Towards Closing the Window on Strongly Interacting Dark Matter: Far-Reaching Constraints from Earth's Heat Flow
aa r X i v : . [ a s t r o - ph ] O c t Towards Closing the Window on Strongly Interacting Dark Matter:Far-Reaching Constraints from Earth’s Heat Flow
Gregory D. Mack,
1, 2
John F. Beacom,
1, 2, 3 and Gianfranco Bertone Department of Physics, Ohio State University, Columbus, Ohio 43210 Center for Cosmology and Astro-Particle Physics, Ohio State University, Columbus, Ohio 43210 Department of Astronomy, Ohio State University, Columbus, Ohio 43210 Institut d’Astrophysique de Paris, UMR 7095-CNRS,Universit´e Pierre et Marie Curie, 98bis boulevard Arago, 75014 Paris, France [email protected], [email protected], [email protected] (Dated: 30 May 2007)We point out a new and largely model-independent constraint on the dark matter scattering crosssection with nucleons, applying when this quantity is larger than for typical weakly interactingdark matter candidates. When the dark matter capture rate in Earth is efficient, the rate ofenergy deposition by dark matter self-annihilation products would grossly exceed the measuredheat flow of Earth. This improves the spin-independent cross section constraints by many orders ofmagnitude, and closes the window between astrophysical constraints (at very large cross sections)and underground detector constraints (at small cross sections). In the applicable mass range, from ∼ ∼ GeV, the scattering cross section of dark matter with nucleons is then bounded from above by the latter constraints, and hence must be truly weak, as usually assumed.
PACS numbers: 95.35.+d, 95.30.Cq, 91.35.Dc
I. INTRODUCTION
There is a large body of evidence for the existence ofdark matter, but its basic properties – especially its massand scattering cross section with nucleons – remain un-known. Assuming dark matter is a thermal relic of theearly universe, weakly interacting massive particles areprime candidates, suggested by constraints on the darkmatter mass and self-annihilation cross section from thepresent average mass density [1]. However, as this re-mains unproven, it is important to systematically testthe properties of dark matter particles using only late-universe constraints. In 1990, Starkman, Gould, Es-mailzadeh, and Dimopoulos [2] examined the possibilityof strongly interacting dark matter, noting that it indeedhad not been ruled out. Many authors since have ex-plored further constraints and candidates. In this litera-ture, “strongly interacting” denotes cross sections signif-icantly larger than those of the weak interactions; it doesnot necessarily mean via the usual strong interactions be-tween hadrons. We generally consider the constraints inthe plane of dark matter mass m χ and spin-independentscattering cross section with nucleons σ χN .Figure 1 summarizes astrophysical, high-altitude bal-loon/rocket/satellite detector, and underground detectorconstraints in the σ χN – m χ plane. Astrophysical limitssuch as the stability of the Milky Way disk constrainvery large cross sections [2, 3]. Accompanying and com-parable limits include those from cosmic rays and thecosmic microwave background [4, 5]. Small cross sec-tions are probed by CDMS and other underground de-tectors [6, 7, 8, 9]. A dark matter (DM) particle can bedirectly detected if σ χN is strong enough to cause a nu-clear recoil in the detector, but only if it is weak enoughto allow the DM to pass through Earth to the detector. log m χ [GeV] -30-25-20-15-10-35-40-45 l og σ χ N [ c m ] U n d e r g r o u n d D e t e c t o r s C o s m i c R a y s M W r e d o n e M W D i s k D i s r u p ti o n XQC IMP 7/8 IMAX SKYLABRRS
FIG. 1: Excluded regions in the σ χN – m χ plane, not yet in-cluding the results of this paper. From top to bottom, thesecome from astrophysical constraints (dark-shaded) [2, 3, 4, 5],re-analyses of high-altitude detectors (medium-shaded) [2, 10,11, 12], and underground direct dark matter detectors (light-shaded) [6, 7, 8, 9]. The dark matter number density scales as1/ m χ , and the scattering rates as σ χN / m χ ; for a fixed scat-tering rate, the required cross section then scales as m χ . Wewill develop a constraint from Earth heating by dark matterannihilation to more definitively exclude the window betweenthe astrophysical and underground constraints. In between the astrophysical and underground limitsis the window in which σ χN can be relatively large [2].High-altitude detectors in and above the atmospherehave been used to exclude moderate-to-strong values ofthe cross section in this region [2, 10, 11, 12]. However,there are still large gaps not excluded. There also issome doubt associated with these exclusions, as some ofthe experiments were not specifically designed to look forDM, nor were they always analyzed for this purpose bypeople associated with the projects. In fact, the exclu-sion from the X-ray Quantum Calorimetry experimentwas recently reanalyzed [12] and it changed substantiallyfrom earlier estimates [11]. If this intermediate regioncan be closed, then underground detectors would set theupper limit on σ χN . That would mean that these detec-tors are generally looking in the right cross section rangeand that DM-nucleon scattering interactions are indeedtotally irrelevant in astrophysics.We investigate cross sections between the astrophysi-cal and underground limits, and show that σ χN is largeenough for Earth to efficiently capture DM. IncomingDM will scatter off nucleons, lose energy, and becomegravitationally captured once below Earth’s escape ve-locity (Section 4). If this capture is maximally efficient,the rate is 2 × (GeV /m χ ) s − . The gravitationally-captured DM will drift to the bottom of the potentialwell, Earth’s core. Self-annihilation results if the DM isits own antiparticle, and we assume Standard Model finalstate particles so that these products will deposit nearlyall their energy in the core.Inside a region in the σ χN – m χ plane that will be de-fined, too much heat would be produced relative to theactual measured value of Earth’s heat flow. The max-imal heating rate obtained via macroscopic considera-tions is ≃ /m χ , while the heat energy from annihila-tions scales as m χ , yielding a heat flow that is inde-pendent of DM mass . The efficient capture we con-sider leads to a very similar heating rate, though itis based on a realistic calculation of microscopic DM-nucleon scattering, as discussed below. DM interactionswith Earth have been previously studied in great detail,e.g., Refs [2, 13, 14, 15, 16, 17, 18, 19, 20, 21], but thoseinvestigations generally considered only weak cross sec-tions for which capture is inefficient.In our analysis, the σ χN exclusion region arises fromthe captured DM’s self-annihilation energy exceedingEarth’s internal heat flow. This region is limited belowby the efficient capture of DM (Section 4), and above bybeing weak enough to allow sufficient time for the DMto drift to the core (Section 5). These two limits definethe region in which DM heating occurs. Why is it im-portant? Earth’s received solar energy is large, about170 ,
000 TW [22], but it is all reflected or re-radiated.The internal heat flow is much less, about 44 TW (Sec-tion 3) [23]. Inside this bounded region for σ χN , DMheating would exceed the measured rate by about twoorders of magnitude, and therefore is not allowed. Wewill show that this appears to close the window noted TABLE I: Relevant heat flow values. The top entries aremeasured, while the lower entries are the calculated potentialeffects of dark matter.Heat Source Heating RateSolar (received and returned) 170,000 TWInternal (measured) 44.2 ± ∼ × − TW above in Fig. 1, up to about m χ ≃ GeV. In orderto be certain of this, however, we call for new analysesof the aforementioned constraints, especially the exactregion excluded by CDMS and other underground detec-tors. Our emphasis is not on further debate of the detailsof specific open gaps, but rather on providing a new andindependent constraint. In Table I, we summarize theheat values relevant to this paper. While the origin ofEarth’s heat flow is not completely understood, we em-phasize that we are not trying to account for any portionof it with heating from DM.There has been some previous work on the heatingof planets by DM annihilation [13, 24, 25, 26, 27, 28,29, 30]. These papers have mostly focused on the Jo-vian planets, for which the internal heat flow valuesare deduced from their infrared radiation [31]. In somecases [13, 24, 25, 26, 27], DM annihilation was invoked toexplain the anomalously large heat flow values of Jupiterand Saturn, while in other cases [28, 29, 30], the low heatflow value of Uranus was used to constrain DM annihi-lation. An additional reason for the focus on these largeplanets is that they will be able to stop DM particles ofsmaller cross section than Earth can (Ref. [27] consid-ered Earth, but invoked an extreme DM clumping factorto overcome the weakly interacting cross section). How-ever, as we argue below in Section 6, the more relevantcriterion is how significant of an excess heat flow couldbe produced by DM annihilation, and this is much morefavorable for Earth. (If this criterion is met, then theranges of excluded cross sections will simply shift for dif-ferent planets.) Furthermore, the detailed knowledge ofEarth’s properties gives much more robust results. In thispaper, we are presenting the first detailed and systematicstudy of the broad exclusion region in the σ χN – m χ planethat is based on not overheating Earth.Our constraints depend on DM being its own antipar-ticle, so that annihilation may occur (or, if it is not, thatthe DM-antiDM asymmetry not be too large). This isa mild and common assumption. The heating due tokinetic energy transfer is negligible. Since the DM ve-locity is ≃ − c , kinetic heating is ∼ − that fromannihilation, and would provide no constraint (Section4). The model-independent nature of our annihilationconstraints arises from the nearly complete insensitivityto which Standard Model particles are produced in theDM annihilations, and at what energies. All final statesexcept neutrinos will deposit all of their energy in Earth’score. (Above about 100 TeV, neutrinos will, too.) Sincethe possible heating rate ( > ∼
40 TW), in effect we onlyrequire that not more than ∼
99% of the energy goes intolow-energy neutrinos, which is an extremely modest as-sumption.Some of the annihilation products will likely be neu-trinos, and these may initiate signals in neutrino detec-tors, e.g., as upward-going muons [2, 32, 33, 34, 35, 36,37, 38, 39]. While the derived cross section limits canbe constraining, they strongly depend on the branchingratio to neutrinos and the neutrino energies. Compre-hensive constraints based on neutrino fluxes for the fullrange of DM masses appear to be unavailable; most pa-pers have concentrated on the 1–1000 GeV range, and afew have considered masses above 10 GeV. We note thatthe constraints for DM masses above about 10 GeVmay require annihilation cross sections above the unitar-ity bound, as discussed below. As this paper is meant tobe a model-independent, direct approach to DM proper-ties based on the DM density alone, we do not includethese neutrino constraints.We review the current DM constraints in Section 2,review Earth’s heat flow in Section 3, calculate the DMcapture, annihilation, and heating rates in Sections 4 and5, and close with discussions and conclusions in Section6.
II. REVIEW OF PRIOR CONSTRAINTS
Figure 1 shows the current constraints in the σ χN – m χ plane. As we will show, the derived exclusion regionfound by the requirement of not overheating Earth usingDM annihilation lies in the uncertain intermediate areabetween the astrophysical and underground constraints. A. Indirect Astrophysical Constraints If σ χN were too large, DM particles in a galactic halowould scatter too frequently with the baryonic disk of aspiral galaxy, and would significantly disrupt it. Usingthe integrity of the Milky Way disk, Starkman et al. [2]restrict the cross section to σ χN < × − ( m χ / GeV)cm . A more detailed study by Natarajan et al. [3] re-quires σ χN < × − ( m χ / GeV) cm . Both of theselimits consider DM scattering only with hydrogen. Asshown below in Eq. (12), the spin-independent DM-nucleon cross section scales as A for large m χ , andthough the number density of helium ( A = 4) is about10 times less than that of hydrogen ( A = 1), taking itinto account could improve these constraints by ≃ ≃
25. Chivukula et al. [40] showed that charged darkmatter could be limited through its ionizing effects on interstellar clouds; this technique could be adapted forstrongly interacting dark matter.Strong scattering of DM and baryons would also af-fect the cosmic microwave background radiation. Addingstronger DM-baryon interactions increases the viscos-ity of the baryon-photon fluid [4]. A strong cou-pling of baryons and DM would generate denser clumpsof gravitationally-interacting matter, and the photonswould not be able to push them as far apart. The peaks inthe cosmic microwave background power spectrum wouldbe damped, with the exception of the first one. The re-sulting constraint is σ χN < × − ( m χ / GeV) cm [4],and is not shown in Fig. 1. These results do take heliuminto account, but do so only using A instead of A . Thispossible change, along with the much more precise cosmicmicrowave background radiation data available currently,calls for a detailed re-analysis of this limit, which shouldstrengthen it.Cosmic ray protons interact inelastically with interstel-lar protons, breaking the protons and creating neutral pi-ons that decay to high-energy gamma rays. A similar sit-uation could occur with a cosmic ray beam on DM targetsinstead [5]. The fundamental interaction is between thequarks in the nucleon and the DM; it is very unlikely thatall quarks will be struck equally, and the subsequent de-struction of the nucleon creates pions. If the DM-nucleoncross section were high enough, the resulting gamma rayswould be readily detectable. From this, Cyburt et al. [5]place an upper limit of σ χN < . × − ( m χ / GeV)cm . Improvements could probably be made easily witha more realistic treatment of the gamma-ray data. B. Direct Detection Constraints
Underground detector experiments have played a largerole in limiting DM that can elastically scatter nuclei,giving the nuclei small but measurable kinetic energies.Due to the cosmic ray background, this type of detector islocated underground. The usual weakly interacting DMcandidates easily pass through the atmosphere and Earthen route to the detector. However, for large σ χN the DMwould lose energy through scattering before reaching thedetector, decreasing detection rates.Albuquerque and Baudis [7] have explored constraintsat relatively large cross sections and large masses us-ing results from CDMS and EDELWEISS. In Fig. 1, wepresent a crude estimate of the current underground de-tector exclusion region. The top line is defined by theability of a DM particle to make it through the atmo-sphere [41] and Earth to the detector without losing toomuch energy [7]. The lower left corner and nearby pointsare taken from the official CDMS papers [6] with theaid of their website [42]. The right edge is taken fromDAMA [9]. As the mass of the DM increases, the num-ber density (and hence the flux through Earth) decreases.At the largest m χ values, the scattering rate within a fi-nite time vanishes. Finally, we have extrapolated eachof these constraints to meet each other, connecting themconsistently. We call for a complete and official analysisof the exact region that CDMS and other direct detectorsexclude. Our focus is on the cross sections in between theunderground detectors and astrophysical limits.To investigate cross sections in this middle range, di-rect detectors must be situated above Earth’s atmo-sphere, in high-altitude balloons, rockets, or satellites.Several such detectors have been analyzed for this pur-pose, though they were not all originally intended tostudy DM. Since these large σ χN limits have in somecases been calculated by people not connected with theoriginal experiments, some caution is required. Never-theless, in Fig. 1 we show the claimed exclusion regions,following Starkman et al. [2] and Rich et al. [10], alongwith Wandelt et al. [11] and Erickcek et al. [12] (includingthe primary references [43, 44, 45, 46]). We are primar-ily in accordance with Erickcek et al. These regions spanmasses of almost 0.1 GeV to 10 GeV, and cross sectionsbetween roughly 10 − cm and 10 − cm . These in-clude the Pioneer 11 spacecraft and Skylab, the IMP 7/8cosmic ray silicon detector satellite, the X-ray QuantumCalorimetry experiment (XQC), and the balloon-borneIMAX. These regions are likely ruled out, but not in ab-solute certainty, and there are gaps between them. ThePioneer 11 region is completely covered by the IMP 7/8and XQC regions, and is therefore not shown in Fig. 1.The region labeled RRS is Rich et al.’s analysis of a sil-icon semiconductor detector near the top of the atmo-sphere, truncated according to Starkman et al., and ad-justed with the appropriate A -scaling as in Eq. (12). III. EARTH’S HEAT FLOW
Heat from the Sun warms Earth, but it is not retained.If all the incident sunlight were absorbed by Earth, theheating rate would be about 170,000 TW [22]. Some of itis reflected by the atmosphere, clouds, and surface, andthe rest is absorbed at depths very close to the surfaceand then re-radiated [31]. Earth’s blackbody tempera-ture would be about 280 K, and it is observed to be be-tween 250 and 300 K, supporting the idea of Earth-Sunheat equilibrium. Internal heating therefore has minimaleffects on the overall heat of Earth [31].Our focus is on this internal heat flow of Earth, as mea-sured underground. Geologists have extensively studiedEarth’s internal heat for decades [47]. To make a mea-surement, a borehole is drilled kilometers deep into theground. The temperature gradient in that borehole isrecorded, and that quantity multiplied by the thermalconductivity of the relevant material yields a heat flux[47, 48].The deepest borehole is about 12 kilometers, which isstill rather close to Earth’s surface. Typical tempera-ture gradients are between 10 and 50 K/km, but thesecannot hold for lower depths. If they did, all rock inthe deeper parts of Earth would be molten, in contra- diction to seismic measurements, which show that shearwaves can propagate through the mantle [48]. Currentestimates place temperature gradients deep inside Earthbetween 0.6 and 0.8 K/km [48].More than 20,000 borehole measurements have beenmade over Earth’s surface. Averaging over the conti-nents and oceans, there is a heat flux of 0.087 ± [23, 47]. Integrating this flux over the surface ofEarth gives a heat flow of 44.2 ± IV. DARK MATTER CAPTURE RATE OFEARTH
The DM mass density, ρ χ = n χ m χ , in the neighbor-hood of the solar system is about 0.3 GeV/cm [1]. Nei-ther the mass nor the number density are separatelyknown. The DM is believed to follow a nonrelativisticMaxwell-Boltzmann velocity distribution with an aver-age speed of about 270 km/s. If a DM particle scatters asufficient number of times while passing through Earth,its speed will fall below the surface escape speed, 11.2km/s. Having therefore been gravitationally captured, itwill then orbit the center of Earth, losing energy witheach subsequent scattering until it settles into a ther-mal distribution in equilibrium with the nuclei in thecore. For the usual weak cross sections Earth is effec-tively transparent, and scattering and capture are veryinefficient. In contrast, we will consider only large crosssections for which capture is almost fully efficient. Notethat for our purposes, the scattering history is irrelevantas long as capture occurs; in particular, the depth in theatmosphere or Earth of the first scattering has no bear-ing on the results. The energies of the individual strucknuclei are also irrelevant, unlike in direct detection ex-periments. We just require that the DM is captured andultimately annihilated. A. Maximum Capture Rate
We begin by considering the maximum possible cap-ture rate of DM in Earth, which corresponds to Earthbeing totally opaque. Although our final calculationswill involve the microscopic scattering cross section ofDM on nuclei, this initial example deals with just themacroscopic geometric cross section of Earth. The fluxper solid angle of DM near Earth is n χ v χ /4 π , where n χ is the DM number density, and v χ is the average DM ve-locity. Since Earth is taken to be opaque, the solid angleacceptance at each point on the surface is 2 π sr. Thusthe flux at Earth’s surface is n χ v χ /2. The capture rateis then found by multiplying by Earth’s geometric crosssection, σ ⊕ = 4 πR ⊕ ≃ . × cm . Since n χ is notknown, this is ( ρ χ /m χ ) σ ⊕ v χ . For v χ = 270 km/s, thismaximal capture rate isΓ maxC = 2 × (cid:18) GeV m χ (cid:19) s − . (1)We will show that our results depend only logarithmi-cally on the DM velocity, and hence are insensitive tothe details of the velocity distribution.This maximal capture rate estimate is too simplistic, asit assumes that merely coming into contact with Earth,interacting with any thickness, will result in DM cap-ture. Instead, we define opaqueness to be limited to pathlengths greater than 0.2 R ⊕ , a value that incorporates thelargest 90% of path lengths through Earth. This reducesthe capture rate, but only by about 2%. We thereforeadopt 0.2 R ⊕ as our minimum thickness to determineefficient scattering. This length, translated into a chordgoing through the spherical Earth, defines the new effec-tive area for Earth. The midpoint of the chord lies ata distance of 0.99 R ⊕ from Earth’s center. Thus, prac-tically speaking, nearly all DM passing through Earthwill encounter sufficient material. The above require-ments exclude glancing trajectories from consideration,for which there would be some probability of reflection from the atmosphere [28, 29]; note also that the exclu-sion region in Section 4 would be unaffected by takingthis into account, since the DM heating of Earth wouldstill be excessive.The type of nucleus with which DM scatters dependson its initial trajectory through Earth. For a minimumpath length of 0.2 R ⊕ , this trajectory runs through thecrust, where the density is 3.6 g/cm [63], and the mostabundant element is oxygen [16]. Choosing this pathlength and density are conservative steps. Any largerpath length would result in more efficient capture, anda higher density and heavier composition (correspond-ing to a larger chord and therefore a different target nu-cleus, such as iron, which is the most abundant elementin the core) would as well. A more complex crust ormantle composition, such as 30% oxygen, 15% silicon,14% magnesium and smaller contributions from other el-ements [16], would stop DM ∼ B. Dark Matter Scattering on Nuclei
When a DM particle (at v χ ≃ − c ) elastically scat-ters with a nucleus/nucleon (at rest) in Earth, it de-creases in energy and velocity. After one scattering witha nucleus of mass m A , DM with mass m χ and initialvelocity v i will have a new velocity of v f v i = r − m A m χ ( m χ + m A ) (1 − cos θ cm ) , (2) −→ m χ ≫ m A r − m A m χ (1 − cos θ cm ) . (3)All quantities are in the lab frame, except the recoil angle, θ cm , which is most usefully defined in the center of massframe (see Landau and Lifschitz [64]). Here and belowwe give the large m χ limit for demonstration purposes,but we use the full forms of the equations for our results.After scattering, the DM has a new kinetic energy, KE χf =12 m χ v i (cid:18) − m A m χ ( m χ + m A ) (1 − cos θ cm ) (cid:19) (4) −→ m χ ≫ m A KE χi (cid:18) − m A m χ (1 − cos θ cm ) (cid:19) . (5)The nucleus then obtains a kinetic energy of KE A = KE χi − KE χf = 12 m χ v i (cid:18) − m A m χ ( m χ + m A ) (1 − cos θ cm ) (cid:19) (6) −→ m χ ≫ m A KE χi m A m χ (1 − cos θ cm ) (7)= m A v i (1 − cos θ cm ) (8)From the kinetic energy, the momentum transfer in thelarge m χ limit is: KE = | ~q | m A = m A v i (1 − cos θ cm ) (9) | ~q | = 2( m A v i ) (1 − cos θ cm ) . (10)In order to maintain consistency with others, we workwith n and σ in nucleon units even though the target wechoose (oxygen) is a nucleus. This means that n A (where A represents the mass number of the target) is n A = nA = ρm N A . (11)In turn, the cross section for spin-independent s-waveelastic scattering is represented as σ χA = A (cid:18) µ ( A ) µ ( N ) (cid:19) σ χN (12) −→ m χ ≫ m A A σ χN . Here A is the mass number of the target nucleus, whichequals m A /m N , and µ ( A or N ) is the reduced mass ofthe DM particle and the target.The A factor arises because at these low momentumtransfers, the nucleus is not resolved and the DM is as-sumed to couple coherently to the net “charge” – thenumber of nucleons. (If this coherence is somehow lost,a factor A would still remain for incoherent scattering.)The momentum transfer q = p m A KE A ≃ m A v i cor-responds to a length scale of ≃
10 fm for oxygen, muchlarger than the nucleus. We find that the correspondingnuclear form factor when the DM mass is comparable tothe target mass is ≃ A factor in Eq. (12) [2]. Ourconstraints could be scaled to represent this case by alsotaking into account the relative abundance of target nu-clei with nonzero spin in Earth, which is of order 1%.Note that if m χ = m A , and θ cm = π , the DM cantransfer all of its momentum to the struck nucleus, losingall of its energy in a single scattering through this scatter-ing resonance [16]. Taking this into account would makeour constraints stronger over a small range of masses, butwe neglect it. The nuclear recoil energy from this reso-nance is then m χ v i . Since v i is on average 270 km/s,this means that the maximum energy transferred from acollision is ∼ − that of the annihilation energy, m χ c . C. Dark Matter Capture Efficiency
From the full or approximate form of Eq. (4), we seethat the DM kinetic energy is decreased by a multiplica- tive factor that is linear in cos θ cm . If, in each indepen-dent scattering, we average over cos θ cm , the average fac-tor by which the kinetic energy is reduced in one or manyscatterings will simply be that obtained by setting cos θ cm = 0 throughout. (For s-wave scattering, the cos θ cm dis-tribution is uniform.)We will define efficient capture so that the heating ismaximized. To be gravitationally trapped, a DM particlemust be below the escape speed of Earth ( v esc = 11.2km/s), or equivalently, its kinetic energy must be lessthan m χ v . After one scattering event, the DM kineticenergy is reduced: KE χf = KE χi f ( m χ ) . (13)In successive collisions, this is compounded until12 m χ v = 12 m χ v i [ f ( m χ )] N scat . (14)Note that for collinear scatterings, the velocity loss inEq, (2) is also speed loss, leading to the same definitionof N scat .Therefore, on average, the number of scatterings re-quired to gravitationally capture the DM is N scat = − v i /v esc )ln h − m A m χ ( m χ + m A ) i (15) −→ m χ ≫ m A m χ m A ln ( v i /v esc ) , (16)where we have set cos θ cm = 0, since this correspondsto the average fractional change in the kinetic energy.Again, for simplicity the same element is taken to be thetarget each time. Note that since the initial DM velocityis inside the logarithm, N scat is insensitive to even largechanges in the assumed initial velocity.The number of scatterings for a given mass is large. ADM particle that has the same mass as the target nucleuswill scatter about 10 times before it is captured. Notethat the required N scat scales as m χ in the large masslimit, becoming very large: for m χ above 16 TeV (10 times the target mass), N scat is already larger than 3000.The actual energy losses in individual collisions are irrel-evant for our analysis, as we require only that the DM iscaptured after many collisions. For large values of N scat ,all scattering histories will be well-characterized by theaverage case.So far, these equations have just been kinematics; therequired N scat for stopping has not yet been made specificto Earth. It becomes Earth-specific by relating N scat tothe path length in Earth L and the mean free path λ : N scat = Lλ = Ln A σ χA . (17)The column density of Earth then defines the requiredcross section to generate N scat scatterings. The short-est path the particle could travel in is a straight line, sowe use that as the minimum. Any other path would belonger, and hence more effective at capture. This there-fore defines the most conservative limit on σ χA . Since wehave fixed cos θ cm to be 0 on average, in fact the pathwill not be completely straight. However, the lab framescattering angles are small.For elastic collisions between two particles, the rangeof scattering angles in the lab frame depends on the twomasses, m and m . There is a maximum scattering an-gle when one mass is initially at rest in the lab frame (inthis case, m ) [64]. If m < m , there is no restriction onthe scattering angle, which is defined in relation to m ’sinitial direction ( m is at rest). However, if m > m ,then sin θ maxlab = m /m . (18)Our main focus is m = m χ > m = m A . For m χ some-what greater than m A , note that the DM scattering anglein the lab frame is always very forward.Combining Eqns. (12), (15), and (17), the minimumrequired cross section to capture a DM particle is σ min χN = m N m A (cid:16) µ ( A ) µ ( N ) (cid:17) ρL N scat ( m χ ) , (19)= − v i /v esc )ln h − m A m χ ( m χ + m A ) i m N (cid:18) m A (cid:16) µ ( A ) µ ( N ) (cid:17) ρL (cid:19) , −→ m χ ≫ m A m χ (cid:18) m N m A (cid:19) ρL ln ( v i /v esc ) . Again, we choose a path length of 0.2 R ⊕ , to select about90% of the path lengths in Earth. Taking this length asa chord through Earth, the location corresponds to thecrust, with an average density of 3.6 g/cm , where themost common element is oxygen. We also choose an in-coming DM velocity of 500 km/s, which effectively selectsthe entire thermal distribution. A slower DM particle ismore easily captured. These parameters give a requiredcross section of σ min χN = − . × − cm (cid:16) µ (1) µ (16) (cid:17) ln h −
16 GeV m χ ( m χ +16 GeV) i (20) −→ m χ ≫ m A . × − cm (cid:16) m χ GeV (cid:17) . (21)Note that we use the unapproximated version, Eq. (20),for our figure, and give the large m χ limit in the equationsfor demonstrative purposes. When m χ is comparable to m A , σ min χN is different from the approximated, large m χ case in an important way.The resulting curve for σ min χN is shown in Fig. 2, asthe lower boundary of the heavily-shaded exclusion re-gion. The straight section of this constraint is easily seenfrom Eq. (21), as the required cross section for our ef-ficient capture scenario scales as m χ , due to the largenumber of collisions required for stopping, as in Eq. (16). log m χ [GeV] -45-40-35-30-25-20-15-10 l og σ χ N [ c m ] FIG. 2: Inside the heavily-shaded region, dark matter anni-hilations would overheat Earth. Below the top edge of thisregion, dark matter can drift to Earth’s core in a satisfac-tory time. Above the bottom edge, the capture rate in Earthis nearly fully efficient, leading to a heating rate of 3260 TW(above the dashed line, capture is only efficient enough to leadto a heating rate of &
20 TW). The mass ranges are describedin the text, and the light-shaded regions are as in Fig. 1.
At lower masses, the curved portion has its minimumat the mass of the target. DM masses close to that ofthe target can be captured with smaller cross sectionsbecause a greater kinetic energy transfer can occur foreach collision. At very low masses, much less than themass of the target, the DM mass dependence in the log-arithm is approximated differently. In this limit, σ χN is ≃ − (GeV /m χ ). As the DM mass decreases, it be-comes increasingly more difficult for the DM to lose en-ergy when it strikes a nucleus. As noted above, the crosssection constraints in the spin-dependent case could bedeveloped, and would shift the results up by 3 or 4 ordersof magnitude. All of the other limits that depend on this A coherence factor would also shift accordingly.The lower edge of the exclusion region is generallyrather sharp, because of these parameters. For example,consider the case of large m χ , where N scat is also large. Ifthe corresponding cross section is decreased by a factor δ ,so is the number of scatterings, and by Eq. (14), the com-pounded fractional kinetic energy loss would only be the1/ δ root of that required for capture. For small cross sec-tions, as usually considered, the capture efficiency is verylow. To efficiently produce heat, the minimum cross sec-tion must result in ∼
90% DM capture. We stress againthat we are not concerned with where the DM is capturedin Earth, so long as it is. The probability for capture can,however, be decreased using Poisson statistics (shown inFig. 2 as the dashed line with the accentuated dip at lowmasses) to yield just 20 TW of heat flow. This extensionand the upper edge of the exclusion region are describedbelow.
V. DM ANNIHILATION AND HEATINGRATES IN EARTHA. Maximal Annihilation and Heating Rates
Once it is gravitationally captured, DM will continueto scatter with nuclei in Earth, losing energy until drift-ing to the core. Once there, because of the large crosssection, the DM will thermalize with the nuclei in thecore. The number of DM particles N is governed by therelation between the capture (Γ C ) and annihilation (Γ A )rates [66]: Γ A = 12 AN = 12 Γ C tanh ( t p Γ C A ) . (22)We neglect the possibility of evaporation [15] for the mo-ment, which will affect our results for low m χ and σ χN ,as we will explain further below. The variable t is the ageof the system. A is related to the DM self-annihilationcross section σ χχ by A = < σ χχ v >V eff , (23)where V eff is the effective volume of the system [66]. Forthe relevant cross sections considered, equilibrium be-tween capture and annihilation is generally reached (seebelow), so the annihilation rate isΓ A = 12 Γ C (24)The effective volume is determined by the method ofGriest and Seckel (1988) [66], which is essentially the vol-ume of the DM distribution in the core. The number den-sity of DM is assumed to be an exponentially decayingfunction, exp( − m χ φ/kT ), like the Boltzmann distribu-tion of molecules in the atmosphere. The temperature ofthe DM in thermal equilibrium is T . The variable φ isthe gravitational potential, integrated out to a radius r ,written as φ ( r ) = Z r GM (˜ r )˜ r d ˜ r ; (25) M (˜ r ) = 4 π Z ˜ r r ′ ρ ( r ′ ) dr ′ . (26)The resulting effective volume using the radius of Earth’souter core, approximately 0.4 R ⊕ , a temperature of 5000K, and a density of 9 g/cm [63], is V eff = 4 π Z R core r e − mχφkT dr. (27)= 1 . × cm (cid:18)
100 GeV m χ (cid:19) Z . √ mχ GeV u e − u du −→ m χ ≫ m A . × cm (cid:18)
100 GeV m χ (cid:19) . (28) For increasing m χ , the integral (without the prefactor) inthe second line of Eq. (27) quickly reaches an asymptoticvalue of about 0.44.At very large masses, the effective volume for annihi-lation becomes very small. For instance, at m χ & GeV, the radius of the effective volume is . m χ = 10 GeV, as reflected in Fig. 2. There-fore, this small effective volume is not a large concern forour exclusion region.We assume that the DM annihilates into primarilyStandard Model particles, which will deposit nearly allof their energy into Earth’s core (with small correctionsdue to particle rest masses and the escape of low-energyneutrinos). When all of the DM captured is efficientlyannihilated, as specified, the heating rate of Earth is inequilibrium with the capture rate:Γ heat = Γ C × m χ = n χ σ eff v χ m χ = ρ χ m χ π (0 . R ⊕ ) (270 km / s) m χ (29)= 3260 TW . This heat flow is independent of DM mass, since the flux(and capture rate, when capture is efficient) scales as1 /m χ , while each DM particle gives up m χ in heat whenit annihilates. The value is much larger than the mea-sured rate of 44 TW we discussed in Section 3. B. Equilibrium Requirements
Does the timescale of Earth allow for equilibrium be-tween capture and annihilation for our scenario? In orderfor Eq. (22) to be in the equilibrium limit, tanh ( t √ Γ C A )must be of order unity. This is true if t √ Γ C A has a valueof a few or greater. Since Earth is about 4.5 Gyr old,we conservatively require that the time taken to reachequilibrium should be less than about 1 Gyr. From this,a realistic annihilation cross section is found. The condi-tion tanh ( t p Γ C A ) = 1; t p Γ C A ≃ few (30)allows the relation < σ χχ v >V eff = A & (few) Γ C t (31) < σ χχ v > & V eff Γ C t . (32)For an efficient capture rate (Eq. 1), the time of 1 Gyr,and the limit of large m χ , this requires an annihilationcross section for equilibrium of < σ χχ v > & − (cid:18) GeV m χ (cid:19) / cm / s . (33)Since this required lower bound is much smaller thanthat of typical weakly interacting DM particles that arethermal relics ( < σ χχ v > ≃ − cm /s [1]), it should beeasily met. One expects large scattering cross sections tobe accompanied by large annihilation cross sections, sothat even the possibility of p-wave-only suppression ofthe annihilation rate should not be a problem.For very large masses, the required annihilation crosssectionm, while small, approaches a quantum mechanicallimit. For example, for m χ & GeV the s-wave crosssection exceeds the unitarity bound [67, 68, 69]. We notethat this may also affect constraints on supermassive DMbased on neutrinos from annihilations [32, 33, 34]. To beconservative, we therefore do not extend our constraintsbeyond this point, though they may still be valid.The timescale also has to be long enough for DM todrift down to the core. If σ χN is too large, the DMwill experience too many scatterings and will not settleinto the core, and thus may not annihilate efficiently.Following Starkman et al. [2], we define the upper edgeof our exclusion region to require a drift time of . σ χN . . × − cm ( m χ / GeV) A ( µ ( A ) /µ ( N )) . . × − cm m χ / GeV A (cid:16) m χ + m N m χ + m A (cid:17) σ χN . −→ mχ ≫ mA . × − cm (cid:16) m χ GeV (cid:17) , (34)for a target of iron. However, a more detailed calcula-tion might relax this requirement. For example, Stark-man et al. [2] show that for large values of the capturecross section and certain other conditions, annihilationmay be efficient enough to occur in a shell, before theDM reaches the core. This would generally still be sub-ject to our constraint on heat from DM annihilation, andhence our exclusion region might extend to larger crosssections than shown. The two features of this drift lineat low mass occur around the mass of the target and themass of a nucleon, due to the various dominances of themass-dependent term in the denominator. The details ofthe shape of this drift line at low masses are irrelevant,because the astrophysical constraints already exclude thecorresponding regions.Aside from drifting to the core, the question of whetherheavy DM can actually get to Earth has been asked [70,71]. The low-velocity tail of the high-mass DM thermaldistribution in the Solar System may be driven into theSun by gravitational capture processes [70, 71], especiallybecause this DM’s velocity is on the order of the orbitalspeed of Earth in the Solar System, which is about 30km/s. However, this would affect only a tiny fraction ofthe full thermal distribution that we require to be effi-ciently captured. C. Annihilation and Heating Efficiencies
We are not picking a specific model for the annihilationproducts, aside from considering only Standard Modelparticles, which will deposit their energy in Earth, withthe exception of neutrinos. Our constraint thus has avery broad applicability. As noted the calculated heatflow if DM annihilation is important is 3260 TW, whichis very large compared to our adopted limit on an uncon-ventional source of 20 TW (or even the whole measuredrate of 44 TW). Typically then, either DM annihilationheating is overwhelming or it is negligible, inside or out-side of the excluded region. As shown in Section 4, thekinetic energy transferred from DM scattering on nucleiis about 6 orders of magnitude less than the energy fromDM annihilations. This contribution to Earth’s heat istoo low to be relevant for global considerations. However,it would be interesting to consider the more localized ef-fect of the kinetic energy deposition in the atmospherefor very large cross sections.There are circumstances in which the heating from DMannihilations can take a more intermediate value, includ-ing down to the chosen 20 TW number. As explainedabove, typically the number of scatterings required togravitationally capture the DM is very large. Therefore,a small decrease in σ χN and the proportionate change inthe expected number of scatterings means that the com-pounded kinetic energy loss is nearly always insufficient.However, at low m χ , the number is small enough thatupward fluctuations relative to the expected number canlead to capture. If N scat collisions typically lead to effi-cient capture for a cross section σ min χN , as defined above, anew and smaller N may be defined by the condition thatthe Poisson probability Prob( N ≥ N scat ) = 20 TW /3260 TW = 1/163. With this N , and its proportionatelysmaller σ χN , upward Poisson fluctuations in the numberof scatterings lead to efficient capture for a fraction 1/163of the incoming flux. Note that this small capture frac-tion is not just the low-velocity tail of the DM thermaldistribution, since we have defined these conditions forthe highest incoming velocities, v i = 500 km/s.The resulting constraint on σ χN is shown by the dashedline that dips below the main excluded region in Fig. 2.The enhanced valley around 16 GeV again arises fromthe ease of capture when the DM mass is near the targetmass. Note that for each mass the required < σ χχ v > isincreased by the same factor that decreased the originalrequired σ min χN . Since most of this exclusion region is al-ready covered by underground detectors, its details maynot be so important.For low DM masses, evaporative losses of DM from thecore due to upscattering by energetic iron nuclei may berelevant [15]. Simple kinematic estimates show that DMmasses below ≃ σ χN is small enough thatscattering is very rare – since otherwise any upscatteredDM will immediately downscatter. From the considera-tions above about Poisson fluctuations in the number of0scatterings, we expect that this should only be relevantbetween the dark-shaded region and the dashed line. VI. DISCUSSION AND CONCLUSIONSA. Principal Results
As summarized in Fig. 1, while very large DM-nucleonscattering cross sections are excluded by astrophysicalconsiderations, and small cross sections are excluded byunderground direct DM detection experiments, there isa substantial window in between that has proven verydifficult to test, despite much effort [2, 10, 11, 12, 13, 15,16, 17, 18, 20, 26, 28, 32, 35, 36, 37, 43, 44, 45, 46, 72,73]. High-altitude experiments have excluded only partsof this window. In this window, DM will be efficientlycaptured by Earth. We point out that the subsequentself-annihilations of DM in Earth’s core would lead toan enormous heating rate of 3260 TW, compared to thegeologically measured value of 44 TW.We show that the conditions for efficient capture, anni-hilation, and heating are all quite generally met, leadingto an exclusion of σ χN over about ten orders of mag-nitude, which closes the window on strongly interact-ing DM between the astrophysical and direct detectionconstraints. These new constraints apply over a verylarge mass range, as shown in Fig. 2. We have beenquite conservative, and so very likely an even larger re-gion is excluded. These results establish that DM in-teractions with nucleons are bounded from above by theunderground experiments, and therefore that these in-teractions must be truly weak, as commonly assumed.This means that direct detection experiments are look-ing in the correct σ χN range when sited underground andmotivates further theoretical study of weakly interactingDM [75, 76, 77, 78]. Furthermore, it means that DM-nucleon scattering cannot have any measurable effects inastrophysics and cosmology, and this has many implica-tions for models with strongly or moderately interactingDM [29, 72, 73, 74] and other astrophysical constraints onthe DM-nucleon interaction cross section [79, 80]. Thisexclusion region also completely covers the cross sectionrange in which strongly interacting dark matter mightbind to nuclei [81].To evade our constraints, extreme assumptions wouldbe required: that DM is not its own antiparticle, or thatthere is a large ( & & B. Comparison to Other Planets
Starkman et al. [2] calculated the efficient Earth cap-ture line for DM, but only had model-dependent results.We have now considered the consequences of annihila-tion in Earth, and have shown that it gives a model-independent constraint. Other planets have been dis-cussed, such as Jupiter and Uranus [13, 24, 25, 26,27, 28, 29, 30], but Earth is the best laboratory. Itis the best understood planet, with internal heat flowdata measured directly underground, from many loca-tions, and Earth’s composition and density profile arewell known [23, 47, 48, 49, 50, 62, 63]. Importantly, therelative excess heat due to DM annihilation would bemuch greater for Earth than the Jovian planets.What about other planets? The maximal heating ratedue to DM scales with surface area, and can be comparedwith the internal heating rates estimated from infrareddata [31]. If a constraint can be set, the minimum crosssection σ min χN that can be probed scales with the planet’scolumn density as ( nL ) − (see Eq. (17)) up to nontrivialcorrections for composition (see Eq. (12)). Note that thecolumn density nL is proportional to the surface grav-ity ∼ GM/R , which varies little between the planets,as noted in Table II [31]. Due to its known (and heavy)composition and well-measured (and low) internal heat,the strongest and most reliable constraints will be ob-tained considering Earth. As an interesting aside, it maythen be unlikely that heating by DM could play a signifi-cant role in explaining the apparent overheating of someextra-solar planets (“hot Jupiters”) [82, 83, 84, 85, 86]. C. Future Directions
While it appears that the window on strongly inter-acting DM is now closed over a huge mass range, moredetailed analyses are needed in order to be absolutelycertain. Our calculations are conservative, and the trueexcluded region is likely to be larger. It would be espe-cially valuable to have new analyses of the astrophysicallimits and the underground detector constraints. Thiswould give greater certainty that no sliver of the windowis still open. For DM masses in the range 1 GeV to 10 GeV, the upper limits on the DM scattering cross sectionwith nucleons from CDMS and other underground exper-iments have been shown to be true upper limits. Thus1
TABLE II: Comparison of potential dark matter constraintsusing various planets [31]. A greater difference between darkmatter and internal heating rates give greater certainty. Theminimum cross section probed scales roughly with the sur-face gravity. Earth is the best for setting reliable and strongconstraints.Planet DM Max. Heating Internal Heat Surface Gravity(TW) (TW) (units of ⊕ )Earth 3.3 × × × × × × × ≤ × × the DM does indeed appear to be very weakly interact-ing, and it will be challenging to detect it. Acknowledgments
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