Galactic Substructure and Energetic Neutrinos from the Sun and the Earth
GGalactic Substructure and Energetic Neutrinos from the Sun and the Earth
Savvas M. Koushiappas and Marc Kamionkowski Department of Physics, Brown University, 182 Hope Street, Providence, RI 02912 ∗ California Institute of Technology, Mail Code 350-17, Pasadena, CA 91125 † We consider the effects of Galactic substructure on energetic neutrinos from annihilation of weakly-interacting massive particles (WIMPs) that have been captured by the Sun and Earth. Substructuregives rise to a time-varying capture rate and thus to time variation in the annihilation rate and result-ing energetic-neutrino flux. However, there may be a time lag between the capture and annihilationrates. The energetic-neutrino flux may then be determined by the density of dark matter in theSolar System’s past trajectory, rather than the local density. The signature of such an effect maybe sought in the ratio of the direct- to indirect-detection rates.
PACS numbers: 95.35.+d,98.35.-a,98.35.Pr,98.85.Ry
Numerous experimental probes have confirmed in-directly the presence of a yet unknown form ofgravitationally-interacting matter in Galactic halos thatcontributes roughly 20% of the total cosmic energy den-sity. It is generally assumed that “dark matter” is inthe form of some yet undiscovered elementary parti-cle. Among the plethora of proposed theoretical par-ticle dark-matter candidates, weakly interacting mas-sive particles (WIMPs) are favored because they provide,quite generally, the correct relic abundance and becausethey may be experimentally accessible in the near fu-ture. WIMPs arise naturally in supersymmetric exten-sions (SUSY) of the Standard Model [1] as well as inmodels with Universal Extra Dimensions (UEDs) [2].The two principle avenues toward dark-matter detec-tion are direct detection (DD) via observation of the re-coil of a nucleus, when struck by a halo WIMP, in a low-background experiment [3, 4]; and neutrino indirect de-tection ( ν ID) via observation of energetic neutrinos fromannihilation of WIMPs that have been captured in theSun (and/or Earth) [5, 6].The DD rate is proportional to the local dark-matterdensity. The ν ID rate is proportional to the rate at whichWIMPs annihilate in the Sun, which in turn depends onan integral of the square of the dark-matter density overthe volume of the Sun. However, the WIMPs depleted inthe Sun by annihilation are replenished by the captureof new WIMPs. In most cases where the ν ID signal islarge enough to be detectable, the timescale for equilibra-tion of capture and annihilation is small compared withthe age of the Solar System. The ν ID rate is then alsodetermined by the local dark-matter density. Since thecapture rate is controlled by the same elastic-scatteringprocess that occurs in DD, the DD and ν ID rates areroughly proportional [7].In this Letter we investigate the effects of Galactic sub-structure on this canonical scenario. Analytic argumentsand numerical simulations suggest that realistic Galactichalos should have significant substructure, remnants ofsmaller halos produced in early stages of the structure-formation hierarchy (which may themselves house rem- nants of even smaller structures, and so on) [8]. Theoreti-cal arguments suggest that the substructure may be scaleinvariant [9] with subhalos extending all the way down tosub-Earth-mass scales [10]. The local dark-matter den-sity of different locations at similar Galactocentric radiiin the Milky Way may thus differ by a few orders ofmagnitude. The analytic descriptions of substructure arerough, and the simulations are limited by finite resolu-tion, and this motivates the pursuit of avenues towardempirically probing the existence of substructure.The purpose of this Letter to show that measurementsof the ratio of DD to ν ID rates can be used to test forGalactic substructure. If there is Galactic substructure,then the dark-matter density at the position of the So-lar System may vary with time. There is a finite timelag between capture and annihilation, and so the cur-rent energetic-neutrino flux may be determined not bythe local dark-matter density, but rather by the densityof dark matter along the past trajectory of the Solar Sys-tem. The ratio for the ν ID/DD rate may thus differ fromthe canonical prediction. A departure from the canonicalratio would thus, if observed, provide information aboutGalactic substructure. Since the equilibration timescalein the Earth is generally different from that in the Sun,additional information might be provided by observationof energetic neutrinos from WIMP annihilation in theEarth.To illustrate, we suppose the WIMP has a scalar cou-pling to nuclei, but the formalism can be easily general-ized to spin-dependent WIMPs. Then the DD rate for aWIMP of mass m χ from a target nucleus of mass m i is[1, 7], R scDD = 2 . × kg − yr − ρ χ, . η c ( m χ , m i ) (cid:16) m χ
100 GeV (cid:17) × (cid:16) m i
100 GeV (cid:17) (cid:18) m i m χ + m i (cid:19) σ , (1)where ρ χ, . is the local dark-matter density in unitsof 0.3 GeV cm − , and η c ( m χ , m i ) (given in Ref. [4])accounts for form-factor suppression. Here, σ is the a r X i v : . [ a s t r o - ph . C O ] S e p cross section for WIMP-nucleon scattering in units of10 − cm .The flux of upward muons induced in a neutrino tele-scope by neutrinos from WIMP annihilation in the Sunis Γ ν, = 7 . × km − yr − ( N/N eq ) ρ χ, . × f ( m χ )[ ξ ( m χ ) / . m χ /
100 GeV) σ , (2)while the corresponding flux from the Earth is obtainedby replacing the prefactor of Eq. (2) by 15 km − yr − .The function f ( m χ ) varies over the range 5 ≤ f ( m χ ) ≤ . ≤ m χ / GeV ≤ ξ ( m χ ) is in the range ∼ . − . N/N eq in Eq. (2) quantifies the number ofWIMPs in the Sun. Once WIMPs are captured in theSun, they accumulate deep within the solar core, wherethey may annihilate to a variety of heavy Standard Modelparticles which then decay to produce high-energy neu-trinos (which may escape the Sun). The time t evolutionof the number N of WIMPs in the Sun, is governed bythe differential equation, dN/dt = C c − C a N , (3)where C c is the capture rate of WIMPs by the Sun, and C a N is twice (because each annihilation destroys twoWIMPs) the effective annihilation rate. If both C c and C a are constant and the initial condition is N ( t = 0) ≡ N , the solution to this equation is N ( t ) = (cid:114) C c C a e t/τ − γe − t/τ e t/τ + γe − t/τ , (4)where γ ≡ − N (cid:112) C a /C c N (cid:112) C a /C c ≤ , (5)and τ = 1 / √ C c C a is the equilibration timecale. After atime t (cid:38) τ , the number N approaches N eq ≡ N ( t (cid:29) τ ) = (cid:112) C c /C a , and the annihilation rate Γ ν becomes equal to(one half) the capture rate, Γ ν = C a N / C c / C c and annihilation coefficient C a ,and thus the equilibration timescale τ , are determined bythe cross sections for WIMPs to annihilate and to scatterfrom nuclei. The equilibration timescale evaluates to τ (cid:12) = 1 . × yrs [ ρ χ, . f ( m χ )( σ A v ) ] − / × ( m χ /
100 GeV) − / σ − / . (6)Here, ( σ A v ) is the annihilation cross section (timesrelative velocity v in the limit v →
0) in units of10 − cm − s − . The equilibration timescale for theEarth is obtained by replacing the prefactor of Eq. (6) FIG. 1: The flux of energetic neutrinos from the Sun versusthe rate for direct detection. Each point denotes a supersym-metric model with the correct relic density and consistent withexperiment. The different symbols indicate the timescale forequilibration between capture and annihilation in the Sun.The horizontal line indicates a flux-threshold target for fu-ture ν ID experiments (IceCube+DeepCore [13] and the verti-cal line a rate-threshold target for DD in a 3-ton liquid-xenondetector [14]. For current bounds see [17]. by 1 . × yr. Using the canonical numbers we haveadopted, the equilibration timescales for the Sun andEarth are both small compared with the age of the So-lar System, but the equilibration timescales may varyby several orders of magnitude over reasonable rangesof the WIMP parameter space (and even more if moreexotic physics, like a Sommerfeld [11] or self-capture en-hancement [12], is introduced). To illustrate, we show inFig. 1 the equilibration timescales, for various DD and ν ID rates, for realistic supersymmetric dark-matter can-didates (using DarkSUSY [15]).Suppose now the WIMP model parameters are deter-mined, e.g., from the LHC and/or by theoretical assump-tion/modeling. Then the unknowns in Eqs. (1) and (2)will be the halo density ρ χ and the number N of WIMPsin the Sun (or Earth). The measured DD rate will thenprovide the local halo density ρ χ . Measurement of the ν ID rate will then determine (
N/N eq ) = tanh( t/τ ) (inboth the Sun and the Earth).For example, suppose the equilibration timescale is τ (cid:12) ≈ years in the Sun, and that the Solar Sys-tem entered a region of density ρ = 100 ρ (cid:12) (where ρ (cid:12) = 0 . − is the smoothed local halo den-sity) a time δt ≈ years ago, e.g., a 10 M (cid:12) halo(see Fig. 2). We would then see a boosted DD rateand a boosted energetic-neutrino flux from the Sun, butthe energetic-neutrino flux from the Earth would becorrespondingly weaker, since the equilibration time in FIG. 2: The neutrino-flux enhancement from an encounter,of duration 10 yr, of the Solar system with a region wherethe dark-matter density is enhanced by a factor of 100 (e.g.a 10 M (cid:12) subhalo). The top panel shows the capture rate(i.e., the DD rate). The bottom panel shows the resultingenergetic-neutrino flux from such an encounter for three equi-libration timescales. the Earth is longer. Now, suppose that the Solar Sys-tem exited this high-density region a million years ago.The DD rate would be at the canonical value, but theenergetic-neutrino fluxes from the Sun/Earth would stillbe boosted. Finally, suppose that the Solar System ex-ited the high-density region 10 years ago. In that case,the DD rate and energetic-neutrino flux from the Sunwould have the canonical values, but the neutrino fluxfrom the Earth would still be boosted, as τ ⊕ > τ (cid:12) .In reality, the capture rate C c ( t ) in Eq. (3) is a functionof time, and the equation for the number of WIMPs inthe Sun or Earth can be integrated numerically to givethe annihilation rate C a N ( t ) / M = M M (cid:12) ,each with a density 1000 βρ (cid:12) [16]. The radius of thesesubhalos would then be R = 0 . M /β ) / . The tran-sit time of the Solar System through such an object is δt ≈ M /β ) / yr. The mean-free time between en-counters with such objects is ¯ t ≈ ( M β ) / yr. Inthis toy model, the dark-matter density (and hence theDD rate) is zero unless the Solar System is within a sub-halo. Fig. 3 illustrates the effects from such a scenariofor M = 1. For equilibration timescales which are oforder τ ∼ ¯ t , the energetic-neutrino signal is depletedcompletely prior to the next encounter, while for longerequilibration timescales, the net effect is an elevated sig-nal at all times. For example, if τ (cid:12) ∼ ¯ t , and τ ⊕ > τ (cid:12) ,the signal from the Earth will be boosted relative to thesignal from the Sun for most of the time. FIG. 3: The neutrino flux in a hypothetical scenario whereall dark matter is in dense objects of 1 M (cid:12) . Different curvescorrespond to equilibration timescales as shown. Short equi-libration timescales (e.g. Sun) almost deplete completely theamount of WIMPs in a time comparable to the timescalebetween interactions. Longer equilibration timescales (e.g.Earth) result in a constant elevated flux.FIG. 4: The net effect on the energetic-neutrino flux of thepresence of 10 − M (cid:12) objects along the solar Galactic radiusduring the whole lifetime of the Solar System. Long equili-bration timescales result in the net build-up of WIMPs andthus an increase in the neutrino flux relative to that whichwould be obtained in a smooth halo. Finally, substructure may also speed up the equilibra-tion between capture and annihilation in cases where thesmooth-halo equilibration timescale is larger than the ageof the Solar System. Suppose the dark matter has asmooth component and some substructure down to verysmall scales, M ∼ − M (cid:12) . The abundance of the small-est subhalos can be inferred by extrapolating the subhalomass function measured in simulations at larger massscales. If we take the density within these 10 − M (cid:12) ob-jects to be ∼
100 times the smooth value, the radius ofthese objects is 10 − pc; the crossing time is 50 years;and the mean time between encounters is roughly a mil-lion years. For equilibration timescales less than the ageof the Sun ( τ (cid:12) (cid:28) t (cid:12) ), the signal will be roughly at theequilibrium value of the smooth component for most ofthe time. However, for long equilibration timescales (e.g.the Earth), the amount of depletion between interactionsis negligible. This effect leads to a continuous build-up ofWIMPs in the Earth, augmented by brief periods of anincreased capture due to interactions with the subhalos.This results in an energetic-neutrino signal that today ishigher than the signal that would be obtained from thesmooth component. This can be understood as follows:For N (cid:28) N eq , the second term in Eq. (3) is small. Whilethe cross section for the Solar System trajectory to inter-sect subhalos is ∝ β − / , the capture rate while in themis ∝ β , thus giving rise to a net increase in the capturerate ∝ β / . Fig. 4 shows the net effect of this speed-up.In summary, we considered the effects of Galactic sub-structure on energetic neutrinos from WIMP annihila-tion in the Sun and the Earth. While DD experimentsdepend on the local dark-matter density and velocity dis-tribution, the energetic-neutrino fluxes from the Sun andthe Earth depend on the past trajectory of the Solar Sys-tem through the clumpy Galactic halo. If experimentalDD and ν ID signals are obtained before the dark-matterparticle-physics parameters are known, then the poten-tial for probing dark matter via the DD/ ν ID ratios willbe compromised by the particle-physics uncertainties. 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