100 GeV WIMP all the way to the − cm − level. A no less dramatic development may occur onthe light WIMP front ( (cid:46)
10 GeV), where the improvementrelative to the current bound may be even greater, as a resultof the M
AJORANA D EMONSTRATOR [2, 3] (see Fig. 1 below).If the light WIMP cross section lies near its current bound, theM
AJORANA D EMONSTRATOR is poised to collect hundredsor even thousands of recoil events. This would open up an in-teresting possibility of not only discovering the DM particle,but also accurately measuring its properties. Here, we explorethis possibility and the issues that arise in connection with it.Light WIMPs are not without theoretical or experimen-tal motivation. For example, on the theoretical side, 1-10GeV DM is typical in models relating the baryon and DMabundances [4–10] (see [11] and references therein for re-cent work). In these models, the DM has a nonzero particle-antiparticle asymmetry, thereby suppressing the indirect sig-natures of DM annihilation . Establishing such low-mass DMat a collider will be difficult since although the Tevatron andthe LHC can detect the missing energy associated with lightDM, they lose sensitivity to the mass of DM when it is muchless than 100 GeV (see e.g., [13]). The missing energy couldthen be attributed to a variety of sources, from novel neu-trino interactions [14] to extra-dimensional particles[15, 16].Hence, low-threshold direct detection experiments such as theM AJORANA D EMONSTRATOR may offer the best way to testthis well-motivated class of models. ∗ Electronic address: [email protected] † Electronic address: [email protected] See, however, [12] for a counterexample.
On the experimental side, the interest of the community hasbeen piqued by the CoGeNT and DAMA experiments, whichhave claimed signs of light dark matter [17, 18]. Most re-cent hints in this mass window come from the CDMS ex-periment [19]. With its remarkable sensitivity, the M
AJO - RANA D EMONSTRATOR is expected to resoundingly refuteor confirm these results. Here, we choose to remain agnos-tic about them and assume that the DM is just at the bor-der of what is allowed by the null results [20–23] (see theblack curves in Fig. 1 for current constraints). If the Co-GeNT/DAMA/CDMS signals are confirmed, the M
AJORANA D EMONSTRATOR will see even more events than we considerbelow.If the M
AJORANA D EMONSTRATOR indeed sees hundredsor thousands of DM events, the experiment will obviously tryto determine the DM mass and cross section from this data.The accuracy of this determination, on very general grounds,is expected to depend on our knowledge of the “beam”, whichin this case is provided by the local dark matter distribution.As we discuss below, its characteristics are uncertain. Twoapproaches to this uncertainty could be taken. One is to relyon models of the Galactic halo (analytical and/or numerical),the other is to try to extract both the DM physics and as-trophysics directly from the data. Previous studies examin-ing the effect of astrophysical uncertainty on mass and crosssection determination [24–33] involve some combinations ofthese approaches (though none deal with such light WIMPs).Below, we highlight both approaches and then concentrateon the second one, which may become feasible with largestatistics and hence the D
EMONSTRATOR may be the mostappropriate place to test it. We will quantify how much si-multaneously fitting for the DM properties and astrophysicsdegrades the determination of the DM mass. We will showthat the culprit responsible for the degradation is a degeneracythat exists between the light WIMP mass and the DM velocity.To avoid any confusion, we stress from the outset that forthe present purpose astrophysics obviously cannot be “inte-grated out”, along the lines of [34, 35]. The power of the“integrating out” technique is that it allows mapping the re-sults of one experiment into another in an astrophysics-free a r X i v : . [ h e p - ph ] J un manner (see [36], [37] and [38] for applications to recent ex-periments). For the purpose of measuring the DM properties,the astrophysics needs to instead be “integrated in”.In the next section we review the basic theory of direct de-tection. In Sect. III we discuss our fit methodology and mock-up of the M AJORANA D EMONSTRATOR . In the first subsec-tion of Sect. IV we present the results of fits to the mass, crosssection and the local velocity dispersion, assuming the “true”(input) velocity distribution is Maxwell-Boltzmann. In sub-section IV B, we examine to what extent light DM is sensitiveto different forms of the velocity distribution by inputing thedistribution from the high-resolution N-body simulation, ViaLactea II. In Sect. IV C we comment on the degeneracy atheavier masses ( m χ ∼ GeV) and in Sect. IV D we brieflytouch on the signatures of streams. We summarize our con-clusions in Sect. V.
II. BASICS
While the nature of WIMP-nucleus scattering is obviouslyunknown, in the vast majority of existing scenarios the scat-tering is mediated by a sufficiently heavy mediator particle(compared to the momentum transfer in the direct detectionprocess). Upon integrating out the exchanged particle, in theusual effective field theory approach, one finds that all thephysics of the DM-nucleus scattering is contained in a set ofhigher dimensional operators. Several studies of the range ofpossibilities have been carried out [39–41]. A complete anal-ysis would in principle examine how each of these operatorsis sensitive to astrophysical uncertainties in turn – a daunting,but eventually necessary task. Here, mainly for clarity, we willvary the astrophysics while restricting ourselves to the sim-plest and most-studied interaction form: isospin-conserving,spin-independent scattering that depend on neither the incom-ing DM velocity nor the exchanged momentum. Such a crosssection could come from a scalar interaction ( qq )( XX ) (suchas from Higgs-exchange for a neutralino, e.g. [42]) or a vec-tor interaction ( qγ µ q )( Xγ µ X ) (arising from the exchange ofa Z (cid:48) vector boson). A complementary study was carried outin [43] where the authors instead fixed the astrophysics andconsidered a set of particle physics choices.To obtain the average differential rate per unit detector massof a WIMP of mass m X scattering on a target nucleus of mass m N , one convolves the cross section with the DM velocitydistribution (see [42, 44] for reviews), dRdE R = ρ (cid:12) m N m X (cid:28) v dσdE R (cid:29) (1) = ρ (cid:12) m N m X (cid:90) ∞ v min ( E R ) d v vf ( (cid:126)v + (cid:126)v e ( t )) dσdE R , where µ N is the DM-nucleus reduced mass, (cid:126)v e ( t ) is the ve-locity of the laboratory observer with respect to the galacticrest frame, f ( v ) is the local DM velocity distribution in therest frame of the galaxy, and ρ (cid:12) the local DM density. Thequantity v min ( E R ) is the minimum DM velocity in the lab frame to produce a nuclear recoil of energy E R ; for elasticscattering, v min ( E R ) = (cid:112) m N E R / µ N .For spin- and momentum-independent interactions, the dif-ferential cross section can be written as dσdE R = m N µ N v σ NSI F ( E R , A ) , (2)where F ( E R , A ) is the nuclear form factor. We use the Helmform factor, which can be found in [45]. The spin-independentDM-nucleus cross section is σ NSI = σ n µ N µ n [ f p Z + f n ( A − Z )] f n , (3)where µ n is the DM-nucleon reduced mass, and f p and f n are the DM couplings to protons and neutrons respectively.Throughout, we will make the standard simplifying assump-tion of isospin-conserving scattering, f p = f n = 1 . Withthese assumptions, the scattering rate simplifies to dRdE R = ρ (cid:12) σ n µ n m X A F ( E R , A ) g ( v min ) , (4)where g ( v min ) is the mean inverse speed, g ( v min ) ≡ (cid:90) ∞ v min f ( (cid:126)v + (cid:126)v e ( t )) v d v. (5)The astrophysical uncertainties in principle affect both thelocal density ρ (cid:12) and the velocity distribution f ( v ) . In thisletter, we adopt the standard fiducial value of the local DMdensity ρ (cid:12) = 0 . GeV/cm and focus on velocities. Withthis framework, the unknown quantities are the DM mass m X ,scattering cross section σ n and f ( v ) or, equivalently, g ( v min ) . A. Velocity Distributions
While an order-of-magnitude estimate of the event rate canoften be obtained by simply using the average velocity of aDM particle in the halo [1], for a quantitative measurementof the DM properties the knowledge of the dark velocity dis-tribution is required. A canonical framework has emerged inthe field, in which the DM halo is taken to be an isothermal,Maxwell-Boltzmann (MB) distribution [46, 47], with a cut-offcorresponding to the escape speed. In the galactic rest-framethe distribution, f MB ( (cid:126)v ) = (cid:26) N e − v /v , v < v esc , v > v esc , (6)is fully specified by just two parameters: its dispersion v andthe local escape speed v esc . The relevant velocity integralEq.(5) and the normalization constant N have a closed formexpression (see Appendix). Assuming an idealized isother-mal halo, with a ρ ∝ r − DM density profile, the dispersion v could further be equated to the circular speed, v e . The cir-cular speed is observable and is measured to be (cid:104) v e ( t ) (cid:105) = 230 km/s [48, 49]. This framework is being used by all experimen-tal collaborations in reporting their results in terms of m X and σ n .We would like to get a sense of the impact of astrophys-ical uncertainties on these results. Even if one chooses tostick with the assumption of an idealized isothermal halo,an important uncertainty comes from the error on the cir-cular speed, which is σ v e (cid:39) km/s [48, 49]. This errorarises from the determination of the distance from the Sunto the Galactic center. Thus, at the minimum, one can con-sider varying the v e and v in unison, within, say, a σ errorbar (see Fig. 1 below) and, further, varying the escape speed,which, according to the recent RAVE survey, is in the region
498 km / s < v esc <
608 km / s at CL [50].Of course, this approach is based on a long list of as-sumptions. It neglects a possible DM core and the gravita-tional effects of the baryons in the Galaxy. The isothermalassumption itself is ad hoc , and indeed detailed N-body sim-ulations appear to be better fit by NFW profiles, the veloc-ities of which show non-Maxwellian features, especially onthe high end [51]. The isotropy assumption for the local ve-locity distribution is also not borne out by the simulations, cf. [52]. Even more distinctive for direct detection is the possi-bility of unequilibrated velocity structures coming from therecent tidal destruction of DM subhalos. These can includestreams [53, 54], as well as more spatially homogeneous tidaldebris in the form of sheets and plumes [55, 56].In view of all this, the possibility of direct experimental determination of the velocity distribution is of great interest.In our analysis, we will use the form in Eq. (6), fix v esc to544 km/s, and vary v e around its fiducial value, 230 km/s. Westress that in this approach, the form of Eq. (6) is simply aparametrization of our ignorance, selected mainly for clarity,and not a physical model. As such, one should not expect that v be correlated with v e . An alternative philosophy would beto use a model of the Galaxy to relate the two, as done in forexample [26].We will further investigate, in Sect. IV B, if the M AJORANA D EMONSTRATOR , given massive statistics, would be able totell a difference between the (averaged) numerical velocitiesand the isothermal model. Finally, we will comment on someimplications from streams in Sect. IV D.
III. EXPERIMENTAL DETAILS AND FIT SETUP
Though the M
AJORANA D EMONSTRATOR [3] is primarilya neutrinoless double-beta decay experiment, the experimen-tal setup is well-suited for DM direct detection as well [2].The D
EMONSTRATOR plans to achieve a very low threshold,in electron-equivalent units, 0.3-0.5 keV ee [2] (or equivalently E NR ≈ . − . keV in nuclear recoil). Such low thresholdsare due to P-type contact detector technology [59], also usedby the CoGeNT experiment. Here we will take 0.3 keV ee to The size of which for an idealized isothermal sphere is a free parameter. X E N O N L U XX E N O N T MAJORANA XENON10 (S2-only)CRESST1DAMIC
FIG. 1: Sensitivity reach of the upcoming and current experi-ments. The current constraints are shown with black curves: 2012XENON100 [22] along with the low-threshold analyses done byXENON10 [21], CRESST1 [57] and DAMIC [58]. For the upcomingexperiments, we use colored curves: solid assuming the velocity dis-persion v = 230 km/s and dashed for v = 170 km/s and v = 290 km/s. The circular speed v e is taken to be equal to v here. Forthe M AJORANA D EMONSTRATOR we assumed a 0.3 keV ee energythreshold and 100 kg-yr exposure. For LUX, we took a 100 kg-yrexposure with E th = 5 keV, and for the XENON1T, a 2.2 ton-yrexposure and E th = 8 . keV (see IV C for more details). The blackcircles indicate points this paper examines in detail. compare with [2]. The collaboration plans to have roughly40 kg of Germanium detectors, deployed in three stages overthe next two years. Here we will assume an exposure of 100kg-yr [2]. The D EMONSTRATOR is now being constructed atthe Sanford Underground Research Facility (SURF) in Lead,South Dakota [3]. They plan to being data-taking in 2013,with the first set of two strings.The DM sensitivity of M
AJORANA D EMONSTRATOR iscrucially affected by the background from cosmogenically ac-tivated tritium. This has been well studied in the thesis [2].Assuming that a detector spends a total of 15 days at the sur-face, the average background rate contributed by decayingtritium is 0.03 counts/day/kg/keV [2]. The tritium spectrumis well-known and extends to around 18 keV ee . This back-ground can of course fluctuate up and fake a WIMP signal.We will assume that this is the only relevant background in thesignal region, and note that other possible backgrounds, in-cluding unknown “surface events” at CoGeNT(see for exam-ple [60]), need not impact the D EMONSTRATOR in the sameway, due to design differences.We also model the Xenon experiments, LUX andXENON1T, in a simplified manner. LUX is currently underconstruction at SURF while XENON1T will be installed atGran Sasso National Laboratory in Italy, with planned datataking to begin in 2015. We mock up the XENON1T experi-ment by assuming a 2.2 ton-yr exposure with an energy win-dow . − . keV and a 40 % acceptance [61]. We havechecked that for heavy masses this mock-up agrees reason-ably well with Fig. 2 of [62]. For the LUX experiment,we take 30000 kg-day exposure, the recoil energy window5-30 keV and 45% efficiency. These choices of the fixedlow energy thresholds are conservative and facilitate compar-isons with existing literature ( e.g. , [30, 33]). The experimentsthemselves, however, may obtain sensitivity to recoil energiesbelow the threshold used here, thereby extending their sen-sitivity to lower mass WIMPs. Thus in comparing with theXENON1T limits in [62] for example, our results reproducewell the exclusion sensitivity at high mass but are less strin-gent than what appears in [62]. We urge the experimental col-laborations to improve on the analysis presented here.For our statistical analysis, we bin events in E R (taking 0.9keV bins for M AJORANA D EMONSTRATOR , and 3 keV binsfor the Xenon experiments). The standard prescription for thelikelihood of parameters is a product of Poisson distributionsacross the energy bins: L ( θ ) = (cid:89) i N ( θ, E i ) N obs ( E i ) N obs ( E i )! e − N ( θ,E i ) (7)where N obs ( E i ) is the observed number of events in the i th energy bin and N ( θ, E i ) is the number of expected eventsfrom parameters θ in the same bin. Given that at M AJORANA D EMONSTRATOR we will be examining cases with large num-bers of events (from the signal plus the background), the Pois-son distributions actually become Gaussians and the overall fitbecomes well-described by a standard χ statistics. We haveverified this explicitly.Moreover, since we are interested in the results obtained onaverage at a given experiment, we follow the method outlinedin the Appendix of [63], and calculate the average χ as (cid:104) χ (cid:105) = N dof + (cid:88) i s i b i (8)where b i and s i are the background and signal expectation inthe i th energy bin. In the case of an exclusion, we calculate thelimit by imposing (cid:104) χ (cid:105) < α CL N dof , where α CL depends onthe desired confidence level. Throughout we will use 95 % CL.When producing confidence regions from input signal spectrawe calculate
L ∝ e −(cid:104) χ (cid:105) / at each parameter point, and thenintegrate the likelihood over the subspace of parameters thatyield of the total likelihood. This method is useful for thenon-Gaussian regions we obtain below. We return to Eq. (7)when we examine the case of XENON1T in Sect. IV C. IV. SIMULATION RESULTSA. The Maxwell-Boltzmann Case with Light WIMPs
As a first example, we illustrate the importance of the un-certainty on the dispersion in setting exclusion limits. We usethe first framework outlined in Sect. II A, namely, vary v e and v in lockstep, assuming the isothermal halo model. Null re-sults at M AJORANA D EMONSTRATOR , LUX and XENON1Twould result in exclusion curves shown in Fig. 1. The solidcurves assume the standard value v = 230 km/s, while thebands show the result of varying v within ± km/s. Forcomparison, the existing exclusion curves are also shown (forfixed v ). As is clear from the figure, the precise value of thevelocity dispersion becomes particularly important for lightWIMP masses, where the sensitivity of the experiments varies by an order of magnitude . Physically, this is due to the factthat the events to which these experiments are sensitive all arecaused by the high-velocity tail of the distribution. The largerthe value of v , the more stringent the bound for light masses.Let us now turn to our main simulation, of an actual WIMPdetection, assuming the (unknown) parametrized velocity dis-tribution of the form in Eq. (6). We will examine in detail twouseful benchmark points: one with input data coming from a 8GeV WIMP with scattering cross section − cm , and theother a 7 GeV WIMP with a − cm cross section. Bothof these are just below the current XENON100 bounds [22],as shown in Fig. 1. At this mass and cross section, the M AJO - RANA D EMONSTRATOR would see nearly 500 events in the 8GeV case and around 3300 events in 7 GeV case.For each of the two points, we generate “data”, using v e = v = 230 km/s and setting the escape speed to v esc = 544 km/s. We then perform a fit on a three-dimensional grid ofpoints, in the ( m X , σ n , v ) space. We keep v esc = 544 km/sand v e = 230 km/s both fixed in the fit. This is done pri-marily for the sake of clarity: to avoid confusion among manyparameters.Let us first examine the 8 GeV case. In Fig. 2 we showthree slices of the full three-dimensional confidence regions atfixed cross sections. In addition to the input value (black), wealso choose one lower value (in green) and one higher value(in red). Marginalizing (integrating likelihood) over the crosssection yields the light blue regions.We see a clear degeneracy between m X and v , which be-comes particularly prominent in the case of larger statistics (inthe right panel). Of course, one intuitively expects a degen-eracy of this sort, since as one raises the fit mass parameterone must simultaneously lower the dispersion in order to pro-duce a reasonable fit to the input spectrum (see for example,[24, 25, 64, 65]). We would like, however, to give a quantita-tively accurate description of it. To this end, let us specializeto the right panel in Fig. 2 and consider a fixed cross sec-tion (the black region, for definiteness). Recall that for lightWIMPs, v min is rather large (only high velocity WIMPs canproduce enough recoil). This implies that the velocity integral g ( v min ) (see the Appendix for the full expression) is well ap- Σ n (cid:61) (cid:180) (cid:45) cm Σ n (cid:61) (cid:45) cm Σ n (cid:61) (cid:180) (cid:45) cm (cid:248) m X (cid:72) GeV (cid:76) v (cid:72) k m (cid:144) s (cid:76) Ge, 100 kg (cid:45) yr, 95 (cid:37) CL Σ n (cid:61) (cid:180) (cid:45) cm Σ n (cid:61) (cid:45) cm Σ n (cid:61) (cid:180) (cid:45) cm (cid:248) m X (cid:72) GeV (cid:76) v (cid:72) k m (cid:144) s (cid:76) Ge, 100 kg (cid:45) yr, 95 (cid:37) CL FIG. 2: Slices of the best-fit regions from the M
AJORANA D EMONSTRATOR in ( m X , σ n , v ) space. Light blue shows the cross section-marginalized regions, whereas the other contours are for fixed cross section as indicated in each plot. The dashed gray curve representthe analytic estimate, Eq. (9), which agrees well with the black contour. The input data is generated from a Maxwellian halo with with v = v LSR = 230 km/s, with DM mass of 8 GeV and cross section − cm ( left ) and 7 GeV and cross section − cm ( right ). proximated by g ( v min ) ∝ (cid:20) − erf (cid:18) v min − v e v (cid:19)(cid:21) , (9)such that directions of constant ( v min − v e ) /v are degenerate.To translate this into a curve in the m X − v plane, we mustfix E R or equivalently v min . Taking 3 keV falls roughly inthe middle of the signal spectrum.We display the analytic estimate based on Eq. (9) as adashed gray line in the right panel of Fig. 2. It nicely re-produces the shape of the black region in Fig. 2 . Note thatthe contours at other cross sections can be understood with ananalogous argument.Notice that the analytical contour bends away from the pre-diction once v is greater than about 350 km/s. This is becauseof the effects of the finite the escape speed: for v (cid:38) kmand v esc = 544 km/s , a significant part of the Maxwellian tailis cut off. (Eq. (9) is obtained neglecting this cutoff.)Let us show how this degeneracy impacts the ability of theexperiment to determine the mass and cross section of darkmatter. To do so, we marginalize out the v dependence byimposing a top hat prior on the velocity dispersion centeredon the true value 230 km/s, for a variety of uncertainties σ v ( ± km/s, ± km/s and from zero to the escape speed). Theresults for both examples are shown in the top two panels ofFig. 3, where we have also shown the regions coming from Note that this fit is better than the explicit parameterization given in [24,25]. fixing the dispersion to 230 km/s (black). The right panel isparticularly instructive. We see that in the fixed astrophysicsassumption one might expect an excellent measurement of themass and cross section, but the measurement deteriorates sig-nificantly as the prior on v get weaker and weaker. The long-tail extending to high masses results from the ability of high-mass/low-velocity dispersion cases to mimic the true data rea-sonably well. The low-mass tail, which originates from theability of low-mass/high-dispersion cases to match the truedata, is considerably shorter.Two comments are in order. First, notice that the degen-eracy of v with σ n is quite weak. This is because changing v alters the signal spectrum in a way that depends on the tar-get mass and does not reduce to a straightforward rescaling ofthe rates. It may be worth noting that this does not open thepath of bringing the “hint” regions favored by different ex-periments [17–19] into closer agreement: the “integrate out”relations between the experiments hold in this case.Second, one may worry that relaxing the prior on v appearsto make some parts of the parameter space less favored. Thishappens because the contours show the most likely range ofparameters, roughly speaking the ∆ χ contours relative to thebest-fit point. If, as the prior is relaxed, a better fit is achievedsomewhere, the ∆ χ contours may shift accordingly. Thiseffect would be absent if we were to plot the goodness of fitcontours (absolute χ ), instead of the parameter estimationcontours.Going one step further, we can marginalize out the crosssection to see how well the WIMP mass, m X , can be deter-mined. The results are shown in the bottom panels in Fig. 3.We first examine the 8 GeV case. Interestingly, when the v (cid:61)
230 km (cid:144) s (cid:177)
25 km (cid:144) s (cid:177)
75 km (cid:144) sAll (cid:248) m X (cid:72) GeV (cid:76) Σ n (cid:72) c m (cid:76) Ge, 100 kg (cid:45) yr, 95 (cid:37) CL v (cid:61)
230 km (cid:144) s (cid:177)
25 km (cid:144) s (cid:177)
75 km (cid:144) sAll (cid:248) m X (cid:72) GeV (cid:76) Σ n (cid:144) (cid:45) c m Ge, 100 kg (cid:45) yr, 95 (cid:37) CL v (cid:61)
230 km (cid:144) s (cid:177)
25 km (cid:144) s (cid:177)
75 km (cid:144) sAll m X (cid:72) GeV (cid:76) (cid:76) (cid:72) m X (cid:76) v (cid:61)
230 km (cid:144) s (cid:177)
25 km (cid:144) s (cid:177)
75 km (cid:144) sAll m X (cid:72) GeV (cid:76) (cid:76) (cid:72) m X (cid:76) FIG. 3: Marginalized 95 % CL regions for the M
AJORANA D EMONSTRATOR experiment with a 100 kg-yr exposure. In the upper panels weimpose a top-hat prior on the dispersion, centered on the true value km/s with varying degrees of uncertainty σ v . In the bottom row, weproject this into the one-dimensional m X distribution. Results from a (8 GeV, − cm ) and (7 GeV, − cm ) input spectra are depictedin the left and right columns respectively. dispersion is varied, the experimental reconstruction prefersWIMP masses around 6 GeV. This is the result of projectingthe “sock-like” region depicted in the panel above: the “an-kle” contains many points along the v direction, which whenprojected onto the m X axis, skews the likelihood toward lowmasses.The same basic degeneracy and the concomitant skewedlikelihood functions carries over for the 7 GeV case, despitethe large increase in statistics. In the right panel of Fig. 3 werepeat the above exercise for the example depicted in the rightpanel of Fig. 2. Again, this example includes a factor ∼ σ region expandsfrom m X = 7 +0 . − . GeV to . +3 . − . GeV in going from fixeddispersion to complete uncertainty.In summary, by inferring the astrophysics ( v ) from thedata, we suffer a significant sensitivity loss to the light WIMPmass. B. Via Lactea II Velocity Distributions
Do our findings depend on the particular choice of the mockspectrum? To check this, we repeat the analysis, but insteadgenerate signal events from the high resolution N-body dark
Via Lactea II scaledMBVia Lactea II raw (cid:248) m X (cid:72) GeV (cid:76) Σ n (cid:144) (cid:45) c m Ge, 100 kg (cid:45) yr, 95 (cid:37)
CL7 GeV, 10 (cid:45) cm (cid:45) (cid:45) cm (cid:45) v (cid:72) km (cid:144) s (cid:76) (cid:76) (cid:72) v (cid:76) FIG. 4: Here we compare the CL regions when for the same ( m X =8 GeV, σ n = 10 − cm ) example when coming from three differ-ent velocity distributions: Maxwell-Boltzmann (black, dashed), ViaLactea II raw distribution (red dotted) and Via Lactea II scaled to v = 220 km/s (blue, solid). Bottom : Likelihood functions of thevelocity dispersion for all three distributions from above for both the7 GeV and 8 GeV examples. Here solid, dashed, and dotted curvesare MB, VL-II scaled and VL-II raw distributions. matter simulations borrowed from the Via Lactea II (VL-II)project [66]. The effect of such distributions on direct detec-tion phenomenology has been previously studied in [52, 67]and constraints from existing experiments derived in [29]. Theeffect of tidal debris in VL-II was first observed in [55] andsubsequently applied to direct detection in [56, 68], thoughhere we focus on the full contents of the velocity distribution.As we will show, the existence of the mass-dispersiondegeneracy does not depend on the input spectrum comingfrom a Maxwell-Boltzmann form of the velocity distribution.Moreover, an experimentalist is unlikely to be able to excludethe Maxwellian hypothesis if the true halo is of a Via Lacteaform. Note that although the unscaled VL-II distribution isnot a realistic approximation of the Milky Way halo, we in-clude it here for comparison. The scaled VL-II distribution is scaled such that its dispersion is 220 km/s.As can be seen in Fig. 4 for the 8 GeV case, the same qual-itative features of mass and dispersion mis-measurement per-sist as well, though differing somewhat in the quantitative de-tails. There we see the small effect induced from the raw VL-II distribution which favors low mass solutions compared tothe other two distributions. An important observation is thatthe best-fit points have χ per degree of freedom that is veryclose to one, indicating that the isothermal fit is good. In-terestingly, the values of m X and σ n obtained in the best-fitpoint are ∼
15% off from the original inputs. Nevertheless,since in an actual experiment the true (“input”) values are un-known, the fact that the distribution is non-Maxwellian willnot be registered.Lastly, we ask how well direct detection can determinethe dispersion of the halo. One way of addressing this is tomarginalize over the mass and cross section to determine thedispersion likelihood function, shown in the bottom panel ofFig. 4. In the 8 GeV case (red curves) the dispersion like-lihood function from D
EMONSTRATOR is extremely broad,with a poor ability to reconstruct the correct dispersion. Itsbest fit is peaked near 150 km/s, but with significant likelihoodextending all the way to dispersions of 600 km/s. The high-statistics 7 GeV case fares a bit better with a best-fit value of173 km/s for the MB and a best-fit of 207 km/s for the scaledVL-II distribution. Further, though there the 7 GeV starts toshow visible separation of the three distributions, it is not suf-ficient for discriminating among velocity distributions.
C. Heavier WIMPs and XENON1T
In the above we have focused on the light WIMP case sincethe lowering of energy thresholds in upcoming experimentscould easily lead to an immediate and huge signal. Significantstatistics in the more canonical heavy WIMP case will requiresignificantly larger experiments such as XENON1T [62].We stress that the energy threshold we have chosenis conservative, and will hopefully be significantly lowergiven that XENON1T will have a larger light yield thanXENON100 [62]. Moreover, if XENON1T (or LUX) canlower their energy thresholds, they will have unprecedentedsensitivity to a hitherto unexplored region of WIMP parame-ter space. Like the XENON10 and XENON100 experiments,XENON1T uses a combination of amplified charge signals(S2) and prompt scintillation light (S1) to discriminate signalfrom background. Such low-thresholds are for example poten-tially achievable with the use of an S2-only analysis such asthe one already performed by XENON10 [21]. Alternatively,XENON1T may be able to use the ionized free electron sig-nal as their primary energy estimator in order to obtain energythreshold approaching a few keV [69]. Using the outer lay-ers of the detector for self-shielding, the 3 tons of total liquidXenon translate to 1.1 tons of fiducial detector mass.As an example, consider a 100 GeV WIMP with a 10 − cm scattering cross section, which produces ∼
100 events at (cid:248)(cid:248)
100 200 300 400 5000100200300400500600 m X (cid:72) GeV (cid:76) v (cid:72) k m (cid:144) s (cid:76) Xe, 2.2 ton (cid:45) yr, 95 (cid:37) CL (cid:248)
100 200 300 400 5000.51.01.52.02.53.0 m X (cid:72) GeV (cid:76) Σ n (cid:144) (cid:45) c m FIG. 5: Here the signal spectrum comes from a v = 230 km/shalo with WIMP mass 100 GeV and cross section − cm . Inboth figures, 68 % (black) and 95 % CL (red and blue) are displayed.
Above : One slice of the best-fit region with the cross section fixed tothe true value, but floating v and m X . Bottom : The ensuing allowedregions coming from the canonical fit with fixed dispersion (black)and v marginalized contour. our XENON1T mock-up. Though the relation describing thedegeneracy at low masses (Eq. 9) no longer holds, we seein the top panel of Fig. 5 that a similar degeneracy persistsat XENON1T. That the analytic description of the degener- Note that here we use a “product of Poissons” likelihood function as inEq. (7) since the number of signal events in each bin can be small, thoughour results to not deviate significantly from a χ analysis. acy does not hold here should not be surprising since v min israther small in this case.When the dispersion is marginalized out (bottom panel ofFig. 5), we see that while the dispersion has an impact on theparameter reconstruction, it is less significant than in the lightWIMP case. In addition, the 95 % CL contours (dashed) en-counter a well known high mass degeneracy [28]. This addi-tional degeneracy appears because at sufficiently high WIMPmass v min no longer contains any information about m X .Thus the only place in the differential scattering rate (Eq. 4)containing the WIMP mass is in the ratio σ n /m X such thatthis combination forms an irreducible degenerate direction inparameter space. This example also offers poor prospects fordispersion determination, yielding at 1 σ , v = 144 +294 − km/s.This skewed parameter determination is again due to the highmass/low dispersion models seen in the top panel of Fig. 5that well mimic the input spectrum.Lastly, one may wonder at what mass m X does the mass-degeneracy relation starts to take effect. In examining runsusing our XENON1T setup, we find that by masses around 20GeV the skewed σ − m X regions (such as Fig. 3) and “sock-like” m X − v regions (such as Fig. 2) start to appear. Thuseven for more canonical “high-threshold” experiments the de-termination of the dispersion can be critical for precise DMmass measurements. D. Streams
Unbound streams of dark matter are generally expectedin the hierarchical process of structure formation [70] aswell as in non-standard halo models, such as the late-infallmodel [71]. The stream coming from the tidal disruptionof the Sagittarius dwarf galaxy is perhaps the most well-studied such example [53, 54, 72, 73], though N-body simu-lations predict a wealth of velocity substructure. One of theirmost dramatic signatures is in annual modulation searches, inwhich they can disrupt the prediction of a sinusoidal variationin time [54, 73].However, even at the level of the nuclear recoil spectrum,dispersionless streams produce rather distinctive phenomenol-ogy, with the stream contribution to g ( v min ) appearing as arelatively sharp step-function. Generally, one expects suchstreams to form a subdominant contribution to the local den-sity, ξ ≡ ρ st /ρ DM < , with the remainder of the local DMpopulation described by an equilibrium distribution such asMaxwell-Boltzmann [74]. Thus suppose when faced with asignal of DM, one is interested in excluding the MB only hy-pothesis. Then when fitting to a stream + MB signal with aMB distribution, we can roughly estimate an expected sen-sitivity to streams scaling as N st > (cid:112) N MB + N bkg , with N st , N MB , and N bkg , being the number of events in the rele-vant energy bin(s) from the stream, MB halo, and backgroundrespectively. Thus, for irreducible backgrounds, the best sen-sitivity to streams occurs when the stream contributes to en-ergy bins where the background dominates the MB signal.There one expects the ability to exclude the MB-only hy-pothesis when N st > (cid:112) N bkg . We simulated spectra for theparameters of the Sagittarius stream, for which v str ≈ km/s [54]. With this scaling, in the case of the 7 GeV, − cm example studied earlier, the M AJORANA D EMONSTRA - TOR would allow 1 σ detection of the Sagittarius stream if itsdensity is ξ (cid:38) .Let us consider the other extreme case, in which the back-ground is completely subdominant to the MB signal which hasbeen recently studied in detail by [73]. There they have foundthat a 10 GeV WIMP capable of explaining the CoGeNT re-sults, would have a systematic mass underestimate and crosssection overestimate [73]. The authors of [73], find that a 5%Sagittarius stream could be detectable at the σ level with a10 kg-yr exposure on the C-4 CoGeNT upgrade. We plan toreturn to this topic in the future, exploring the implications forthe M AJORANA D EMONSTRATOR . V. DISCUSSION AND CONCLUSIONS
At the outset of this paper we set out to ask a basic questionthat we may well be faced with shortly: If DM is a light WIMPwith a large cross section, will we have sufficient informationto infer both the particle physics and the local astrophysics ofDM? We have found that an accurate reconstruction of boththe mass and the dispersion will be unlikely even when thesignal consists of many thousands of events. This is primar-ily due to a previously unexplored degeneracy between theWIMP mass and the local velocity dispersion. Indeed, theuncertainty on the average velocity of the local WIMP popu-lation is likely to be the largest factor inhibiting an accuratemass measurement.In addition, we have found that this conclusion holds if thesignal spectra came from an isothermal halo or one describedby the high resolution Via Lactea-II simulation. We stressas well that despite these simulations including tidal debriswhich is made up of unequilibrated structure [55, 56], an ex-periment like the M
AJORANA D EMONSTRATOR will not beable to infer a deviation from the standard isothermal halo. The phenomenologically more distinctive streams, however,could be well-identified with sufficient statistics. Of course,improved sensitivity to streams and other tidal debris wouldcome from the inclusion of modulation or directional infor-mation (for recent work see [75–80]), which we have ignoredin this first analysis.In view of our findings, it is clear that dark matter mass de-termination could well hinge on our ability to better constrainthe velocity distribution of the Milky Way by other means.Such an improvement could come about from improved an-alytic modeling, for example including self-consistent distri-butions that take into account the relevant astrophysical dataalong the lines of [33, 81], as well as from more detailed andaccurate numerical simulations of dark matter halos. We planto return to these questions in future work.
Acknowledgments
We would like to thank Elena Aprile, Michael Marino, An-tonio J. Melgarejo, and Peter Sorenson for helpful correspon-dence. This work was supported by the LANL LDRD pro-gram.
Appendix: The Velocity Integral
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