Integrating out Astrophysical Uncertainties in the Search for Dark Matter

Integrating out astrophysical uncertainties in the search for Dark Matter

Patrick Fox

arXiv:1011.1915 [hep-ph] PF, Liu and Weiner

Thursday, February 10, 2011 Integrating out astrophysical uncertainties in the search for Dark Matter or, Dark Matter when astrophysics Don’t Matter

Patrick Fox

arXiv:1011.1915 [hep-ph] PF, Liu and Weiner

Thursday, February 10, 2011 Plan

•Direct detection and the standard assumptions •Comparing multiple experiments, extracting g(v) •Some case studies •e.g. CoGeNT vs XENON •CRESST vs CDMS/XENON •DAMA vs XENON •Future ideas, a nice plot, and how to use it •Conclusions

Thursday, February 10, 2011 distribution1, f(v, t), by,

∞ f(v, t) g(v ,t)= dv . (2) min v !vmin

There is a minimum speed that the DM must have in order to deposit recoil energy ER in the detector. For elastically scattering WIMPs this minimum velocity is

MT ER vmin = . (3) " 2µ2

Making a comparison between diﬀerent experiments is confused by the fact that it is not a single velocity that contributes to the scattering rate at a particular ER. Rather, all particles with velocities greater than vmin will contribute, making it impossible to map rates into velocity space.

However, we can considerHere we a use related the space reduced – vmin mass-space, defined whose elementswith respect are the to sets the of independent of their mass (and thus, have several candi- all particles with velocitiesincoming greater particles, than vmin. Because all particles with adequately high dates of different masses with similar abundances, using for example the WIMPless miracle [43]). velocities contribute, it is reasonable to consider a mappingmχmN between ER and vmin through Direct Detection µ . (1) The event rate of dark matter scattering [44], differen- (3). ≡ mχ + mN vmax tial in ER, is determined by This simple relationshipdR allowsN usT M to compareT ρ results from3 diffferent("v, directv"2E ) detection ex- The recoil energy of the collision is ER = q /2mN! with vmax = 2 d "v σ(ER) dR NT ρχi 3 d σi $vi periments without making an assumption about! the distribution of DM velocities in the = d $vi fi($vi(t)) | | , (7) dER 2mχ2 µ 2 vmin2 v dE m dE q = p + p! 2pp! cos θcom . (2) R i χi vi,min R Milky Way’s halo, provided one can relate the scattering− cross sections at the various ex- * + periments. In the standardThe recoil cases of of energy SI or SDE DMR, velocity the nuclearv and scattering cos θlab crossare section related can where the sum is over different species of WIMPs, m N & be relatedSI to or the SD nucleonic by,dark (in matter: this case the proton) cross section as Amp is the nucleus mass with mp the proton mass and

2 2 A the atomic number. The recoil energy depends on the µ2 v(f Z +mfχ (A Zm))χ σ (E )=σ pδ n −v F 2(2Em) E cos θ (4) kinematics of the collision, as described above. Given SI R p µ2 χ f 2 R N ! R lab nχ 2 mχ!p − mχ! our assumption of no significant time variation in the 2 2 a2 S (E )+a !a S (E )+a2 S (E ) σp µ p pp R p mnNpn! R n nn R rate, f($vi(t)) f($vi), and thus we are effectively ne- σSD(ER)= E 1+ + δ + δ =0,. (3)(5) 2J +1µ2 R a2 χ N → nχ #− mχ p $ glecting the Earth’s motion around the Sun. This is a " # ! $ % reasonable approximation so long we are probing veloci- allowing comparison of different experiments, we have defined µ as the DM-nucleon re- Define δ δχ + δN . If δ > 0, we cannχ safely perform an ties larger than Earth’s velocity in the Sun’s frame, i.e., ≡ duced mass.Kinematicsexpansion of DM in δscattering:/m 1 to obtain v 30 km/s. Typically the maximum speed is taken # max ! Let us suppose we have two experiments to compare, with targets T1,2 with masses M1,2. to be vmax = vearth + vesc, the galactic escape velocity 1 mN ER v = + δ (1). (1) (4) boosted into the Earth frame. However, vmax is ulti- We assume the first has a signal whichmin appears over an energy range [Elow,Ehigh]. This √2mN ER µ mately determined by the (unknown) details of the dark low high # $ energy range correspond to vmin ranges [vmin,vmin ], using (3). matter velocity distribution in Earth frame. which taking δ 0 is the well-known result for in- N → Given our assumption of no direction dependent signal, 1 It is usually assumed thatelastic the DM dark follows matter a Maxwell-Boltzmann (iDM) [40–42]. distribution By (in the“safe” galactic we frame), mean with Thursday, February 10, 2011 we can carry out the angular integral in Eq. (7), reduc- 2 v2/v2 characteristic speed v0,that in which our case upper (again in bound the galactic on frame)vmin,f( whichv) v e− is in0 . the far non- ∝ ing it to a one dimensional integral where we introduce relativistic regime, automatically implies δ mχ,mN 1 | | # the quantity f1(v)= dΩf($v). The differential rate to allow scattering4 to be kinematically possible. becomes Up to higher order terms in δ/m, we obtain an expres- , sion for the recoil energy dR NT ρχi mN 2 = 2 FN (ER) dER µi mχi µ µ2 i 2 2 2 2 * vmax ER +2ER (δ µv cos θlab)+ 2 δ = 0 (5) mN − m dv v f (v )¯σ (v ,E ) , (8) N × i i i1 i i i R +vi,min The recoil energy is unique for a given fixed scattering where we have written relative velocity v and nucleus recoil angle θlab and can be solved by the usual quadratic formula, dσ m i = F 2 (E ) N σ¯ (v ,E ) (9) dE N R µ v2 i i R µ 2 2 R i i ER = µv cos θlab δ (6) mN − 2 in terms of the nuclear form factor FN (ER). There are &'2 2 1/2 (2 2 1/2 (µv cos θlab) µv cos θlab 2δ . several possible forms for the scattering cross section ± − σ¯ (v, E ), depending on the interaction, ) i R This result has the well known' feature that the( smallest 2 2 recoil energies come from maximizing v cos θlab, corre- σi0 2 sponding physically to head-on collisions at the highest σi0F (ER) σ¯ (v, E )= χi . (10) velocities available. i R σ (v)F 2 (E )  i0 χi R  σi0(v, ER)  III. EVENT DISTRIBUTIONS The different forms forσ ¯correspond to functional forms of known dark matter scattering that contain velocity Our basic assumptions consist of assuming the scat- and/or recoil energy dependence. The first possibility, tering process is off only one type of nuclei. We will, a constant independent of v and ER is the well-known however, remain general with respect to the possibility of isotropic (s-wave) cross section that results at lowest multiple WIMPs with different masses, abundances, and order in the non-relativistic expansion from many dark cross sections. One might think it requires a large coin- matter models. cidence to have several dark matter particles with cross sections large enough to produce events in an experiment. However, there are well known counterexamples where it can be natural to have the abundance of particles to be 1 The velocity distribution is normalized such that d3vf(v) = 1. ! 3 per unit detector mass at a DM direct detection experiment is given by [22] dR N m ρ f("v,"v ) = T N χ d3"v E σ F 2(E ) , (2.1) dE 2 µ2 m v N R R Nχ χ !vmin where m Am is the nucleus mass with m the proton mass and A the atomic number; N ≈ P P F (ER) is the nuclear form factor and accounts for the fact that the cross section drops as one moves away from zero momentum transfer; the two-parameter Fermi charge distribution is used to calculate F (ER) throughout this paper [23]; NT is the number of target nuclei per unit mass, given by N = N /A with Avogadro’s number, N =6.02 1026 kg−1; σ is the T A A × N cross section to scatter of a nucleus, and µNχ is the reduced mass of the DM-nucleus system. 3 The DM mass is mχ and we take the local DM density to be ρχ =0.3 GeV/cm . The velocity of the dark matter onto the (Earth-borne) target is "v. The Earth’s velocity in the galactic frame, "vE, is the sum of the Earth’s motion around the Sun [22] and the Sun’s motion in the galaxy [24]. We assume the WIMP velocity distribution is Maxwell-Boltzmann with velocity Direct Detection [see also, Drees and Shan] dispersion v0 = 220 km/s. Thus, Is a signal a measurement of particle physics or astrophysics? 1 2 2 f("v,"v )= e−("v+"vE ) /v0 . (2.2) E 2 3/2 The only way( πwev0 )have of probing our local DM distribution As a function of time in the galactic frame, the Earth’s velocity is v 227+14.4 cos [2π t−t0 ] f(v) has considerable E ≈ T nd 2 km/s, with T = 1 year and t is around June 2 . The DM velocity distribution3 is−( cut-ov/v0) ff 0 f(v) ∝ d ve " # uncertainty −1 at the galactic escape velocity. Thus, the upper limit of the integration in (2.1)v0 = 220 is given km s by " v

streams ! "v + "v v , and the lower• limit, since we will consider elasticf scatters, is given by | E| ≤ esc •voids •escape velocity mN ER vmin = 2 . (2.3) functional form 2 µ v v • $ Nχ 0 esc (differences in the large v The current allowed range for the galactic escape velocity [25] is 498 km/s v 608 velocity tail) ≤ esc ≤ km/s. For concreteness we set vesc = 500 km/s. Increasing this value slightly increases our allowed parameter space, but the general features remain unchanged. Because of different energy detection efficienciesThursday, for February di 10,ff 2011erent detectors, a quench factor fq is introduced to relate the observed recoil energy, E¯R, to the actual recoil energy ER, ER = E¯R/fq. This allows one to convert Eq. (2.1) to the experimental differential spectrums as dR/dE¯R =1/fq dR/dER. For example, we take the quench factor fq =0.085 for the iodine element in the DAMA experiment. In the usual calculation the nuclear cross section σN is related to the nucleon scattering cross section, σp, by, 2 2 (Zfp +(A Z)fn) µNχ σN = 2− 2 σp , (2.4) fp µnχ where fp,n are the coupling strengths of DM to protons and neutrons and µnχ is the DM- nucleon reduced mass. Here however, we wish to work explicitly with the nuclear scattering cross section, and leave relating it to the microscopic Lagrangian to later, section 3. In

– 3 – per unit detector mass at a DM direct detection experiment is given by [22] dR N m ρ f("v,"v ) = T N χ d3"v E σ F 2(E ) , (2.1) dE 2 µ2 m v N R R Nχ χ !vmin where m Am is the nucleus mass with m the proton mass and A the atomic number; N ≈ P P F (ER) is the nuclear form factor and accounts for the fact that the cross section drops as one moves away from zero momentum transfer; the two-parameter Fermi charge distribution is used to calculate F (ER) throughout this paper [23]; NT is the number of target nuclei per unit mass, given by N = N /A with Avogadro’s number, N =6.02 1026 kg−1; σ is the T A A × N cross section to scatter of a nucleus, and µNχ is the reduced mass of the DM-nucleus system. 3 The DM mass is mχ and we take the local DM density to be ρχ =0.3 GeV/cm . The velocity of the dark matter onto the (Earth-borne) target is "v. The Earth’s velocity in the galactic frame, "vE, is the sum of the Earth’s motion around the Sun [22] and the Sun’s motion in the galaxy [24]. We assume the WIMP velocity distribution is Maxwell-Boltzmann with velocity Direct Detection [see also, Drees and Shan] dispersion v0 = 220 km/s. Thus, Is a signal a measurement of particle physics or astrophysics? 1 2 2 f("v,"v )= e−("v+"vE ) /v0 . (2.2) E 2 3/2 The only way( πwev0 )have of probing our local DM distribution As a function of time in the galactic frame, the Earth’s velocity is v 227+14.4 cos [2π t−t0 ] f(v) has considerable E ≈ T nd 2 km/s, with T = 1 year and t is around June 2 . The DM velocity distribution3 is−( cut-ov/v0) ﬀ 0 f(v) ∝ d ve " # uncertainty −1 at the galactic escape velocity. Thus, the upper limit of the integration in (2.1)v0 = 220 is given km s by " v

streams ! "v + "v v , and the lower• limit, since we will consider elasticf scatters, is given by | E| ≤ esc •voids •escape velocity mN ER vmin = 2 . (2.3) functional form 2 µ v v • $ Nχ 0 esc (differences in the large v The current allowed range for the galactic escape velocity [25] is 498 km/s v 608 velocity tail) ≤ esc ≤ km/s. For concreteness we set vesc = 500 km/s. Increasing this value slightly increases our allowed parameter space, but the generalBut, we features have remainmore than unchanged. one experiment! Because of different energy detection efficienciesThursday, for February di 10,ff 2011erent detectors, a quench factor fq is introduced to relate the observed recoil energy, E¯R, to the actual recoil energy ER, ER = E¯R/fq. This allows one to convert Eq. (2.1) to the experimental differential spectrums as dR/dE¯R =1/fq dR/dER. For example, we take the quench factor fq =0.085 for the iodine element in the DAMA experiment. In the usual calculation the nuclear cross section σN is related to the nucleon scattering cross section, σp, by, 2 2 (Zfp +(A Z)fn) µNχ σN = 2− 2 σp , (2.4) fp µnχ where fp,n are the coupling strengths of DM to protons and neutrons and µnχ is the DM- nucleon reduced mass. Here however, we wish to work explicitly with the nuclear scattering cross section, and leave relating it to the microscopic Lagrangian to later, section 3. In

– 3 – Two(+) experiments allow us to test particle physics independently of astrophysics In the past all experiments were “projected” into x-sec--mass plane. Confused discovering DM with measuring3 DM

!39.0 !39.0 ! CDMS!Si !39.5 CDMS Si !39.5 " COGENT " COGENT !N 40.0 !N 40.0 Σ Σ ! ! ! 10 10 40.5 !40.5 log log !41.0 !41.0 ! XENON XENON 41.5 !41.5 CDMS ! CDMS 42.0 !42.0 4 6 8 10 12 4 6 8 10 12 mDM !GeVThursday, February" 10, 2011 mDM !GeV"

FIG. 1: The regions in the elastic scattering cross section (per nucleon), mass plane in which dark matter provides a good fit to the CoGeNT excess, compared to the region that can generate the annual modulation reported by DAMA at 90% confidence (darker grey regions). In this figure, we have adopted v0 = 270 km/s and use two values of the galactic escape velocity: vesc = 490 km/s (left) and vesc = 650 km/s (right). In calculating the DAMA region, we have neglected the lowest energy bin (the effect of this is shown in later figures) and treated channeling as described in Ref. [26]. If a smaller fraction of events are channeled in DAMA than is estimated in Ref. [26], the DAMA region will move upward, toward the yellow regions (near −39.5 2 σN ≈ 10 cm , which include no effects of channeling), improving its agreement with CoGeNT. Also shown is the 90% C.L. region in which the 2 events observed by CDMS can be produced. If the escape velocity of the galaxy is taken to be relatively large, this region can also approach those implied by CoGeNT and DAMA. Constraints from the null results of XENON10 and the CDMS silicon analysis are also shown. For the XENON10 constraint, we have used the lower estimate of the scintillation efficiency (at 1σ) as described in Ref. [27]. events can be fit very well by a 10 GeV dark matter particle with an elastic scattering cross section of 7 1041 cm2. In Fig. 1, we confirm this conclusion,∼ where we show the parameter space region in which elastically∼ scattering× dark matter can accommodate the CoGeNT excess at 90% confidence. In this figure, we have used v0 = 270 km/s and vesc = 490 km/s. Here, and throughout this paper, 90% (99%) confidence regions are defined as contours of 2 2 χ = χmin +4.61 (9.21), while constraints from null experiments are defined as 90% limits based on the maximum gap [28] method. To carry out this fit, we have assumed that the background is well described by an exponential plus constant, and we have required bin-by-bin that the background not exceed the amplitude of the dark matter signal. Without a constraint on dark matter signal to background, the entire spectrum is well fit by a pure exponential background. Tighter constraints on the amplitude of the background will correspond to the dark matter signal region shifting to larger cross sections. We fit the data in 0.05 keV-electron-equivalent (keVee) bins from threshold at 0.4 keVee to 1.8 keVee where the dark matter signal is negligible. Peaks in the data (consistent with a background from radioactive tin) at 1.1 and 1.29 keVee are fit by Gaussians of relative height 0.4 and with width consistent with the experimental resolution at those energies (0.0774 and 0.078 keVee respectively). We can see that for appropriate choices of the halo model and the fraction of channeled events in DAMA, the CoGeNT region can be consistent at 90% C.L. with the DAMA signal and the null results XENON and CDMS-Si. Some consistency between the preferred region for CDMS with DAMA and CoGeNT can also be found. We now turn to discussing in detail how this occurs. The DAMA experiment [1] observes an annual modulation in their count rate, which can be parameterized as R = R0 + S1cos[ω(t t )]. (5) i i i − 0 0 The subscript i in this expression denotes different energy bins. The constant term Ri is composed of both a signal component coming from dark matter initiated processes, and a background component arising from other sources of 0 0 0 0 1 nuclear recoil: Ri = bi + Si . The expressions for Si and Si are obtained by integrating Eq. (4) over a given energy bin. Channeling is a potentially important but difficult-to-predict theoretical effect which can significantly change the interpretation of DAMA’s signal, especially when comparing this signal to the results of other direct detection experi- Two(+) experiments allow us to test particle physics independently of astrophysics In the past all experiments were “projected” into x-sec--mass plane. Confused discovering DM with measuring3 DM

!39.0 !39.0 v $ 230, v $ 600 ! CDMS!Si 0 esc !39.5 CDMS Si !39.5!39.5 " " 2 COGENT " COGENT !N 40.0 !N 40.0!40.0 SIMPLE Σ cm Σ ! ! !

! n 10 10 40.5 !40.5! Σ 40.5 Xenon log 10 CDMS!Si log ! 41.0 !41.0!41.0 XENON XENON ! Log 41.5 !41.5!41.5 CDMS ! 6 8 10 CDMS12 14 16 42.0 !42.0 4 6 8 10 12 4 6 8 MΧ10GeV12 mDM !GeVThursday, February" 10, 2011 mDM !GeV"

FIG. 1: The regions in the elastic scattering cross section (per nucleon), mass plane in which dark matter provides a good fit to the CoGeNT excess, compared to the region that can generate the annual modulation reported by DAMA at 90% confidence (darker grey regions). In this figure, we have adopted v0 = 270 km/s and use two values of the galactic escape velocity: vesc = 490 km/s (left) and vesc = 650 km/s (right). In calculating the DAMA region, we have neglected the! lowest energy" bin (the effect of this is shown in later figures) and treated channeling as described in Ref. [26]. If a smaller fraction of events are channeled in DAMA than is estimated in Ref. [26], the DAMA region will move upward, toward the yellow regions (near −39.5 2 σN ≈ 10 cm , which include no effects of channeling), improving its agreement with CoGeNT. Also shown is the 90% C.L. region in which the 2 events observed by CDMS can be produced. If the escape velocity of the galaxy is taken to be relatively large, this region can also approach those implied by CoGeNT and DAMA. Constraints from the null results of XENON10 and the CDMS silicon analysis are also shown. For the XENON10 constraint, we have used the lower estimate of the scintillation efficiency (at 1σ) as described in Ref. [27]. events can be fit very well by a 10 GeV dark matter particle with an elastic scattering cross section of 7 1041 cm2. In Fig. 1, we confirm this conclusion,∼ where we show the parameter space region in which elastically∼ scattering× dark matter can accommodate the CoGeNT excess at 90% confidence. In this figure, we have used v0 = 270 km/s and vesc = 490 km/s. Here, and throughout this paper, 90% (99%) confidence regions are defined as contours of 2 2 χ = χmin +4.61 (9.21), while constraints from null experiments are defined as 90% limits based on the maximum gap [28] method. To carry out this fit, we have assumed that the background is well described by an exponential plus constant, and we have required bin-by-bin that the background not exceed the amplitude of the dark matter signal. Without a constraint on dark matter signal to background, the entire spectrum is well fit by a pure exponential background. Tighter constraints on the amplitude of the background will correspond to the dark matter signal region shifting to larger cross sections. We fit the data in 0.05 keV-electron-equivalent (keVee) bins from threshold at 0.4 keVee to 1.8 keVee where the dark matter signal is negligible. Peaks in the data (consistent with a background from radioactive tin) at 1.1 and 1.29 keVee are fit by Gaussians of relative height 0.4 and with width consistent with the experimental resolution at those energies (0.0774 and 0.078 keVee respectively). We can see that for appropriate choices of the halo model and the fraction of channeled events in DAMA, the CoGeNT region can be consistent at 90% C.L. with the DAMA signal and the null results XENON and CDMS-Si. Some consistency between the preferred region for CDMS with DAMA and CoGeNT can also be found. We now turn to discussing in detail how this occurs. The DAMA experiment [1] observes an annual modulation in their count rate, which can be parameterized as R = R0 + S1cos[ω(t t )]. (5) i i i − 0 0 The subscript i in this expression denotes different energy bins. The constant term Ri is composed of both a signal component coming from dark matter initiated processes, and a background component arising from other sources of 0 0 0 0 1 nuclear recoil: Ri = bi + Si . The expressions for Si and Si are obtained by integrating Eq. (4) over a given energy bin. Channeling is a potentially important but difficult-to-predict theoretical effect which can significantly change the interpretation of DAMA’s signal, especially when comparing this signal to the results of other direct detection experi- Two(+) experiments allow us to test particle physics independently of astrophysics 1) Make hypothesis about DM e.g. elastically scattering DM with mass 100 GeV and x-sec 10-43 cm2 2) Use experiment A to extract astrophysics i.e. rho x g(v) 3) Use these extracted astrophysics properties to predict result at experiment B 4) Compare to B’s measurement/bound 5) Rule in or out each particle physics hypothesis Doesn’t allow extraction of “unique” x-sec, mass Need relatively large statistics ~10’s events Experiments must run over same part of year Other uncertainties (nuclear, atomic etc) not addressed) Thursday, February 10, 2011 Enter vmin space

vmax dR NT MT ρ 3 f("v, v"E ) = 2 d "v σ(ER) dER 2mχµ !vmin v g(v) " v ! f

= MT ER Recoil energy uniquely determines vmin 2 ! 2µ minimum DM velocity

Thursday, February 10, 2011 This brings to the central point of our efforts: to make a comparison between two ex- This brings to the central point of our efforts: to make a comparison between two experiments one must first determine whether the vmin space probed by the two experiments periments one must first determine whetheroverlaps. the v Asmin aspace matter probed of practical by the course, two a experiments given experiment has a lower energy threshold overlaps. As a matter of practical course,Emin a given, which experiment can be translated has a into lower a lower energy bound threshold on the vmin range. If experiment 1 has data for the differential rate of DM scattering in their experiment, dR /dE at energies E(1) Emin, which can be translated into a lower bound on the vmin range. If experiment 1 has 1 R i (2) this can be used to predict a rate at energy Ei at experiment(1) 2, dR2/dER, or vice versa if data for the differential rate of DM scattering in their experiment, dR1/dER at energies Ei experiment 2 has the signal. Thus, we have this can be used to predict a rate at energy E(2) at experiment 2, dR /dE , or vice versa if Using vmin spacei 2 R (1) (1) low high (2) (2) experiment 2 has the signal. Thus, we have [Elow,Elow] [vmin,vmin ] [Elow,Ehigh], (6) Experiment 1 ↔ Experiment 2 ⇐⇒ ⇐⇒

(1) (1) lowwherehigh (2) (2) [Elow,Elow] [vmin,vmin ] [Elow,Ehigh], (6) ⇐⇒ ⇐⇒ (1) µ2M [E(2) ,E(2) ]= 2 T [E(1) ,E(1) ]. (7) low high 2 (2) low high where µ1MT 1500 (1) low high We2 can invert (1) to solve for g(vmin) limited to the range vmin [v ,v ] (2) (2) µ2MT (1) (1) ∈ min,1 min,2 [Elow,Ehigh]= [Elow,Ehigh]. (7) 1000 2 (2) 2 µ1MT 2mχµ dR1 g(vmin)= (8) NAκ mp ρσ(ER) dE1 700 low high We can invert (1) to solve for g(vmin) limited to the range vmin [vmin,1,vmin,2] v km s This then allows us to explicitly∈ state the expected rate for experiment two, again 2 re- min 500 2 (2) (2) stricted2mχ toµ the energydR1 range dictated by the appropriate velocity range i.e. E [Elow,Ehigh]. g(vmin)= (8) ∈ NAnalogousAκ mp ρσ( toE theR) dE energy1 mapping above, we have a rate mapping, ! # " 300 CRESST DAMA"Na dR 2dR This then allows us to explicitly stateCoGeNT the expected rate for experiment1 two,g(v again) re-2 , (9) 200 min dE1 ⇐⇒ ⇐⇒ dE2 stricted to the energy range dictated by the5 appropriate10 velocity range15 i.e. E 20 [E(2) ,E(2) ]. ∈ low high with mΧ GeV Analogous to the energyThursday, mappingFebruary 10, 2011 above, we have a rate mapping, (2) 2 2 (2) dR2 κ µ1 σ2(E2) dR1 µ1 MT (E2)= E2 . (10) (1) 2 (2) (1) dR1 dRdE2 R κ µ µ2 M dER 2 ! " 2 1 T #µ2 MT $ g(vmin) , σ1 (1) E2 (9) FIG. 1: vmin thresholds for various experiments. Solid bands are CRESSTµ2 OxygenM band, 15- dE dE 2 T 1 ⇐⇒ ⇐⇒ 2 ! " 40 keV (red, top), DAMA Na band 6.7-13.3 keV (green, middle), CoGeNT Ge 1.9-3.9 keV (blue, with Equations (7), (8) and (10) are the central results of this paper. They make no astrophysical bottom). Constraints are Xenon 1, 2assumptions, and 5 keV (dashed, but only dotted, rely upon and dot-dashed, the assumption thick that blue an), and actual signal has been observed. (2) 2 2 (2) dR2 κ µ1 σ2(E2) dR1 µ1 MT CDMS-Si 7 and(E 102)= keV, (dot-dashed andWe dashed, now focus thin red on). the SI case,E2 since. there are a greater(10) number of experiments probing (1) 2 (2) (1) dER κ µ µ2 M dER 2 2 1 T #µ2 MT $ σ1 this(1) scenario,E2 but the analysis for SD is similar. In this (SI) case we can use (5) to rewrite µ2M signals, some without. The possible! 2 comparisonsT " between these various experiments will be 2 Equationsthe (7), subject (8) and of the (10) subsequent are the central sections. resultsSince Usingg(v of), (11) by this its scattering definition,paper. They is rates a monotonically make can be no compared astrophysical decreasing between function of vmin, one can in principle go to lower energies as well, but one may only place a lower bound on the predicted rate, rather than make a assumptions,experiments. but only However, rely upon to comparethe assumption totrue actual prediction. that experimental an actual data signal the has relative been exposures, observed. effi- We nowciencies focus and on other the SI detector-specific case, since there factors are must a greater be correctly number taken of into experiments account. In probing the next 5 this scenario,section but we the describe analysis in detail for SD the is experimental similar. In parametersthis (SI) case necessary we can for use the (5) comparisons to rewrite in the rest of the paper. 2 Since g(v), by its definition, is a monotonically decreasing function of vmin, one can in principle go to lower energies as well, but one may only place a lower bound on the predicted rate, rather than make a III. APPLICATIONS: A COMPARISON OF EXISTING EXPERIMENTS true prediction.

The important consequences of (10) are immediately obvious. In principle, one can com- 5 pare a positive signal at one experiment with one at another, or test the compatibility of a null result with a positive one. Unfortunately, ideal circumstances will rarely present them- selves: additional backgrounds can complicate the extraction of g(v), resolution can smear signals, or uncertainties in atomic physics (such as quenching factors) can complicate issues, making a precise extraction of the true ENR and hence vmin impossible. Furthermore, the signal may appear as a modulation (as in DAMA) limiting access to g(v) to a summer/winter

7 This brings to the central point of our efforts: to make a comparison between two ex- This brings to the central point of our efforts: to make a comparison between two experiments one must first determine whether the vmin space probed by the two experiments periments one must first determine whetheroverlaps. the v Asmin aspace matter probed of practical by the course, two a experiments given experiment has a lower energy threshold overlaps. As a matter of practical course,Emin a given, which experiment can be translated has a into lower a lower energy bound threshold on the vmin range. If experiment 1 has data for the differential rate of DM scattering in their experiment, dR /dE at energies E(1) Emin, which can be translated into a lower bound on the vmin range. If experiment 1 has 1 R i (2) this can be used to predict a rate at energy Ei at experiment(1) 2, dR2/dER, or vice versa if data for the differential rate of DM scattering in their experiment, dR1/dER at energies Ei experiment 2 has the signal. Thus, we have this can be used to predict a rate at energy E(2) at experiment 2, dR /dE , or vice versa if Using vmin spacei 2 R (1) (1) low high (2) (2) experiment 2 has the signal. Thus, we have [Elow,Elow] [vmin,vmin ] [Elow,Ehigh], (6) Experiment 1 ↔ Experiment 2 ⇐⇒ ⇐⇒

(1) (1) lowwherehigh (2) (2) [Elow,Elow] [vmin,vmin ] [Elow,Ehigh], (6) ⇐⇒ ⇐⇒ (1) µ2M [E(2) ,E(2) ]= 2 T [E(1) ,E(1) ]. (7) low high 2 (2) low high where µ1MT 1500 (1) low high We2 can invert (1) to solve for g(vmin) limited to the range vmin [v ,v ] (2) (2) µ2MT (1) (1) Xenon w/ ∈ min,1 min,2 [Elow,Ehigh]= [Elow,Ehigh]. (7) 1000 2 (2) 2 µ1MT 2mχ1,2,5µ keVdR1 g(vmin)= (8) NAκ mp ρσ(ER) dE1 700 low high thresholds We can invert (1) to solve for g(vmin) limited to the range vmin [vmin,1,vmin,2] v km s This then allows us to explicitly∈ state the expected rate for experiment two, again 2 re- min 500 2 (2) (2) stricted2mχ toµ the energydR1 range dictated by the appropriate velocity range i.e. E [Elow,Ehigh]. g(vmin)= CDMS-Si(8) w/ ∈ NAnalogousAκ mp ρσ( toE theR) dE energy1 mapping above, we have a rate mapping, ! # " 300 CRESST 7,10 keV " DAMA Na 2 dR1 thresholdsdR2 This then allows us to explicitly stateCoGeNT the expected rate for experiment two,g(v again) re-, (9) 200 min dE1 ⇐⇒ ⇐⇒ dE2 stricted to the energy range dictated by the5 appropriate10 velocity range15 i.e. E 20 [E(2) ,E(2) ]. ∈ low high with mΧ GeV Analogous to the energyThursday, mappingFebruary 10, 2011 above, we have a rate mapping, (2) 2 2 (2) dR2 κ µ1 σ2(E2) dR1 µ1 MT (E2)= E2 . (10) (1) 2 (2) (1) dR1 dRdE2 R κ µ µ2 M dER 2 ! " 2 1 T #µ2 MT $ g(vmin) , σ1 (1) E2 (9) FIG. 1: vmin thresholds for various experiments. Solid bands are CRESSTµ2 OxygenM band, 15- dE dE 2 T 1 ⇐⇒ ⇐⇒ 2 ! " 40 keV (red, top), DAMA Na band 6.7-13.3 keV (green, middle), CoGeNT Ge 1.9-3.9 keV (blue, with Equations (7), (8) and (10) are the central results of this paper. They make no astrophysical bottom). Constraints are Xenon 1, 2assumptions, and 5 keV (dashed, but only dotted, rely upon and dot-dashed, the assumption thick that blue an), and actual signal has been observed. (2) 2 2 (2) dR2 κ µ1 σ2(E2) dR1 µ1 MT CDMS-Si 7 and(E 102)= keV, (dot-dashed andWe dashed, now focus thin red on). the SI case,E2 since. there are a greater(10) number of experiments probing (1) 2 (2) (1) dER κ µ µ2 M dER 2 2 1 T #µ2 MT $ σ1 this(1) scenario,E2 but the analysis for SD is similar. In this (SI) case we can use (5) to rewrite µ2M signals, some without. The possible! 2 comparisonsT " between these various experiments will be 2 Equationsthe (7), subject (8) and of the (10) subsequent are the central sections. resultsSince Usingg(v of), (11) by this its scattering definition,paper. They is rates a monotonically make can be no compared astrophysical decreasing between function of vmin, one can in principle go to lower energies as well, but one may only place a lower bound on the predicted rate, rather than make a assumptions,experiments. but only However, rely upon to comparethe assumption totrue actual prediction. that experimental an actual data signal the has relative been exposures, observed. effi- We nowciencies focus and on other the SI detector-specific case, since there factors are must a greater be correctly number taken of into experiments account. In probing the next 5 this scenario,section but we the describe analysis in detail for SD the is experimental similar. In parametersthis (SI) case necessary we can for use the (5) comparisons to rewrite in the rest of the paper. 2 Since g(v), by its definition, is a monotonically decreasing function of vmin, one can in principle go to lower energies as well, but one may only place a lower bound on the predicted rate, rather than make a III. APPLICATIONS: A COMPARISON OF EXISTING EXPERIMENTS true prediction.

7 Comparing experiments

e.g. mχ = 10 GeV

Approx. range O Na Si Ar Ge Xe

CoGeNT (Ge): 2 - 4 4.3 - 8.6 3.9 - 7.8 3.6 - 7.2 3.0 - 6.0 2 - 4 1.3 - 2.5 DAMA (Na): 6 - 13 6.6 - 14 6 - 13 5.5 - 12 4.6 - 10 3.1 - 6.7 1.9 - 4.2 CRESST (O): 15 - 40 15 - 40 14 - 36 12 - 33 10 - 28 6.9 - 19 4.3 - 12

TABLE I: Conversionall of energies energy ranges in (all keV, in keV)undoing between quenching various experiments/targets etc for a 10 GeV DM particle, using the expression in (7). thresholdsNow are generallywe know limited who to heavier compares masses. with whom, how do Finally, we see that the CRESSTwe compare results are completelythem? tested by the low-threshold

XENON10Thursday, analysis, February 10, 2011 CDMS-Si (even with a 10 keV) threshold. While the nominal threshold, depending on the details of , of XENON10 ( 5 keV) and XENON100 ( 6 keV) is too Leff ∼ ∼ high, both experiments can probe down to 4 keV with moderately reduced sensitivity, and energy smearing will given XENON sensitivity to the CRESST signal. With these ranges in hand, we can proceed to compare the experiments directly. We shall see that if the potential signal is large enough, g(v) can be extracted directly, even if f(v) cannot be extracted with any reliability. In such cases, we can make slightly stronger statements involving the spectra. However, even if g(v) cannot be reconstructed, we can still make signiﬁcant statements by integrating over the relevant velocity range.

A. Application I: Employing Spectra in Near-Ideal Situations (CoGeNT)

We consider first the situation when there is sufficient data to be able to extract a recoil spectrum, CoGeNT is a example of such an experiment, because the putative signal is quite large. We concentrate on the events below 3.2 keVee where the DM signal should be largest and there are few cosmogenic backgrounds. In this range, in addition to the possible DM signal at low energies, the data contains several clear cosmogenic peaks and a constant background above the peaks. We average the [1.62-3.16 keVee] bins as an estimate of the constant background and subtract this from the bins in the [0.42-0.92 keVee] range, which we then consider as the DM signal, after this subtraction there are 92 signal events before efficiency correction. This allows us to determine g(v) or, equivalently, predict the rate at any other experiment in the equivalent energy range. One can easily observe from its definition that g(v) is monotonically decreasing as a function of v (see, for instance the

12 Direct Detection without bias

vmax dR NT MT ρ 3 f("v, v"E ) = 2 d "v σ(ER) dER 2mχµ !vmin v g(v)

dR NT MT ρσ = 2 g(v) dER 2µ mχ

= MT ER Recoil energy uniquely determines vmin 2 ! 2µ minimum DM velocity

Thursday, February 10, 2011 Direct Detection without bias

vmax dR NT MT ρ 3 f("v, v"E ) = 2 d "v σ(ER) dER 2mχµ !vmin v g(v)

dR NT MT ρσ = 2 g(v) dER 2µ mχ Target speciﬁc

= MT ER Recoil energy uniquely determines vmin 2 ! 2µ minimum DM velocity

Thursday, February 10, 2011 Direct Detection without bias

vmax dR NT MT ρ 3 f("v, v"E ) = 2 d "v σ(ER) dER 2mχµ !vmin v g(v)

dR NT MT ρσ = 2 g(v) dER 2µ mχ Target Target speciﬁc independent

= MT ER Recoil energy uniquely determines vmin 2 ! 2µ minimum DM velocity

Thursday, February 10, 2011 For instance, [38] argued that an independent comparison for the iodine spin-independent explanation of DAMA could be made by studying the comparable range of energy at a This brings to the central pointXenon of our target, e givenfforts: their kinematical to make similarity. a comparison It was pointed out between in [39] that there two is an ex- overlap in velocity space between the 1keVee signal at CoGeNT and the 7 keVr threshold periments one must first determine whether the vmin space∼ probed by the two experiments at CDMS-Si. With positive results at two experiments, a measurement of the WIMP mass overlaps. As a matter of practical course,can be done a without given assuming experiment a halo model has [40]. Finally, a lower [41] studied energy the possibility threshold of extracting f(v) from dark matter experiments in the future when large signals have been Emin, which can be translated intofound. a lower bound on the vmin range. If experiment 1 has In this paper, we take a different approach. Rather than attempt to find the physical dataThis for brings the diff toerential the central rate of point DM of scattering our efforts: in their to make experiment, a comparisondR /dE betweenat energies two ex- E(1) function f(v), or study variations in it, we attempt to directly1 map experimentalR signals fromi (2) one detector to another. We do this by focusing on integral quantities, namely g(vmin)= thisperiments can be one used must to predict first determine a rate at whether energy theEi vminatspace experiment probed 2, bydR the2/dE twoR, experiments or vice versa if dvf(v)/v and dv vg(v). We determine the robustness of constraints by considering vmin experimentoverlaps. As 2 a has matter the signal. of practical Thus, course,!the we relationship have a given between! experiment recoil energy and hasvmin aspace, lower rather energy than actual threshold velocity space. Although in our approaches we will gain less information about astrophysics, we can compare Emin, which can be translated into a lower bound on the vmin range. If experiment 1 has (1) (1) experiments evenhigh when f(v) cannot be(2) reliably(2) extracted. low (1) [Elow,Elow] [vmin,vmin ] [Elow,Ehigh], (6) data for the differential rate of DM⇐⇒ scattering in their⇐⇒ experiment, dR1/dER at energies Ei II. vmin RANGES(2) AND ASTROPHYSICS-INDEPENDENT SCATTERING this can be used to predict a rate at energy Ei at experiment 2, dR2/dER, or vice versa if where RATES experiment 2 has the signal. Thus, we have Our approach2 will(1) be simple: we will endeavor to map an energy range in a given ex- (2) (2) µ M (1) (1) [E ,Eperiment]= into the2 haloT velocity[E space,,E and from]. there into any other experiment we wish to(7) (1) (1)low high low high2 (2) low (2) high(2) [E ,E ] compare[v to.,vµ InM this] way, we[ canE determine,E what], energy ranges of experiments can(6) be di- low low ⇐⇒ min min1 T⇐⇒ low high rectly compared. In optimal situations, we will be able to extract g(v), while in less optimal situations we will only be able to discuss total rates. low high whereWe can invert (1) to solve for g(vmin) limited to the range vmin [vmin,1,vmin,2] We begin with the differential rate at a direct detection∈ experiment, which for elastically scattering DM2 is(1) given by,2 (2) (2) µ M2m µ(1) (1)dR [E ,E ]= 2 T [χE ,E ].1 (7) lowg(vminhigh)= low dR highNT MT ρ (8) 2 (2) = σ(E ) g(v ) , (1) Nµ Mκ m ρσ(E ) dE 2 R min Comparing experimentsA1 T p dERR 2m1χµ where µ is the DM-nucleus reduced mass, and N = κN m /M is the number of target T lowA p highT 2 ThisWe then can allows invertHW (1) us problem: to to solve explicitly for solveg(v statemin for) g(v)limited the expected to the range ratev formin experiment[v ,v two,] again re- scattering sites per kg with NA Avogadro’s number∈ andmin,κ the1 massmin, fraction2 of the detector that is scattering DM. The function g(vmin) is related to the integral of the(2) DM speed(2) stricted to the energy range dictated by the appropriate2 velocity range i.e. E [E ,E ]. 2mχµ dR1 ∈ low high g(vmin)= (8) Analogous to the energy mapping above,NAκ wemp haveρσ(E aR) ratedE1 mapping, 3 (10) in a simple form 2 This then allows us to explicitlydR state1 the expected ratedR for2 experiment two, again re- (2) 2 2 (2) g(vmin) dR2 C, T F2 (E2) dR1 µ1 MT (9) (E2)= E2(2), (2) (11) (1) (2) (1) dE ⇐⇒ ⇐⇒dER dE µ2 M dER 2 stricted to the energy range dictated1 by the appropriate velocity2 CT range2 1 T i.e. E#µ2 MT[E $ ,E ]. F (1) E2 low high 1 µ2M ! 2 T " ∈ withAnalogous to the energy mapping above, wewhere have we have a introduced rate mapping, a target specific coefficient The master formula (SI): (i) (i) (i) (i) (i) 2 CT = κ fp Z + fn (A Z ) . (12) (10) in a simple form dR(2)1 2 dR2 2 (2) − dR2 κ µ1 g(σv2min(E)2) dR1, µ1%MT & (9) (E2)= In certain situations differential rates may notE2 be available. and instead it is only possible(10) dE(1)1 ⇐⇒2 (2)⇐⇒ dE2 (1) dER κ µ µ2toM compare total rates,dER this is the2 situation at present with CRESST. In general the total dR C(2)2 σ F 21(E T) E dR µ#2 Mµ2(2)MT $ 2 T 1 2 2rate2 at(1) a particular2 1 experiment1 withT energy — and corresponding velocity — thresholds of (E2)= µ M E2 , (11) with (1) 2 (2)T (1) dER µ2 (ME ,vlow ) anddE (ER ,vhigh),2 can be expressed as, CT 2 ! 1 lowT min " high#minµ2 MT $ F1 2 (1) E2 µ M vhigh (2) 2 2 T 2NA2ρ mp κ(2) Equations (7), (8)dR and2 (10) areκ theµ central!σ2(E results2) "dR of1 thisR = µ paper.M Theydv #(ER) makeσ(ER(v))vg no(v) . astrophysical(13) 1 m1χ MTT (E2)= 'vElow 2 . (10) Thursday, February 10, 2011 (1) 2 (2) (1) where we havedE introducedR κ a targetµ specificµ2 MFor coe theffi particularcientdE caseR of SI2 on which we are focused this becomes, assumptions, but only rely upon2 the assumption1 T that# anµ2 actualMT signal$ has been observed. σ1 2 (1) E2 µ M 2N ρσ m µ2C vhigh 2 T R = A p p T dv #(E )F 2(E (v))vg(v) , (14) (i) ! " m µ2 f 2 2 M R R We now focus on the SI case, since(i) there(i) are( ai)! greaterχ n(χi) p "! numberT " 'vlow of experiments probing CT = κ fp Z + fn (A Z ) . (12) Equations (7), (8) and (10) are the central resultswhere #(ER of) an this an− energy-dependent paper. They efficiency. make To compare no astrophysical two experiments, we must this scenario, but the analysis for SD% is similar.extract the energy In thisdependent (SI)& terms case from the we integral. can So while use we (5) make to no assumptions rewrite In certain situations differential rates may not be available and instead it¯ is¯ only2 (1) possible2 (2) assumptions, but only rely upon the assumptionabout g(v), that we evaluate an actual the form factor signal at a value hasE2 been= E1µ2M observed.T /µ1MT where the ra- ¯ 2 ¯ ¯ 2 ¯ tio #2(E2)F2 (E2)/#1(E1)F1 (E1) is minimized or maximized, depending on whether we are 2 to compare total rates, this is the situation at present with CRESST. In general the total SinceWe nowg(v), focus by its on definition, the SI is case, a monotonically since thereconsidering are decreasing a a putative greater function signal number or constraint. of vmin of Thus experiments, comparisons one can of in rates principle probing at two experiments go to thislower scenario,rate energies at a butparticular as well, the analysisbut experiment one may for SDonly with is place energy similar.may a lower — then and In be simply bound this corresponding compared (SI) on theby case taking predicted we ratiosvelocity can of CT rate,with use — the thresholds (5) rather form to factor rewrite than evaluated of make at the a conservative value E¯, true prediction.low high (Elow,vmin) and (Ehigh,vmin ), can be expressed as, (2) (1) # (E¯ )F 2(E¯ ) C M µ2 R 2 2 2 2 T T 2 R . (15) 2 2 ≤ # (E¯ )F 2(E¯ ) (1) (2) µ2 1 Since g(v), by its definition, is a monotonicallyvhigh decreasing function of1 v1 1 ,1 oneCT M canT 1 in principle go to 2NAρ mp κ min lower energies as well, butR one= may only place aIn lower order5dv to# bound( determineER)σ on(E whatR the( comparisonsv)) predictedvg(v) can. be rate, made rather between experiments, than make(13) we must a ex- mχ MT vlow true prediction. ' amine the relevant velocity space they probe. We re-emphasize that the signal at energy For the particular case of SI on which weE arelow < focused E < Ehigh is this sensitive becomes, to all particles with velocity greater than vmin(E,MN ,Mχ) through the integral g(vmin). A separate experiment with threshold E˜ will offer constraints

2 independentvhigh of astrophysics if the resulting minimum velocityv ˜ < v2. The optimal limits are 2NAρσpmp µ CT 5 2 reached whenv ˜ < v1. We illustrate this in Fig. 1 for an ensemble of experiments, some with R = 2 2 dv #(ER)F (ER(v))vg(v) , (14) mχ µnχ fp MT vlow ! "! " ' 6 where #(ER) an an energy-dependent eﬃciency. To compare two experiments, we must extract the energy dependent terms from the integral. So while we make no assumptions ¯ ¯ 2 (1) 2 (2) about g(v), we evaluate the form factor at a value E2 = E1µ2MT /µ1MT where the ra- ¯ 2 ¯ ¯ 2 ¯ tio #2(E2)F2 (E2)/#1(E1)F1 (E1) is minimized or maximized, depending on whether we are considering a putative signal or constraint. Thus comparisons of rates at two experiments may then be simply compared by taking ratios of CT with the form factor evaluated at the conservative value E¯,

# (E¯ )F 2(E¯ ) C(2) M (1) µ2 R 2 2 2 2 T T 2 R . (15) 2 ≤ # (E¯ )F 2(E¯ ) (1) (2) µ2 1 1 1 1 1 CT MT 1 In order to determine what comparisons can be made between experiments, we must ex- amine the relevant velocity space they probe. We re-emphasize that the signal at energy

Elow < E < Ehigh is sensitive to all particles with velocity greater than vmin(E,MN ,Mχ) through the integral g(vmin). A separate experiment with threshold E˜ will oﬀer constraints independent of astrophysics if the resulting minimum velocityv ˜ < v2. The optimal limits are reached whenv ˜ < v1. We illustrate this in Fig. 1 for an ensemble of experiments, some with

6 There’s an app for that

Thursday, February 10, 2011 difference, or a lower bound on its mean. Finally the signal may be of such low statistics that a reliable inference on the shape of g(v) will be impossible, as is expected in many experiments before scaling to larger targets or running for longer exposures. Nonetheless, in light of these challenging issues, there remain meaningful comparisons that can be made between experiments. Especially since these transformations preserve all information in the signal, we should be able to make the strongest possible relative statements without invoking additional assumptions about the halo. Such results are especially interesting in view of recent results that may pertain to light WIMPs. Since light WIMPs probe the highest part of the velocity distribution, where deviations from Maxwellian properties are the most likely, our approach is especially relevant. We consider three potential signals: the CoGeNT low-energy excess [7], the DAMA annual modulation signal [42] and the recently reported Oxygen-band events at CRESST [43] and for constraints: XENON10 (both conventional analyses and S2-only) and CDMS-Si; we describe the relevant parametersCoGeNT necessary[Collar for et comparison al. (1002.4703)] between the various experiments 3 below. signal region •Avoid cosmogenic peaks CoGeNT •Large number of events, 92 events after background The CoGeNT experiment [7] consists of a low noise germaniumsubtraction detector with 330 g of flat background fiducial mass which has reported data for 56 days of exposure.•11 CoGeNTbins reports recoil energies that range from 0.4 keVee to 12 keVee, but we•Assume focus here DM on only the eventscouples to ∼ ∼ proton (e.g. kinetic mixing) between 0.4 keVee and 3.2 keVee. The observed electron equivalentgives most energy lenient is related bounds to 1.12 the nuclear recoil energy by, Eobs =0.2(Er/keV) , so that the range of nuclear recoils of interest is 1.9 12 keV. In this range there are two cosmogenic peaks whose position − and width are well understood, aPredict relatively deconvoluted flat spectrum above rate these at peaksXENON10... and a clear excess at energies below the peaks. It is this low energy excess that may be due to a DM Thursday, February 10, 2011 signal and, rather than assume a particular functional form and fit, we extract it from the data by taking the data below the first peak (Eee < 1 keVee) subtracting from it the FIG. 3: Low-energy spectrum after all cuts, prior to efficiency average of the high bins (Eee > 1.6 keVee). Thus our “signal” region, shown in Fig. 1, is corrections. Arrows indicate expected energies for all viable 0.42 keVee (1.9 keV)cosmogenicAxion-Like Particle (ALP) dark matter limits from these region (their L-shell/K-shell EC ratio is 1/8 [5]). A spectra. The energy region employed to extract WIMP third possibility, quantitatively discussed below, consists limits is 0.4-3.2 keVee (from threshold to full range of of recoils from unvetoed muon-induced neutrons. the highest-gain digitization channel). A correction is Fig. 4 (top) displays the extracted sensitivity in spin- applied to compensate for signal acceptance loss from independent coupling (σSI) vs. WIMP mass (mχ). For 2 cumulative data cuts (solid sigmoid in Fig. 3, inset). mχ in the range ∼7-11 GeV/c the WIMP contribu- In addition to a calculated response function for each tion to the model acquires a finite value with a 90% WIMP mass [1], we adopt a free exponential plus a confidence interval incompatible with zero. The bound- constant as a background model to fit the data, with aries of this interval define the red contour in Fig. 4. two Gaussians to account for 65Zn and 68Ge L-shell However, the null hypothesis (no WIMP component in EC. The energy resolution is as in [1], with parameters the model) fits the data with a similar reduced chi- 2 σn=69.4 eV and F =0.29. The assumption of an irre- square χ /dof =20.4/20 (for example, the best fit for 2 2 ducible monotonically-decreasing background is justified, mχ = 9 GeV/c provides χ /dof =20.1/18 at σSI = given the mentioned possibility of a minor contamination 6.7 × 10−41cm2). It has been recently emphasized [6] from residual surface events and the rising concentration that light WIMP models [1, 8, 9] provide a common ex- CoGeNT and XENON10

mχ = 10 GeV

Xe Xenon ER keVr 0.5 1. 1.5 2. 2.5 3. " 12 "

dru ! " !

30 dru 10 ! Ge Ge R Xe Xe

25 R dR dE dR

8 dE $

20 $ Ge R Xe R E

1 6 # E 1 2 15 # 2 Ge Xe F 4 10 F

2 5 Xenon

CoGeNT 0 0 1 2 3 4 5

Ge CoGeNT ER keVr

FIG. 2: The extracted CoGeNT signal (left and bottom axes) and the rate it is mapped to on a ! " Xenon target (top and right axes) for mχ = 10 GeV (rescaled by form factors at the corresponding Thursday, February 10, 2011 energies F 2 (EXe),F2 (EGe) 1). The dashed line is the lower bound on the rate at low energies, Xe R Ge R ∼ using the monotonically falling nature of g(vmin).

discussion in [41]), and thus the value at the low end of this range is a lower bound for lower values of v. This is not especially relevant for our analysis here, but would be likely relevant in situations where the other experiments could probe lower energies as well.

Since we will compare this with the XENON10 experiment, we choose fp = 1 and fn = 0, which is motivated from light mediators mixing with the photon, since it will give the most lenient bounds. Using (11) we can map the CoGeNT signal onto a Xenon target, and study the signal that would arise at XENON10. We show this in figure 2. What is remarkable about this figure is that – once the CoGeNT signal is specified – the expected rate on a Xenon target is completely unambiguous (and similarly on any other target). This involves no assumptions about the halo escape velocity, velocity dispersion, or even the assumption that the velocity distribution is Maxwellian, but requires only an input of the WIMP mass. After taking into account exposure and the detector efficiencies (MIN, MED and MAX cases described above) we can predict the total number of events predicted by the CoGeNT

13 6. Dark matter exclusion limits

Having obtained significant insight on the low energy acceptance η, the actual acceptance of the 50% acceptance box and the detector resolution, we turn to calculation of dark matter exclusion limits. As in Sec. 3 and Sec. 4, we use the and curves given by Case 2 as a central value. Qy Leff Noting that any reasonable curve must also lead to reasonable values for , we choose the Leff Qy set of curves shown in Fig. 8 (dashed) as a lower bound. This is conservative considering that lower values of would be very difficult to reconcile with measurements [9]. We also consider a spline Qy through the measured values of [10] as an upper bound for , as shown in Fig. 8 (dotted). While Leff higher values of could also be considered reasonable, we find it more interesting to consider Leff a case which coincides with a published measurement. Each of these cases results in excellent agreement with the XENON10 nuclear recoil band, similar to that shown in Fig. 3.

10 CoGeNT and XENON108 y 6 Q 4 2 Predicted number of events at0 XENON10 (who saw 0 in relevant energy range) 0.3 0.2 eff L 0.1 [Sorensen (1007.3549)] 0 2 5 10 20 50 100 keVr 500 Figure 8: Qy and Leff values used to calculate central (solid curve, also shown in Fig. 1 as Case 2) most conservative (dashed) and upper (dotted) 90% C.L. bounds. Each curve is extrapolated as necessary to reproduce the measured nuclear recoil band shown in Fig. 3. The dotted curve is truncated at 1 keV. Other 100 data as indicated in Fig. 1.

50 We calculate predicted diﬀerential event rates as a function of nuclear recoil energy on a xenon target from vesc

events dR ρχσn 2 2 f(v) . = NT MN 2 A F (Enr) dv, (6.1) dEnr 2mχµne !vmin v 10

Num where the number of target nuclei in the detector is NT , the mass of the target nucleus is MN and −3 5 its atomic number is A. We assume the standard local dark matter density ρχ =0.3 GeV cm , with dark matter particle mass mχ and cross section (per nucleon) σn. The reduced mass µne is for the nucleon dark matter particle system. The nuclear form factor F (E ) accounts for a − nr loss of coherence as momentum transfer to the nucleus increases. We use the Helm form factor 1 parameterization F (E ) = 3j (qr )/qr exp( (qs)2/2). We take the effective nuclear radius nr 1 n n · − r = r2 5s2, with r =1.2A1/3 fm and the skin thickness s = 1 fm [29]. The momentum 3.0 5.0 7.0 n 010.0− 015.0 20.0 transfer" to the nucleus is just q = √2MN Enr. mΧ GeVWe assume the dark matter halo velocity distribution to be Maxwellian, and perform the Thursday, February 10, 2011 −1 integration over f(v)/v following [30]. We take the velocity dispersion to be v0 = 230 km s , and as in [31] set the rotational speed of the local standard of rest vrot = v0. Considering the sun’s − FIG. 3: The number of events predicted at XENON10velocity! " [30] by implies the possible an earth-halo DM velocity signal of 243 at km CoGeNT s 1. The lower for limit of the integral in Eq. 6.1 3 cases of , MIN (dashed red), MED (solid green) and MAX (dotted blue). The black line is Leff the 90% C.L. upper limit on the number of events allowed by XENON10 data. – 13 – events (if they are indeed coming from elastically scattering DM), we show this in Fig. 3. Since there were no events at XENON10 in the energy range corresponding to the CoGeNT MIN range we see that independent of all astrophysical assumptions, only for Leff are CoGeNT and XENON10 are consistent at the 90% C.L. In the MIN case, mχ < 11 GeV allows Co- GeNT to evade XENON10. For MED and MAX cases the predicted signal at XENON10 would be too large by a significant amount, excluding the elastic SI WIMP scattering interpretation by more than an order of magnitude. Because of the uncertainties associated with extraction of the value of at low energies, Leff additional attempts have been made to probe the low energy region with Xenon experiments. In particular, [47, 55] examined data from XENON10, and used only the ionization signal (S2), which is typically larger than S1 and can allow a more reliable signal at low energies. The value of the charge yield (drift electrons per keV) was extracted from Monte Carlo. Using the values there, the equivalent energy range for CoGeNT is approximately 8 13 ∼ electrons, above the 7 electron threshold. Assuming a value of Qy = 4 electrons/keV for instance, the threshold of 7 electrons at XENON10 only captures a portion of the signal predicted by CoGeNT. While the 7 electron cutoff corresponds to a particular value of energy in principle, Poisson

14 6. Dark matter exclusion limits

10 CoGeNT and XENON108 y 6 Q 4 2 Predicted number of events at0 XENON10 (who saw 0 in relevant energy range) 0.3 0.2 eff L 0.1 [Sorensen (1007.3549)] 0 2 5 10 20 50 100 keVr 500 Figure 8: Qy and Leff values used to calculateMAX central (solid curve, also shown in Fig. 1 as Case 2) most conservative (dashed) and upper (dotted) 90% C.L. bounds. Each curve is extrapolated as necessary to reproduce the measured nuclear recoil band shown in Fig. 3. The dotted curve is truncated at 1 keV. Other 100 data as indicated in Fig. 1. MED

50 We calculate predicted diﬀerential event rates as a function of nuclear recoil energy on a xenon target from vesc

events dR ρχσn 2 2 f(v) . = NT MN 2 A F (Enr) dv, (6.1) dEnr 2mχµne !vmin v 10

Num where the number of target nuclei in the detector is NT , the mass of the target nucleus is MN and −3 5 its atomic number is A. We assume the standardMIN local dark matter density ρχ =0.3 GeV cm , with dark matter particle mass mχ and cross section (per nucleon) σn. The reduced mass µne is for the nucleon dark matter particle system. The nuclear form factor F (E ) accounts for a − nr loss of coherence as momentum transfer to the nucleus increases. We use the Helm form factor 1 parameterization F (E ) = 3j (qr )/qr exp( (qs)2/2). We take the effective nuclear radius nr 1 n n · − r = r2 5s2, with r =1.2A1/3 fm and the skin thickness s = 1 fm [29]. The momentum 3.0 5.0 7.0 n 010.0− 015.0 20.0 transfer" to the nucleus is just q = √2MN Enr. mΧ GeVWe assume the dark matter halo velocity distribution to be Maxwellian, and perform the Thursday, February 10, 2011 −1 integration over f(v)/v following [30]. We take the velocity dispersion to be v0 = 230 km s , and as in [31] set the rotational speed of the local standard of rest vrot = v0. Considering the sun’s − FIG. 3: The number of events predicted at XENON10velocity! " [30] by implies the possible an earth-halo DM velocity signal of 243 at km CoGeNT s 1. The lower for limit of the integral in Eq. 6.1 3 cases of , MIN (dashed red), MED (solid green) and MAX (dotted blue). The black line is Leff the 90% C.L. upper limit on the number of events allowed by XENON10 data. – 13 – events (if they are indeed coming from elastically scattering DM), we show this in Fig. 3. Since there were no events at XENON10 in the energy range corresponding to the CoGeNT MIN range we see that independent of all astrophysical assumptions, only for Leff are CoGeNT and XENON10 are consistent at the 90% C.L. In the MIN case, mχ < 11 GeV allows Co- GeNT to evade XENON10. For MED and MAX cases the predicted signal at XENON10 would be too large by a significant amount, excluding the elastic SI WIMP scattering interpretation by more than an order of magnitude. Because of the uncertainties associated with extraction of the value of at low energies, Leff additional attempts have been made to probe the low energy region with Xenon experiments. In particular, [47, 55] examined data from XENON10, and used only the ionization signal (S2), which is typically larger than S1 and can allow a more reliable signal at low energies. The value of the charge yield (drift electrons per keV) was extracted from Monte Carlo. Using the values there, the equivalent energy range for CoGeNT is approximately 8 13 ∼ electrons, above the 7 electron threshold. Assuming a value of Qy = 4 electrons/keV for instance, the threshold of 7 electrons at XENON10 only captures a portion of the signal predicted by CoGeNT. While the 7 electron cutoff corresponds to a particular value of energy in principle, Poisson

14 produced as a result of a dedicated neutron (gamma) calibration, the result is generally referred to as the nuclear (electron) recoil band. Typical examples can be found in [1, 19]. For nuclear recoils, the measured quantity S1 (in units of photoelectrons) is related to the nuclear recoil energy Enr via 1 S1 S E = e . (2.1) nr · L · S Leff y n 1 For XENON10, the light yield Ly =3.0 photoelectrons/keVee for 122 keV photons, and the scintillation quenching of electron and nuclear recoils due to the electric field Ed =0.73 kV/cm are CoGeNTSe =0. 54and [14] andXENON10Sn =0.95 [9]. These quantities depend on the applied electric field, which is (S2 similaronly for analysis) XENON100 (Ed[see=0 P. Sorensen’s.53 kV/cm) talk at IDM2010] and higher for ZEPLIN III (Ed =3.9 kV/cm). This construction of in terms of S1 and the ‘standard candle’ L is necessary because the probability Leff y Usingof only detecting ionization a single scintillation signal (S2) photon avoids is significantly confusions less thanin one. Physically, Ly/Se represents measuringthe total S1, detection and allows efficiency a forlower primary threshold scintillation photons. In contrast, the measured quantity S2 (also in units of photoelectrons) is related to the nuclear Doesn’trecoil allow energy signal/background Enr via discrimination Result of monte carlo best fit S2 Equivalent of Leff is charge yield, Qy Enr = , (2.2) y E = 730 V/cm Q where is the number of detected electrons per keV. It is possible to express in absolute terms Qy Qy because single electrons are measured with high efficiency and relative ease. Both of these quantities PRC 81 025808 (2010) energy scaling obtained from scattering angle are reported in [9], in which the true nuclear recoil energy is known from the initial neutron energy and the scattering angle.Nucl. Instr. Note Meth. A 601 that 339 (2009) in contrastenergy scaling to obtained from, S1, viais Leff not referred to zero electric field. Leff Qy This may lead to some residual dependence on the electric field.

2.2 Monte carlo simulationThis of work signalenergy scaling production obtained from S2 and measurement Because and govern signal production from nuclear recoils in liquid xenon, they must also Qy Leff dictate the shape of the nuclear recoilsystematic band. uncertainty This observation, which has not previously been used to constrain measured values of and , is the premise of Sec. 3. Given and as a Qy Leff Qy Leff function of nuclear recoil energy, it is relatively easy to generate a nuclear recoil data set via monte Thursday, February 10, 2011 carlo simulation.The Qy curve allows The us to resultsassign a keVr of value this to every simulation event, based on its will S2 be compared with the measured XENON10 nuclear recoil band [15], for various combinations of y and eff . In the interest of clarity, the NOTE: conservative 1! stat. uncertainties given by !Ne Q L following description of the simulation will therefore use numerical examples from the XENON10 detector [16]. Peter Sorensen, LLNL 10 IDM, Montpellier FR 26 July 2010 For each simulated event with nuclear recoil energy Enr, which was modeled in 0.25 keV steps, the simulation output is a value of S1 and S2 as would be seen by the detector. This was accom- plished as follows: given E , the expected number of detected electrons N was calculated from nr e Qy and subjected to Poisson fluctuations. The number of S2 photoelectrons was then obtained from N =N 25 photoelectrons/electron, taking care to account for the measured width (σ/µ =0.30) S2 e · of the single electron distribution [16]. The expected number of S1 photoelectrons NS1 was calculated from , modeled as a discrete random hit pattern on the photomultipliers, and also Leff subjected to Poisson fluctuations 2. Both S1 and S2 were subjected to Gaussian fluctuations in the size of a single photoelectron measured by the photomultipliers, using the average σ/µ =0.58 [16]. Finally, the S1 signal was required to satisfy an n 2 coincidence requirement in the photomulti- ≥ pliers [19], within a 300 ns time window [12]. The primary scintillation signal was assumed to have a pulse shape as shown in Fig. 10. The scintillation decay time is discussed in the Appendix.

1The suﬃx ‘ee’ refers to electron recoil equivalent energy. The suﬃx ‘r’ is usually appended to indicate nuclear recoil equivalent energy, to emphasize the quenching of the electronic signal from nuclear recoils. In this work we always mean nuclear recoil energy unless stated otherwise, and will reserve the ‘keVr’ unit to explicitly indicate measured nuclear recoil energy. This is discussed in detail in Sec. 4.1. 2Strictly speaking, the probability for a scintillation photon to produce a photoelectron is governed by binomial statistics. In practice, Poisson statistics are a very good approximation.

– 3 – 13 7 70 3.6

12 6 50 "

" 5 11 keV # 30 5 4 electrons 20 ! 10 10

electrons 3 ! 10 9 y Q 3.6 Threshold 5 2 8

1 20 7 30 6 8 10 12 14 16 18 20 6 8 5.1kg-days10 12 14 16of 18exposure20 S2 threshold is 7 electrons mΧ GeV PoissonmΧ GeV ﬂuctuations can be 1 event seen important FIG. 4: (left) The number of! events" predicted (labels on contours), by CoGeNT,! " at XENON10 for an

S2 only analysis [47, 55] for various S2 thresholds, assuming a constant value Qy = 4 electrons/keV. Threshold = 7 e

(right) The signal above threshold of 7 electrons, but assuming di13ﬀerent constant values of charge 7 70 3.6 yield, Qy. 12 6 50 "

" 5 ﬂuctuations smear this. Nonetheless, an interesting question11 is the expected rate on the keV target used by [47, 55], with 5.1 kg d of eﬀective exposure. This is most easily phrased in # 30 5 4

electrons 20 ! 10 10 terms of the question of what charge yield can make these experiments consistent. Assuming

electrons 3 ! 10 a constant charge yield over the energies in question, we can9 calculate the likelihood based y Q 3.6 Threshold 5 on Poisson fluctuations of events appearing in the XENON10 experiment, which we show 2 8 in 4. One sees that one would require a charge yield of roughly Q < 2.4 electrons/keV for y 1 20 ∼ 30 consistency, much lower than the value of Qy 7 extracted7 by [47, 55]. Whether such a ≈ 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 significant difference is reasonable will no doubt be subject to a great deal ofm discussionΧ GeV [56]. mΧ GeV

Thursday, February 10, 2011 B. Application II: Total Rate Comparisons inFIG. Sub-Optimal 4: (left) The Situations number of! (CRESST) events" predicted (labels on contours), by CoGeNT,! " at XENON10 for an

S2 only analysis [47, 55] for various S2 thresholds, assuming a constant value Qy = 4 electrons/keV. The above situation with CoGeNT is close to ideal:(right low backgrounds,) The signal above high statistics, threshold good of 7 electrons, but assuming different constant values of charge energy resolution and calibration. In contrast, there are often situations with significantly yield, Qy. less ideal characteristics. In particular, it may be that not enough is known about the backgrounds, or the data itself, to be able to extractfluctuations a recoil spectrum smear for this. DM, but Nonetheless, we shall an interesting question is the expected rate on the see it is nonetheless possible to say something about thetarget total used number by of [47, DM 55], scatters. with 5.1 This kg is d of effective exposure. This is most easily phrased in terms of the question of what charge yield can make these experiments consistent. Assuming 15 a constant charge yield over the energies in question, we can calculate the likelihood based on Poisson fluctuations of events appearing in the XENON10 experiment, which we show in 4. One sees that one would require a charge yield of roughly Qy < 2.4 electrons/keV for ∼ consistency, much lower than the value of Q 7 extracted by [47, 55]. Whether such a y ≈ significant difference is reasonable will no doubt be subject to a great deal of discussion [56].

B. Application II: Total Rate Comparisons in Sub-Optimal Situations (CRESST)

The above situation with CoGeNT is close to ideal: low backgrounds, high statistics, good energy resolution and calibration. In contrast, there are often situations with signiﬁcantly less ideal characteristics. In particular, it may be that not enough is known about the backgrounds, or the data itself, to be able to extract a recoil spectrum for DM, but we shall see it is nonetheless possible to say something about the total number of DM scatters. This is

15 APPLICATION:CRESST APPLICATION:CRESSTCRESST (dealing with the unknown) • What!!"#$# do you!"#$%&%'(") do when you don’t#%"&"'#()*+*,&#)-, haveDon’t a spectrum? know a spectrum, Or but can reverse engineer a total background?• What!!"#$# do you!"#$%&%'(") do when you don’t#%"&"'#()*+*,&#)-, haverate a from spectrum? what hasOr been presented in conferences APPLICATION:CRESST[eg Seidel’s talk at IDM2010] background? !"#"$#%& '()*+,"-. "/"0#1 !"#"$#%&2 '()*+,"-.*3)42 "/"0#1•2What!!"#$# do you!"#$%&%'(") do when you don’t#%"&"'#()*+*,&#)-, have a spectrum? Or 2 *3)42 2 3( **)52 3background? 3( **)52 3 !"#"$#%& '()*+,"-. "/"0#1 36 **)72 8 2 *3)42 2 36 **)72 8 !"#$%&%'(") 44 *2)22 3 3( **)52 3 !"#$%&%'(") 44 *2)22 3 84 *2)22 8 36 **)72 8 84 *2)22 8 82 *6)*2 3 !"#$%&%'(") 44 *2)22 3 82 *6)*2 3 89 *9)42 8 84 *2)22 8 89 *9)42 8 2* 6)72 7 82 *6)*2 3 2* 6)72 7 Seidel, IDM 2010 talk 22 33)32 4 89 *9)42 8 Seidel,*#+,"-'./0 IDM 2010 talk 22 33)32 4 2* 6)72 7 *#+,"-'./0 #%#:; 43 #%#:; 43 22 33)32 4 Thursday, February6")/,-/#.,%&(,#/7(812"-+'9 10, 2011 Seidel, IDM 2010 talk !/$#(1(2#/0 6")/,-/#.,%&(,#/7(812"-+'9 *#+,"-'./0 !/$#(1(2#/0 :4#81/;-"/8-%'8%9#'8#. #%#:; 43 :4#81/;-"/8-%'8%9#'8#. 6")/,-/#.,%&(,#/7(812"-+'9 ! 3-&#,4%'2/'#5/0 ! !/$#(1(2#/0 ! 3-&#,4%'2/'#5/0 ! :4#81/;-"/8-%'8%9#'8#. ! 3-&#,4%'2/'#5/0 ! (10) in a simple form

(2) 2 2 (2) dR2 CT F2 (E2) dR1 µ1 MT (E2)= E2 , (11) (1) (2) (1) dER µ2 M dER 2 CT 2 1 T #µ2 MT $ F (1) E2 1 µ2M ! 2 T " where we have introduced a target speciﬁc coeﬃcient

2 C(i) = κ(i) f Z(i) + f (A(i) Z(i)) . (12) T p n − % & In certain situations differential rates may not be available and instead it is only possible to compare total rates, this is the situation at present with CRESST. In general the total rate at a particular experiment with energy — and corresponding velocity — thresholds of low (10) in a simplehigh form (Elow,vmin) and (Ehigh,vmin ), can be expressed as, (2) 2 2 (2) dR2 CT F2 (E2) dR1 µ1 MT (E2)= v E2 , (11) (1) high (2) (1) 2dENAR ρ mp κ µ2 M dER 2 CT 2 1 T #µ2 MT $ F (1) E2 R = 1 µ2Mdv #(ER)σ(ER(v))vg(v) . (13) m M 2 T χ T 'vlow! " where we have introduced a target specific coefficient For the particular case of SI on which we are focused this becomes, (i) (i) (i) (i) (i) 2 CT = κ fp Z + fn (A Z ) . (12) 2N ρσ m µ2C vhigh − R = A p p %T dv #(E &)F 2(E (v))vg(v) , (14) In certainm situationsµ2 f 2 differentialM rates may not be availableR and insteadR it is only possible ! χ nχ p "! T " 'vlow to compareCRESST total rates,(dealing this is the with situation the at presentunknown) with CRESST. In general the total where #(E ) an an energy-dependent efficiency. To compare two experiments, we must R rateWork at a particular out: experiment with energy — and corresponding velocity — thresholds of (E ,vlow ) and (E ,vhigh), can be expressed as, extract the energylow•22min dependent events highabove termsmin 15 keV from (from the integral.7/9 detectors), So while with we(all maken’s no assumptions vhigh and gamma’s attributed2NAρ mp κto these detectors)¯ ¯ 2 (1) 2 (2) about g(v), we evaluate theR = form factor at adv value#(ER)σE(ER(=v))Evg(vµ) .M /µ M (13)where the ra- m M 2 1 2 T 1 T •280 kg-days exposureχ T 'vlow ¯ 2 ¯ ¯ 2 ¯ tio #2(E2)F2 (ForE2 theIntegrate)/# particular1(E1)F over1 case(E 1energies of) SI is on minimized which to get we are total or focused maximized, rate this becomes, depending on whether we are

2 vhigh considering a putative signal2NA orρσ constraint.pmp µ CT Thus comparisons2 of rates at two experiments R = 2 2 dv #(ER)F (ER(v))vg(v) , (14) mχ µnχ fp MT vlow may then be simply compared! by taking"! ratios" ' of CT with the form factor evaluated at the where #(ER) an an energy-dependent eﬃciency. To compare two experiments, we must conservative valueEvaluateE¯, at energy that makes comparison most conservative extract the energy dependent terms from the integral. So while we make no assumptions (2) ¯ (1) ¯ 2 (1) 2 (2) about g(v), we evaluate the form¯ factor2 ¯ at a value E2 = E21µ2MT /µ1MT where the ra- #2(E2)F2 (E2) CT MT µ2 ¯ 2 ¯ R ¯ 2 ¯ R . (15) tio #2(E2)F2 (E2)/#1(E2 1)F1 (E1)¯ is minimized2 ¯ or(1) maximized,(2) 2 depending1 on whether we are ≤ #1(E1)F1 (E1) C M µ1 considering a putative signal or constraint. ThusT comparisonsT of rates at two experiments

In order to determinemayThursday, then February be 10, simply 2011 what compared comparisons by taking can ratios be of madeCT with between the form factor experiments, evaluated at the we must ex- conservative value E¯, amine the relevant velocity space they probe. We re-emphasize that the signal at energy # (E¯ )F 2(E¯ ) C(2) M (1) µ2 R 2 2 2 2 T T 2 R . (15) Elow < E < Ehigh is sensitive to 2all≤ #particles(E¯ )F 2(E¯ ) with(1) (2) velocityµ2 1 greater than vmin(E,MN ,Mχ) 1 1 1 1 CT MT 1 ˜ through the integralIn order tog determine(vmin). Awhat separate comparisons experiment can be made with between threshold experiments,E will we must oﬀer ex- constraints amine the relevant velocity space they probe. We re-emphasize that the signal at energy independent of astrophysics if the resulting minimum velocityv ˜ < v2. The optimal limits are Elow < E < Ehigh is sensitive to all particles with velocity greater than vmin(E,MN ,Mχ) reached whenv ˜ < v1. We illustrate this in Fig. 1 for an ensemble of experiments, some with through the integral g(vmin). A separate experiment with threshold E˜ will oﬀer constraints independent of astrophysics if the resulting minimum velocityv ˜ < v2. The optimal limits are 6 reached whenv ˜ < v1. We illustrate this in Fig. 1 for an ensemble of experiments, some with

6 CRESST and CDMS-Si/XENON10

CRESST predicts CRESST expects

200 XENON10(MED) 100 10.0 CRESST (our estimate) XENON10(MIN) 50 5.0 CRESST

at CDMS-Si

events 20 CDMS-Si 2.0 . events XENON10(MED)

Num 1.0 10 on UL 0.5

5 " CDMS-Si (allowed) 90 2 XENON10(MIN)0.2 5 10 15 20 5 10 15 20

mΧ GeV mΧ GeV

! " ! " FIG. 5: LH plot: the CRESST prediction for the total number of events at CDMS-Si (solid red) and XENON10,CRESST for O-bandMIN (dashed events black) at and oddsMED with(dotted CDMS black), theand dotted XENON (blue) line is the Leff Leff 90% C.L. upper limit on the numberPerhaps of events signal allowed not by CDMS-Si.eDM? RH plot: the 90% C.L. upper limit on the number of events at CRESST as predicted by CDMS-Si (solid red) and XENON10, Thursday, February 10, 2011 again for MIN (dashed black) and MED (dotted black), the dotted (blue) line is the number of Leff Leff events we estimate above background in CRESST. the case for the CRESST data, which we estimate has 15 events above background between 15 and 40 keV (see the discussion in III). We use (15) to compare the CRESST integrated rate to the null results of both CDMS-Si and XENON10, Fig. 5. When comparing the two experiments we take into account eﬃciencies and form factors so as to be as conservative as possible, as explained after (15). As is clear from Fig. 5 any sizeable signal in this range is highly incompatible with both the XENON10 and CDMS-Si results. While some have criticized the calibration at the lowest energies for CDMS-Si [54], the lowest energy relevant for 15 keV Oxygen recoils is above 10 and typically 11 keV on Silicon, depending on the WIMP mass. Thus, these constraints are likely quite stable to future modiﬁcations, making elastic WIMP scattering very unlikely to be the explanation of the CRESST anomalous events.

IV. OTHER APPLICATIONS AND FUTURE RESULTS

DAMA also has extracted a recoil spectrum, possibly associated with DM, but in this case it is for the modulating part of the DM signal, i.e. DAMA allows extraction of g(v, t). We can repeat the exercise of translating from one experiment to another to get a prediction

16 1.0 1.0

1.0 1.0 0.8 0.8

0.8 0.8 Modulation 0.6 Modulation 0.6 on on limit limit

Modulation 0.6 Modulation 0.6 0.4 0.4 on on lower lower " "

limit limit 0.2 0.2 0.4 0.4 90 90 lower lower 0.0 0.0 " 0.2 " 0.2 5 10 15 20 5 10 15 20 90 90 mΧ GeV mΧ GeV

0.0 0.0 5 10 15 20 5 10 15 20 ! " ! " mΧ GeV mΧ GeV FIG. 6: The 90% C.L. lower limit on the modulation fraction allowed by XENON10 data, for a

! " ! " quench factor in sodium of 0.3 (LH plot) and 0.45 (RH plot) and for 3 cases of eff , MIN (dashed FIG. 6: The 90% C.L. lower limit on the modulation fraction allowed by XENON10 data, for a L red), MED (solid green) and MAX (dotted blue). quench factor in sodium of 0.3 (LH plot) and 0.45 (RH plot) and for 3 cases of , MIN (dashed Leff red), MED (solid green) and MAX (dotted blue). for the size of the modulating signal at XENON10. Since XENON10 took its data in the winter and saw no events in the region corresponding to DAMA’s 2-6 keVee, this places an for the size of the modulating signal at XENON10. Since XENON10 took itsupper data limit in of the 2.3 events in the winter which in turn places a lower bound on the amount of winter and saw no events in the region corresponding to DAMA’s 2-6 keVee,modulation this places the an DM signal must have in order not be ruled out by XENON10’s null result. upper limit of 2.3 events in the winter which in turn places a lower bound onWe the present amount this of lower bound on the modulation fraction3 in Fig. 6 for two choices of the modulation the DM signal must have in order not be ruled out by XENON10’squench null factor result. in sodium, qNa =0.3, 0.45. Thus, irrespective of astrophysics, in order for DAMA to be consistent with XENON10 the modulation fraction has to be larger than 20% We present this lower bound on the modulation fraction3 in Fig. 6 for two choices of the and in most cases almost 100% for the standard assumption of qNa =0.3. For the more quench factor in sodium, qNa =0.3, 0.45. Thus, irrespective of astrophysics, in order for extreme choice of qNa =0.45 the modulation may be smaller but for DM heavier than 10 DAMA to be consistent with XENON10 the modulation fraction has to be larger than 20% GeV it again has to be above 20%. and in most cases almost 100% for the standard assumption of q =0.3. For the more Na An interesting relationship between CoGeNT and DAMA can be made here. The modula- extreme choice of qNa =0.45 the modulation may be smaller but for DM heavier than 10 DAMA and modulationtion at DAMA can be applied to CoGeNT through (15). In doing so, one finds a modulation GeV it again has to be above 20%. O(0.8 0.9cpd/kg) expected at CoGeNT. With a quenching factor q =0.3, this is ex- − Na An interesting relationship between CoGeNT and DAMA canXENON10 be made here. sawpected no The events modula- to overlap in winter the ⇒ L-shell upper peaks, bound which, (2.3) in decaying away, would make a rising signal Corresponding lower bound on % modulation at DAMA tion at DAMA can be applied to CoGeNT through (15). In doing so, one findsdi affi modulationcult to extract. The modulation in the signal range we cannot predict.

O(0.8 0.9cpd/kg) expected at CoGeNT. With a quenching factor qNa =0.3,On this the is other ex- hand, if qNa =0.45, then the energy range of CoGeNT overlaps that of

− 1.0 1.0 pected to overlap the L-shell peaks, which, in decaying away, would makeDAMA. a rising The signal 0.8 0.9cpd/kg modulation amplitude would then be visible over the 5cpd/kg 0.8 −0.8 ∼ in that range (i.e., a modulated amplitude of 20 %, or 40% peak-to-peak), which should diﬃcult to extract. The modulation in the signal range we cannotModulation 0.6 predict. Modulation 0.6 ∼ ∼ on on

limit 0.4 limit 0.4 On the other hand, if qNa =0.45, then the energy range of CoGeNT overlaps3 that of S W lower We deﬁne the modulationlower fraction as S−+W where S,W denote the summer and winter event rate respec- " 0.2 " 0.2 DAMA. The 0.8 0.9cpd/kg modulation amplitude would then90 be visible over thetively.5cpd/kg 90 − 0.0 ∼ 0.0 5 10 15 20 5 10 15 20 in that range (i.e., a modulated amplitude of 20 %, or 40% peak-to-peak),mΧ GeV which should mΧ GeV ∼ ∼ 17 ! " ! " FIG. 6: The 90% C.L. lower limit on the modulation fraction allowed by XENON10 data, for a 3 S W We deﬁne the modulation fraction as S−+W where S,W denote the summer and winter event rate respec- quench factor in sodium of 0.3 (LH plot) and 0.45 (RH plot) and for 3 cases of eff , MIN (dashed tively. Thursday, February 10, 2011 L red), MED (solid green) and MAX (dotted blue).

17 for the size of the modulating signal at XENON10. Since XENON10 took its data in the winter and saw no events in the region corresponding to DAMA’s 2-6 keVee, this places an upper limit of 2.3 events in the winter which in turn places a lower bound on the amount of modulation the DM signal must have in order not be ruled out by XENON10’s null result. We present this lower bound on the modulation fraction3 in Fig. 6 for two choices of the quench factor in sodium, qNa =0.3, 0.45. Thus, irrespective of astrophysics, in order for DAMA to be consistent with XENON10 the modulation fraction has to be larger than 20% and in most cases almost 100% for the standard assumption of qNa =0.3. For the more extreme choice of qNa =0.45 the modulation may be smaller but for DM heavier than 10 GeV it again has to be above 20%. An interesting relationship between CoGeNT and DAMA can be made here. The modulation at DAMA can be applied to CoGeNT through (15). In doing so, one ﬁnds a modulation O(0.8 0.9cpd/kg) expected at CoGeNT. With a quenching factor q =0.3, this is ex- − Na pected to overlap the L-shell peaks, which, in decaying away, would make a rising signal diﬃcult to extract. The modulation in the signal range we cannot predict.

On the other hand, if qNa =0.45, then the energy range of CoGeNT overlaps that of DAMA. The 0.8 0.9cpd/kg modulation amplitude would then be visible over the 5cpd/kg − ∼ in that range (i.e., a modulated amplitude of 20 %, or 40% peak-to-peak), which should ∼ ∼

3 S W We deﬁne the modulation fraction as S−+W where S,W denote the summer and winter event rate respectively.

17 Here we use the reduced mass defined with respect to the independent of their mass (and thus, have several candi- incoming particles, dates of different masses with similar abundances, using for example the WIMPless miracle [43]). mχmN µ . (1) The event rate of dark matter scattering [44], differen- ≡ mχ + mN Onwards... tial in ER, is determined by The recoil energy of the collision is E = q2/2m with R N! dR N ρ vmax d σ $v = T χi d3$v f ($v (t)) i| i| , (7) 2 2 2 dE m i i i dE q = p + p! 2pp! cos θcom . (2) R i χi vi,min R iDM and other “non-standard”− models of* DM + The recoil of energy ER, velocity v and cos θlab are related where the sum is over different species of WIMPs, m N & by, Amp is the nucleus mass with mp the proton mass and A the atomic number. The recoil energy depends on the 700 v2 m m χ χ kinematics of the collision, as described above. Given δχ v 2mN ! ER cos θlab 2 mχ! − mχ! our assumption of no significant time variation in the ! mN ! rate, f($vi(t)) f($vi), and thus we are effectively ne- 500 ER 1+ + δχ + δN =0. •(3)vmin-ERglecting the relation Earth’s→ motion no around the Sun. This is a " − mχ! s # " # $ % reasonable approximation so long we are probing veloci- km ! Define δ δχ + δN . If δ > 0, we can safely performlonger an ties larger one-to-one than Earth’s velocity in the Sun’s frame, i.e.,

min ≡ v expansion in δ/m 1 to obtain v 30 km/s. Typically the maximum speed is taken # max ! •compareto be vmax = experimentvearth + vesc, the galactic escape velocity 300 1 mN ER boosted into the Earth frame. However, v is ulti- vmin = + δ . (4) max √2m E µ to itself? N R # $ mately determined by the (unknown) details of the dark matter velocity distribution in Earth frame. 0 which taking20 δN 400 is the well-known60 80 result for in- Given our assumption of no direction dependent signal, elastic dark matter→ (iDM) [40–42]. By “safe” we mean Er keV we can carry out the angular integral in Eq. (7), reduc- that our upper bound on vmin, which is in the far non- ing it to a one dimensional integral where we introduce relativistic regime, automatically implies δ mχ,mN 1 ! " | | # the quantity f1(v)= dΩf($v). The differential rate to allow scattering to be kinematically possible. becomes Up to higher order terms in δ/m, we obtain an expres- , sion for the recoil energy dR NT ρχi mN 2 = 2 FN (ER) dER µi mχi µ µ2 i 2 2 2 2 * vmax ER +2ER (δ µv cos θlab)+ 2 δ = 0 (5) mN − m dv v f (v )¯σ (v ,E ) , (8) N × i i i1 i i i R +vi,min The recoil energy is unique for a given fixed scattering where we have written relative velocity v and nucleus recoil angle θlab and can Thursday, Februarybe 10, solved 2011 by the usual quadratic formula, dσ m i = F 2 (E ) N σ¯ (v ,E ) (9) dE N R µ v2 i i R µ 2 2 R i i ER = µv cos θlab δ (6) mN − 2 in terms of the nuclear form factor FN (ER). There are &'2 2 1/2 (2 2 1/2 (µv cos θlab) µv cos θlab 2δ . several possible forms for the scattering cross section ± − σ¯ (v, E ), depending on the interaction, ) i R This result has the well known' feature that the( smallest 2 2 recoil energies come from maximizing v cos θlab, corre- σi0 2 sponding physically to head-on collisions at the highest σi0F (ER) σ¯ (v, E )= χi . (10) velocities available. i R σ (v)F 2 (E )  i0 χi R  σi0(v, ER)  III. EVENT DISTRIBUTIONS The different forms forσ ¯correspond to functional forms of known dark matter scattering that contain velocity Our basic assumptions consist of assuming the scat- and/or recoil energy dependence. The first possibility, tering process is off only one type of nuclei. We will, a constant independent of v and ER is the well-known however, remain general with respect to the possibility of isotropic (s-wave) cross section that results at lowest multiple WIMPs with different masses, abundances, and order in the non-relativistic expansion from many dark cross sections. One might think it requires a large coin- matter models. cidence to have several dark matter particles with cross sections large enough to produce events in an experiment. However, there are well known counterexamples where it can be natural to have the abundance of particles to be 1 The velocity distribution is normalized such that d3vf(v) = 1. ! 3 A new plot

mΧ % 10 GeV 10!22 CoGeNT CDMS-Si CDMSGei 10!23 # 1 !24 ! 10 day !

% !25

v 10 $

g XENON(MED) p Χ

Σ !26

m 10 Ρ

10!27 XENON(MIN)

10!28 200 400 600 800 1000 v km s

FIG. 7: A comparison of measurements and constraints of the astrophysical observable g(v) [see ! " # relevant expressions in (1),(2),(8)] for mχ = 10 GeV: CoGeNT (blue), CDMS-Si (red, solid), Thursday, February 10, 2011 CDMS-Ge (green, dot-dashed), XENON10 - MIN (purple, dashed), and XENON10 - MED Leff (gray, dotted). CoGeNT values assume the events arise from elastically scattering dark Leff matter, while for other experiments, regions above and to the right of the lines are excluded at 90% conﬁdence. The jagged features of the CDMS-Ge curve arise from the presence of the two detected events.

To determine this plot, in the presence of a positive signal, one needs merely to read off g(v) from (8). In the absence of a (clear) signal, there is always a certain element of choice in how one quantifies a constraint. However, one can exploit the fact that g is a monotonically decreasing function, so for our constraints, we simply assume that g(v) is constant below v, and assume a Poisson limit on the integral of (8) from the experimental threshold to v. However, other techniques could also be used. This approach with a g v plot has numerous advantages over the traditional m σ − χ − plots. It makes manifest what the relationships between the different experiments are in terms of what vmin-space is probed, and shows (for a given mass) whether tensions exist. Moreover, the quantity g(v) is extremely tightly linked to the data, with only a rescaling

19 In addition to standard σ − m plots because..

A more direct comparison of data than x-sec--m plots Easy to derive from data For eDM (and single target experiments) need only show for one mass Ultimately allows measurement of g(v) Consistency of g(v) determines allowed DM params

Thursday, February 10, 2011 Conclusions-theory

•There are large astrophysical uncertainties •Should analyse data independent of them •With multiple experiments should compare g(v) •Allows mapping of experimental results between experiments •Under particle physics assumption can compare multiple experiments, test consistency •Ultimately ﬁnd region of consistent parameter space

Thursday, February 10, 2011 Conclusions-applications

•CoGeNT only consistent with XENON10 for low Leff •For the S2 XENON10 analysis consistency requires a very low Qy •For DAMA and XENON10 to be consistent requires >80% modulation for the standard Na quench factor •For higher quenching factors modulation can be lower •CRESST seems unlikely to be eDM due to constraints from CDMS-Si and XENON10

Thursday, February 10, 2011