DSF/21/2008, IFIC/08-48 Radio constraints on dark matter annihilation in the galactic halo and its substructures E. Borriello1,∗, A. Cuoco2† and G. Miele1,3‡ 1Università “Federico II”, Dipartimento di Scienze Fisiche, Napoli, Italy & INFN Sezione di Napoli 2Department of Physics and Astronomy, University of Aarhus, Ny Munkegade, Bygn. 1520 8000 Aarhus Denmark 3Instituto de F´ısica Corpuscular (CSIC-Universitat de València), Ed. Institutos de Investigación, Apartado de Correos 22085, E-46071 València, Spain. (Dated: January 22, 2009) Annihilation ofDarkMatterusually producestogetherwithgamma rayscomparableamountsof electrons and positrons. The e+e− gyrating in the galactic magnetic field then produce secondary synchrotron radiation which thus provides an indirect mean to constrain the DM signal itself. To 9 this purpose, we calculate the radio emission from the galactic halo as well as from its expected 0 0 substructures and we then compare it with the measured diffuse radio background. We employ a 2 multi-frequencyapproach using datain therelevant frequency range 100 MHz–100 GHz, aswell as the WMAP Haze data at 23 GHz. The derived constraints are of the order hσAvi=10−24cm3s−1 n for a DM mass m = 100GeV sensibly depending however on the astrophysical uncertainties, in χ a particular on the assumption on the galactic magnetic field model. The signal from single bright J clumpsisinsteadlargely attenuatedbydiffusioneffectsandoffersonlypoordetectionperspectives. 2 2 PACSnumbers: 95.35.+d,95.85.Bh,98.70.Vc ] h I. INTRODUCTION medium the high energy e+ and e− release secondary p - low energy radiation, in particular in the radio and X- o rayband, that, hence, canrepresenta chance to look for r CosmologyandAstrophysicsprovidenowadaysacom- t DM annihilation. Furthermore, while the astrophysical s pelling evidence of the existence of Dark Matter (DM) uncertainties affecting this signal are similar to the case a [ [1, 2]. Nevertheless, its nature still remains elusive, and of direct e+, e− detection, the sensitivities are quite dif- Dark Matter constituents have escaped a direct detec- ferent, and, in particular, the radio band allows for the 2 tion in laboratory so far. Promising candidates are DM discriminationoftinysignalseveninabackgroundmany v particles produced in thermal equilibrium in the early order of magnitudes more intense. 0 universe, the so-called Weakly Interacting Massive Par- 9 Indirect detection of DM annihilation through sec- ticles (WIMPs). Theoretically, models of WIMPs natu- 9 ondary photons has received recently an increasing at- rally arise, for example, in SUSY as the Lightest Super- 2 tention, exploring the expected signature in X-rays [10, . symmetric Particle or as the Lightest Kaluza-Klein Par- 9 11, 12], at radio wavelengths [13, 14, 15, 16] , or both ticle inthe frameworkofextra-dimensions. These candi- 0 [17, 18, 19]. In the following we will focus our anal- dates are self-conjugate and can thus annihilate in cou- 8 ysis on the radio signal expected from the Milky Way 0 ples to produce as final states: neutrinos, photons, elec- (MW) halo and its substructures. It is worth noticing : trons, light nuclei (as wells as their antiparticles), etc., v that the halo signal has been recently discussed in Ref.s which can in principle be detected. Xi [20,21,22]inconnectiontothe WMAPHaze,whichhas Among the indirect DM detection channels, gamma- been interpreted as a signal from DM annihilation. In r a rayemissionrepresentsoneofthemostpromisingoppor- this concern we will take in the following a more con- tunity due to the very low attenuationin the interstellar servative approach, by assuming that the current radio medium, and to its high detection efficiency. See for ex- observationsareentirelyastrophysicalinorigin,andthus ampleRef.s [2,3,4]forareviewofthis extensivelystud- derivingconstraintsonthepossibleDMsignal. Themain iedissue. Theexpectedneutrinodetectionratesaregen- pointwillbetheuseoffurtherradioobservationsbesides erallylowalthoughforthcomingkm3 detectorsoffersome the WMAP ones, in the wide frequency range 100 MHz- promising prospect [5, 6]. Finally, positrons and protons 100 GHz, and a comparison of the achievable bounds. strongly interact with gas, radiation and magnetic field Furthermore, the model dependence of these constraints in the galaxy and thus the expected signal sensibly de- on the assumed astrophysical inputs will be analyzed. pends onthe assumedpropagationmodel [7, 8, 9]. How- Wewillalsodiscussthedetectionperspectivesofthesig- ever,during the processofthermalizationin the galactic nal coming from the brightest DM substructures in the forthcoming radio surveys. The paper is organizedasfollows: insectionII we will discuss the astrophysical inputs required to derive the ∗eborriello@na.infn.it †cuoco@phys.au.dk DM signal such as the structure of the magnetic field, ‡miele@na.infn.it theDMspatialdistributionandtheradiodataemployed 2 to derive the constraints. In section III we describe in Simulations, however, predict a DM distribution sum detail the processes producing the DM radio signal ei- ofasmoothhalocomponent,andofanadditionalclumpy ther whenit is originatedfrom the halo orfrom the sub- onewithtotalmassesroughlyofthesameorderofmagni- structures. InsectionIVwepresentanddiscussourcon- tude. Hereafterwewillassumeforthe massoftheMilky straints,while insectionVwe analyzethe detectionsen- Way M = M +M = 2×1012M , where M and MW h cl ⊙ h sitivity to the signal coming from the single DM clump. M denote the total mass contained in the host galactic cl In section VI we give our conclusions and remarks. halo and in the substructures (subhaloes) distribution, respectively. The relative normalization is fixed by im- posingthatsubhaloesintherange107M ,1010M have ⊙ ⊙ a total mass amounting to 10% of M [23]. Current II. ASTROPHYSICAL INPUTS MW numerical simulations can resolve clumps with a minimum mass scale of ∼ 106M . However, for WIMP par- A. Dark matter distribution ⊙ ticles, clumps down to a mass of 10−6M are expected ⊙ [30, 31]. We will thus consider a clump mass range be- OurknowledgeoftheDMspatialdistributionongalac- tween 10−6M and 1010M . ⊙ ⊙ tic and subgalactic scales has greatly improved thanks Finally, to fully characterize the subhalo population to recent high resolution zoomed N-body simulations wewillassumeamassdistribution∝m−2 andthatthey cl [23, 24, 25, 26]. These simulations indicate that for arespatiallydistributedfollowingtheNFWprofileofthe the radial profile of the galactic halo the usual Navarro- main halo. The mass spectrum number density of sub- Frank-White (NFW) distribution [27] haloes, in galactocentric coordinates~r, is thus given by ρ(r)= ρh , (1) dncl (m ,~r)=A mcl −2 r −1 1+ r −2 , 2 cl rrh 1+ rrh dmcl (cid:18)Mcl(cid:19) (cid:18)rh(cid:19) (cid:18) rh(cid:19) (2) (cid:16) (cid:17) where A is a dimensional normalization constant. The still worksas a good approximationoverall the resolved above expression assumes some approximations: for ex- scales. We will thus use this profile in the following. ample, a more realistic clump distribution should take Note, anyway,that this choice is quite conservative with into account tidal disruption of clumps near the galac- respect to other proposed profiles like the Moore profile tic center. Numerical simulations suggest also that the [28], which exhibits an internal cusp ∝ r−1.5 that would radial distribution could be somewhat anti-biased with giveinprincipleadivergentDMannihilationsignalfrom respect to the host halo profile. However, with our con- the center of the halo. Observationally the situation is servativeassumptionstheDMannihilationsignalisdom- more uncertain: Baryons generally dominate the gravi- inated by the host halo emission within up to 20◦−30◦ tational potential in the inner kpc’s and fitting the data from the galactic center so that the details of the sub- thus requires to model both the baryon and DM com- dominantsignalfromthe clumps havejust aslightinflu- ponent at the same time. The NFW profile is in fair ence on the final results. Recent results also show that agreement with the observed Milky Way rotation curve massdistributionseemstoconvergetom−1.9ratherthan [29], although, depending on the employed model, it is cl m−2 [25]. This would also produce only a minor change possibletofindanagreementformanydifferentDMpro- cl in the following results. files (see also [2] and references therein). We emphasize, Following the previous assumptions the total mass in however, that the various profiles differ mainly in the DM clumps of mass between m and m results to be halocenter(for r < 1kpc) wherethe uncertainties,both 1 2 ∼ innumericalsimulationsandfromastrophysicalobserva- m2 dn cl M(m ,m ) = d~r m (m ,~r) dm tions are maximal. Thus, our analysis which explicitly 1 2 cl dm cl cl excludes the galactic center, does not crucially depend Z Zm1 cl c on the choice of the profile. = 4π ln(1+c )− h Ar3M h 1+c h cl AproblemrelatedtotheprofileofEq.1isthatthemass (cid:20) h(cid:21) enclosedwithin the radiusr is logarithmicallydivergent. m2 (cid:0) (cid:1) × ln M , (3) A regularizationprocedure is thus required to define the m cl (cid:18) 1(cid:19) halo mass. Following the usual conventions we define where c ≡ r /r denotes the host halo concentration; the mass of the halo as the mass contained within the h vir h while their number is virial radius r , defined as the radius within which the vir mean density of the halo is δ = 200 times the mean m2 dn vir cl N(m ,m ) = d~r (m ,~r) dm ccoristmicoallogcoicsamlomloogdiceall(Ωdens≃ity0.3ρ,cΩr wh≃ic0h.,7f[o1r])aissetqaunadlartdo 1 2 Z Zm1 dmcl cl cl m Λ c ρcr ≃5×10−6GeVc−2cm−3. Theparametersdescribing = 4π ln(1+ch)− 1+hc Arh3Mcl the halo are then determined imposing the DM density (cid:20) h(cid:21) to be equal to ρS =0.365 GeV c−2 cm−3 near the Solar × Mcl − Mcl . (cid:0) (cid:1)(4) System, at a galactocentric distance of RS =8.5 kpc. (cid:18)m1 m2 (cid:19) 3 Imposing the normalization condition M(107M ,1010M ) = 10%M , we finally get ⊙ ⊙ MW for the mass due to the entire clumps distribution: M =M(10−6M ,1010M )∼53.3%M , (5) cl ⊙ ⊙ MW while for the number of these clumps we obtain N(10−6M ,1010M )∼2.90×1017. (6) ⊙ ⊙ Finally by using the previous constraints one can fix the values of free parameters r , ρ and A, hence obtaining h h r = 14.0 kpc, that corresponds to a halo concentration h of c = 14.4, ρ = 0.572GeV c−2 cm−3 and A = 1.16× h h 10−19 kpc−3 M−1. ⊙ A further piece of information is required to derive the annihilationsignalfromthe clumps, namely how the DM is distributed inside the clumps themselves. We will assume that each clump follows a NFW profile as the main halo with r and ρ replacing the correspond- cl cl ing quantities of Eq.1. However, for a full character- ization of a clump, further information on its concentration c is required. Unluckily, numerical simulations cl are not completely helpful in this case, since we require information about the structure of clumps with masses down to 10−6M , far below the current numerical res- ⊙ olution. Analytical models are thus required. In the FIG. 1: Sky map of galactic foregrounds at the frequency of current cosmologicalscenario [1] structures formed hier- 23 GHz (top), and of the residual map showing the WMAP archically, via gravitational collapse, with smaller ones Haze (bottom). forming first. Thus, naively, since the smallest clumps formed when the universe was denser, a reasonable ex- pectation is ccl ∝ (1+zf), where zf is the clump for- B. Galactic Magnetic Field mation redshift. Following the model of ref. [32] we −α will thus assume ccl = c1 Mmc⊙l with c1 = 38.2 and The MW magnetic field is still quite uncertain espe- α = 0.0392. With this con(cid:16)centr(cid:17)ation the integrated DM cially near the galactic center. The overall structure annihilation signal from all the substructures dominates is generally believed to follow the spiral pattern of the overthe smoothhalo componentonly at about 30◦ from galaxy itself with a normalization of about ∼ 1µG near the galactic center (see section III), so that the con- the solar system. Eventually, a toroidalor a dipole com- straintsonthe DMsignaldonotcruciallydependonthe ponent is considered in some model. unresolvedclumpssignal,comingbasicallyonlyfromthe We will consider in the following the Tinyakov and smooth halo component. However, given the large un- Tkachevmodel(TT)[34]whichisafairrepresentativeof certainties in the models, larger contributions from the the available descriptions of MW magnetic field. Within unresolvedpopulationofclumpsareinprinciplepossible this model, the fieldshowsthe typicalspiralpattern,an considering a different parametrizationof the concentra- exponentialdecreasealongthez axisanda1/Rbehavior tion (see for example the various models considered in inthegalacticplane. Thefieldintensityintheinnerkilo- [25, 33]). We will not investigate further this possibility parsecsisconstantatabout7µG.Wewillusetheslightly here. An enhancementofthe clumps signalis alsopossi- modified parametrization of this model as described in bleconsideringdifferentchoicesoftheclumpprofileother [36]. Highernormalizationsarein principlepossible con- than the NFW: Differently from the case of the halo, in sideringmorecomplexstructuresasforexampleadipole fact, the clump signal depends sensibly from the chosen or a toroidal component [35]. Indeed some recent anal- profile and a Moore profile or an Isothermal profile can yses [37, 38] including new available data seems to favor in principle enhance the signal of several orders of mag- the presence of these further structures. We will thus nitude. Alsointhis casewechoosetoquoteconservative consider as possible also an “high normalization model” constraints and we will not consider these possibilities that we simply parameterize as a constant 10 µG field. further. This choice is also motivated by a comparison with the 4 results of [20, 21] where the same magnetic field is used. [44] fails in finding the feature. We will anyway show in Further, beside the regular component, the galactic the following for comparison the constraints derived us- magnetic field presents a turbulent random component. ingtheHazeresidualmapattheWMAPfrequencyof23 The r.m.s. intensity of this component is generally ex- GHz [45]. A map of the Haze and ofthe 1 GHz emission pectedofthesameorderofmagnitudeoftheregularone, is shown in Fig.1. We will see however that within our but both its spatial distribution and its spectrum are conservative approach comparable or better constraints poorly known, thus here we neglect its effects. Naively, can be obtained thanks to the use of multi-frequency in- this random component is expected to affect the syn- formation. For a given DM mass, in fact, 23 GHz is chrotron maps that we will show in the following pro- generally not the best frequency to use and better con- ducing a blurring of the otherwise regular pattern. Also, straints are instead obtained using observations at lower therandomcomponentcontributestoincreasetheoverall frequencies even without further foreground modeling. normalizationof the field. Thus without this component Definitely, a detailed foreground modeling at all radio the synchrotronsignalis slightly underestimated so that frequencies would clearly give much stronger constraints we can regard this choice as conservative. onthe DM signaland/oreventually confirmthe DM nature of the WMAP haze. To this purpose consistent progress will be achieved in the next years with the new C. Radio Data highqualitydatacomingfromthePLANCKmissionand from low frequency arrays like LOFAR and SKA. In the following we will derive constraints on the DM emission comparing the expected diffuse emission from thesmoothhaloandtheunresolvedpopulationofclumps III. DM SYNCHROTRON SIGNAL with all sky observation in the radio band. In the frequency range 100 MHz-100 GHz where the DM synchrotron signal is expected, various astrophysical pro- A. Particle Physics cesses contribute to the observed diffuse emission. Com- petingsynchrotronemissionisgivenbyCosmicRayelec- InastandardscenariowhereWIMPsexperienceanon trons accelerated in supernovae shocks dominating the exotic thermal history, a typical mass range for these radioskyupto∼10GHz. AthigherfrequenciestheCos- particles is 50GeV< m <1TeV, while a simple esti- mic Microwave Background (CMB) and its anisotropies ∼ χ ∼ mate for their (thermally averaged) annihilation cross represent the main signal. However, thanks to the very sectionyieldshσ vi=3×10−27cm3s−1/Ω h2 [3],giv- sensitive multi-frequency survey by the WMAP satel- A cdm inghσ vi≈3×10−26cm3s−1 forΩ h2 ≈0.1asresult- lite, this signal (which represents thus a background for A cdm ing from the latest WMAP measurements [1]. However, DM searches) can be modeled in a detailed way and can this naive relation can fail badly if, for example, coanni- thus be removed from the observed radio galactic emis- hilationsplayaroleintheWIMPthermalizationprocess sion [39]. Other processes contributing in the 10-100 [46], and a much wider range of cross sections should be GHz range are given by thermal bremsstrahlung (free- consideredviable. In this work we consider values of m freeemission)ofelectronsonthegalacticionizedgas,and χ from about 10 GeV to about 1 TeV, and hσ vi in the emission by small grains of vibrating or spinning dust. A range (10−26-10−21)cm3s−1 In the following our approach will be to compare the The e+e− annihilation spectrum for a given super- DM signal with the observed radio emission where only symmetricWIMP candidate canbe calculatedforexam- the CMB is modeled and removed. For this purpose we ple with the DarkSUSY package [47]. However,the final usethecodedescribedin[40]wheremostoftheradiosur- spectrumhasonlyaweakdependence ontheexactanni- vey observations in the range 10 MHz-100 GHz are col- hilationprocesswiththechannelsχχ→ZZ,W+W−,qq¯ lectedandaschemetoderiveinterpolated,CMBcleaned giving basically degenerate spectra. For leptonic chan- sky maps at any frequency in this range is described. nels like the τ+τ− decaying mode the spectrum differs A more aggressive approach would be of course to try significantly, although this channel has generally a quite tomodelandsubtractalsotheremainingemissions(syn- low branching ratio. For simplicity we will assume here- chrotron, free-free, dust) in order to compare the ex- after full decay into qq¯ channel, hence e− (e+) will be pected DM signal with the residual radio map. This emittedbydecayingmuons(anti-muons)producedinpi- is indeed the approach followed in [41, 42] where resid- onsdecays. Inthisframework,theresultinge+,e− spec- ual maps at the 5 WMAP frequencies are derived using trum can be written as a convolution, namely spatial templates for the various expected astrophysical components. The residual maps then exhibit the feature called the WMAP Haze, which has been indeed in- dN mχc2 dN(µ) e e (E )= dE (E ,E ) terpreted as radio emission related to DM annihilation dE e µ dE e µ [20,43]. However,thefitprocedureusedfortheHazeex- e ZEe e traction is crucial, and using more degrees of freedom to Eµ/ξ dNµ(π) × dE W (E ) (E ) (7) π π π π modeltheforegroundsasperformedbytheWMAPteam dE ZEµ µ 5 with ξ =(m /m )2 , where B ∼ 10µG translates into a minimum valid frequency µ π ν = 10–100 MHz, thus below the frequency window we dN(µ) 2 5 3 E 2 2 E 3 are going to explore (see Eq.20 below). e e e (E ,E )= − + ,(8) e µ dE E 6 2 E 3 E e µ " (cid:18) µ(cid:19) (cid:18) µ(cid:19) # dN(π) 1 m2 B. Electrons equilibrium distribution µ (E )= π , (9) dE π E m2 −m2 µ π π µ Dark matter annihilation injects electrons in the 1 15 mχc2 23 Eπ 2 galaxy at the constant rate W (E )= 1− (.10) π π m c216 E m c2 χ (cid:18) π (cid:19) (cid:18) χ (cid:19) 2 1 ρ(r) dN e Q(E ,r)= hσ vi . (12) Inparticular,Eq. (8)istheelectron(positron)spectrum e 2 m A dE produced in the muon (anti-muon) decay µ− → e−ν ν¯ (cid:18) χ (cid:19) e µ e (µ+ → e+ν¯ ν ). Eq. (9) stands for the µ− (µ+) spec- Ontheotherhand,theinjectedelectronslooseenergyin- µ e trumfromπ− →µ−ν¯ (π+ →µ+ν )decayprocess,and, teracting with the interstellar medium and diffuse away µ µ finally,Eq. (10)providesareasonableanalyticalapprox- from the production site. In the limit in which convec- imation of the spectrum of pions from qq¯hadronization tionandreaccelerationphenomenacanbe neglected,the [48]. It is worth noticing that to be more accurate Eq. evolution of the e+e− fluid is described by the following (10) should be substituted by a numerical calculation, diffusion-loss equation [49, 50, 51] which however results not necessary for the aim of the ∂ dn dn ∂ dn present paper as discussed in the following. e = ▽~ · K(E ,~r)▽~ e + b(E ,~r) e e e In this approximation the final electron (positron) ∂tdEe (cid:20) dEe(cid:21) ∂Ee(cid:20) dEe(cid:21) spectrum can be castin a simple polynomial formof the +Q(E ,~r), (13) e ratio E /m c2: e χ where dn /dE stands for the number density of e+, e− e e dNe 1 Ee j per unit energy, K(Ee,~r) is the diffusion constant, and dE (Ee)= m c2 aj m c2 , (11) b(Ee,~r) represents the energy loss rate. The diffusion e χ j∈J (cid:18) χ (cid:19) length of electrons is generally of the order of a kpc (see X section V) thus for the diffuse signal generated all over where J ={−3,−1,0,1,2,3}andthe coefficients a are 2 2 2 j the galaxy, and thus over many kpc’s, spatial diffusion listed in Table I. can be neglected. This is not the case for the signal The main advantage of using the above analytical ap- comingfromasingleclumpforwhichtheemittingregion proximation instead of a more accurate numerical input is much smaller than a kpc. We will further analyze this is that, as will be clear in the next section, most of the point in section V. By neglecting diffusion, the steady observablesfortheradioemissionwillbeexpressedinan state solution can be expressed as analyticalformaswell. This,inturn,isofhelpforabet- ter understanding of the physical results. Nevertheless, dn τ mχc2 the difference with the complete numerical calculation dEe(Ee,~r)= E dEe′ Q(Ee′,~r), (14) turns to be small, arising only for quite low electron en- e e ZEe ergies, and thus for very low radio frequencies. At low whereτ =E /b(E ,~r)isthecoolingtime,resultingfrom e e energies, in fact, the analytical form has an asymptotic the sum of several energy loss processes that affect elec- behavior ∝ Ee−1.5 while the numerical spectrum has a trons. Inthefollowingwewillconsidersynchrotronemis- turn down. From a comparison with the numerical out- sionandInverseCompton Scattering (ICS) offthe back- put from DarkSUSY for a 100 GeV WIMP with 100% groundphotons(CMB andstarlight)only,whicharethe branching ratio into b¯b the analytical form is a fair ap- fasterprocessesandthustheonesreallydrivingtheelec- proximationuntilEe ≈1GeV,whichforamagneticfield trons equilibrium. Other processes, like synchrotron self absorption, ICS off the synchrotron photons, e+e− annihilation, Coulomb scattering over the galactic gas and TABLE I: a values bremsstrahlung are generally slower. They can become j relevant for extremely intense magnetic field, possibly coefficient analytical numerical present in the inner parsecs of the galaxy [14], and thus a−3/2 168591−1−ξ3ξ/2 0.456 will be neglected in this analysis. a −66 1 −5.37 For synchrotron emission the energy loss is given by −a1/2 25ξ2−1781ξ++ξ81ξ/12/2+9 10.9 (for ex. see [52]) bsyn(Ee) = 4/3cσTγ2β2UB with UB = 0 36 (1−ξ)ξ1/2 B2/2µ the magnetic energy density so that the time a1/2 91−1ξ−−ξ1/2 −6.77 scale o0f the energy loss is: a − 3 5ξ2−42ξ+72ξ1/2−35 0.969 a2 1 2385ξ2−27(01ξ−+ξ4)2ξ41/ξ21/2−189 −0.185 τ =τ0 B −2 Ee −1 (15) 3 189 (1−ξ)ξ1/2 syn syn µG GeV (cid:18) (cid:19) (cid:18) (cid:19) 6 20 10-20 z(cid:32)0 10 10-22 -1L z H -1 3 cpk 0 (cid:43)(cid:47) Solar -2ms y c System V 10-24 e G H 2 y (cid:16)10 nsit e d x u fl 10-26 1 (cid:16)20 (cid:16)20 (cid:16)10 0 10 20 x kpc 10-28 1 10 102 103 0.75 0.75 (cid:43) (cid:47) yy(cid:32)(cid:32)00 frequency HGHzL c (cid:47) pk 0 0 (cid:43) z FIG.3: Synchrotronfluxdensityfromthreedifferentclumps. (cid:16)0.75 (cid:16)0.75 (cid:16)20 (cid:16)10 0 10 20 Theneutralinoparametersaremχc2 =100GeVandhσAvi= x kpc 3×10−26 cm3s−1, while the relevant clumps parameters are shown in table II. 0 % 100 % (cid:43) (cid:47) b we have ICS FIG. 2: Projections of the galaxy in the xy and xz planes −1 showing the fractional synchrotron contribution to the e+e− τ(E ,~r) = Ee µ(~r)τ0 , (17) total energy losses for TT model [34] of GMF and Galprop e GeV syn (cid:18) (cid:19) tmoo2d0el%[5r3e]aocfhIiSnRgFit.sTmheaxsyimncuhmroatrtotnhleoscseenstceornotfritbhuetemuapgnmeotsict B(~r) 2 τs0yn Urad(~r) −1 µ(~r) = + , (18) arms. In the remaining regions, included the galactic center, µG τ0 eV/cm3 "(cid:18) (cid:19) ICS # ICS is dominating. withthefunctionµ(~r)enclosingthewholespatialdepen- dence. with τ0 =3.95×1017s. syn By substituting the above expressions into Eq.(14) we Similarly, for Inverse Compton emission the energy get the following equilibrium distribution for electrons loss is given by b (E ) = 4/3cσ γ2β2U . The rel- ICS e T rad egvaalancttircaduinaitfoiornmbcaocnktgrribouuntidonfocroInCsiSstiisnggiovfetnhbeyCaMnBexwtirtah- dne = hσAviτs0yn µ(~r) ρ(r) 2 UCMB = 8π5(kT)4/15(hc)3 ≈ 0.26 eV/cm3, the opti- dEe 2mχc2 (cid:18)mχ (cid:19) cal/infraredextragalacticbackgroundandthe analogous m c2 −k E k−1 χ e spatially varying galactic contribution, the Interstellar × bk , (19) GeV GeV Radiation Field (ISRF). For the latter we use as tem- k∈K (cid:18) (cid:19) (cid:18) (cid:19) X plate the Galprop distribution model [53] which reduces being K=J ∪{−1},b =−a /(k+1), if k 6=−1, while to the extragalacticone at highgalactocentricdistances. k k b = a /(j+1). In this model, the ISRF intensity near the solar posi- −1 j∈J j tion is about 5 eV/cm3, and reaches values as large as P 50 eV/cm3 in the inner kpc’s. With this model the ICS C. Synchrotron spectrum is alwaysthe the dominantenergylossprocess,alsonear the galactic center (see Fig.2). We thus have The synchrotron spectrum of an electron gyrating in U (~r) −1 E −1 a magnetic field has its prominentpeak atthe resonance τ =τ0 rad e , (16) ICS ICS eV/cm3 GeV frequency (cid:18) (cid:19) (cid:18) (cid:19) with τI0CS =9.82×1015s. ν =ν B Ee 2 , (20) Finally,consideringboththeenergylossesbtot =bsyn+ 0 µG GeV (cid:18) (cid:19)(cid:18) (cid:19) 7 10-13 1GHz mΧ=100GeV b=0 <ΣAv>=3´10-26cms-1 10-14 -1Lsr10-15 -1-1sHz10-16 radoiobseemrvisesdion -2m Vc10-17 Ge H x flu10-18 total 10-19 clumps fluxdensityHGeVcm-2s-1Hz-1L NFW 10-20 10-18 10-21 10-24 10-27 -180 -135 -90 -45 0 45 90 135 180 lHdegreesL FIG. 4: Sky map at the frequency of 1 GHz for a realization FIG. 5: DM synchrotron profile for the halo and unresolved of clumps distribution. For each clump, the circle radius is substructures and their sum at 1 GHz for m = 100 GeV proportional to thelogarithm of radio flux. and hσ vi = 3×10−26cm3s−1. The astrophyχsical observed A emission atthesamefrequencyisalsoshown. Thegrayband indicatestheangularregionwithinwhichtheDMsignalfrom with ν0 =3.7×106Hz. This implies that, in practice, a the host halo dominates over the signal from substructures δ–approximationaroundthepeaksworksextremelywell. modeled as in section IIA. Usingthisfrequencypeak approximation,thesynchrotron emissivity can be defined as withd thedistancebetweentheobserverandtheclump. cl dn dE (ν) This can be rewritten as e e j (ν,~r)= (E (ν),~r) b (E (ν),~r). (21) ν e syn e dEe dν I (ν,R~ )=I0µ(R~ ) A B(R~cl) 1−k2 ν k2(23) Thisquantityisthenintegratedalongthelineofsightfor ν cl ν cl k µG Hz ! the various cases to get the final synchrotron flux across Xk (cid:16) (cid:17) the sky. mχc2 −k ν0 −k2−1 A (m )=b , (24) k χ k GeV Hz (cid:18) (cid:19) (cid:16) (cid:17) with 1. Single clump signal I0 ν = According to the description of previous section IIA, GeV cm−2s−1 Hz−1 let us consider a clump of mass mcl, whose center of m c2 −3 hσ vi mass is placed at R~ and with a sufficiently small size. =2.57×10−12 χ A cl 100GeV 10−26cm3s−1 In this case it is possible to neglect the spatial variation (cid:18) (cid:19) 3 −2 2 of µ(~r) inside the clump itself, and thus the flux Iν can × rcl dcl ρcl . (25) be calculated as: kpc kpc GeVc−2cm−3 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 1 I (ν,R~ )= d~rj (ν,R~ +~r) , (22) ν cl 4πd2 ν cl Fig. 3 shows some examples of signal, produced by cl Z three clumps of our simulation. An important feature to notice is that the synchrotronsignal sensibly depends clump d r ρ B fluxdensity at 1 GHz on the magnetic field both in the normalization and in cl cl cl # kpc kpc GeVc−2cm−3 µG GeVcm−2s−1Hz−1 the covered frequency range. In particular, the signal frequency cutoff, remnant of the energy spectrum cutoff 1 14.2 0.180 6.51 0.0962 1.70×10−25 near m , depends on B following Eq.20. 2 4.71 0.181 6.50 0.320 4.55×10−23 χ 3 5.50 0.188 6.404 3.08 2.66×10−21 Fig. 4showsinsteadthepositionsandradiointensities for a realization of the clumps distribution with masses TABLE II:Parameters of theexample clumpschosen in Fig. m > 107M . It can be seen that all the clumps with cl ⊙ 3. a non negligible signal lie near the galactic plane where 8 most of the galactic magnetic field is concentrated. Few from the substructures, instead, we have to sum over all clumps are visible at high latitude just because of pro- haloes jection effects, being located very near and slightly up or below the solar position with respect to the galactic d2Iunr dn plane. ν =cosb dm dss2 cl (m ,~r)Ires(ν,~r), dldb cl dm cl ν Z Z cl (27) 2. Diffuse signals with Iνres given by Eq.(23) and~r=~r(s,l,b). Interestingly, the sum of the two diffuse contributions Thediffusehalosignalissimilarlygivenbytheintegral can be rewritten as along the line of sight of Eq.(21) d2I cosb ∞ d2IDM cosb ν = j ds, (26) ν = jDMds, (28) dldb 4π ν dldb 4π ν Z0 Z where (l,b) are coordinates on the sphere and s the line of sight coordinate. To calculate the total contribution where 1 m c2 −3 hσ vi ρ /GeVc−2cm−3 2 ρ /GeVc−2cm−3 jDM = χ A h + CL ν 4(cid:18)GeV(cid:19) cm3s−1 ((cid:20)(r/rh)(1+r/rh)2(cid:21) (r/rh)(1+r/rh)2 ) B(~r) 1−k/2 ν k/2 ×µ(~r) A (m ) GeVcm−3s−1Hz−1sr−1. (29) k χ µG Hz Xk (cid:18) (cid:19) (cid:16) (cid:17) Thus, from the point of view of DM annihilation the has basically no influence for the constraints on the DM unresolved clumps signal behaves like a further smooth signal. NFW component with the same scale radius of the halo In Fig.7 we show the radio constraints on the DM an- profile, but with a different effective density ρCL = nihilation signal in the mχ–hσAvi plane for various fre- 0.604GeVc−2cm−3, and with an emissivity simply pro- quencies and various choices of the foreground. Several portional to the density profile instead of its square. comments are in order. First, we can see that, as ex- We seethatthe halocomponentdominatesinthe cen- pected, the use of the haze at 23 GHz gives about one tral region of the galaxy, where orderofmagnitudebetterconstraintswithrespecttothe synchrotron foregrounds at the same frequency. How- r r 2 ρ2 ever, using also the information at other frequencies al- 1+ < h ⇒ r <4.39kpc (30) most the same constraints can be achieved. This infor- r r ρ h (cid:18) h(cid:19) CL mationinparticulariscomplementarygivingbettercon- straints at lower DM masses. This is easily understood which corresponds to a disk of radius 27.3 degrees (see since a smaller DM mass increases the annihilation sig- fig. 5). nal (∝ m−2) at smaller energies, and thus smaller syn- χ chrotron frequencies. In particular, the constraints im- proveofaboutoneorderofmagnitudeatm ∼100GeV χ IV. DM ANNIHILATION CONSTRAINTS from23GHzto1GHzwhileonlyamodestimprovement is achieved considering further lower frequencies as 0.1 ThepatternandintensityoftheDMradiomapresult- GHz. Thissaturationoftheconstraintsisduetothefre- ing from the sum of the contributions from the smooth quency dependence of the DM signal, that below 1 GHz halo and unresolved clumps is shown in Fig. 6 for becomes flatter than the astrophysical backgrounds so m = 100 GeV and hσ vi = 3×10−26cm3s−1. Similar that the fraction of contributionfrom DM is maximalat χ A maps are obtained at different frequencies and different about 1 GHz. Further, the constraints show a threshold m and hσ vi to obtain DM exclusion plots. For our behavior given basically by Eq.20 which settles a maxi- χ A analysis we use a small mask covering a 15◦×15◦ region mum emitted radio frequency for a given DM mass m . χ around the galactic center where energy loss processes This threshold behavior is, for example, clearly seen at otherthansynchrotronandICSstartpossiblytoberele- 23GHz in the rightpanelofFig.7 where only for masses vant. We include the galactic plane although this region above ∼40 GeV the cross section is constrained. 9 in fact, the smaller values of U reduces the ICS losses rad enhancing in turn the synchrotron signal. It should be said however that, while the magnetic field normalization is still quite uncertain, the ISRF is instead more constrainedandalargevariationwithrespecttotheGal- prop model seems unlikely. The constraints shown in Fig.7 extend down to ∼10 GeV, which is somewhat the mass limit for a con- servativeanalysis. Itisclearthatforlowmassesthecon- straintscomemoreandmorefromlowerfrequencies: For exampleforaWIMPof30GeVthedataat100MHzare 2 orders of magnitude more constraining than the data at 10 GHz. However, extremely low frequencies are not experimentally accessible. For a WIMP of 1 GeV, from FIG. 6: Sky map of the galactic radio signal generated by Eq.20 with a magnetic field of O(µG) only frequencies theDMsmoothhaloandunresolvedclumpsatthefrequency < 10 MHz would be useful to place constraints on the of 1GHz for m = 100 GeV and hσ vi = 3×10−26cm3s−1. ∼ χ A DMsignal. Althoughobservationsatthisfrequencyexist The peculiar shape of the signal, pinched approximately at [40], in general the survey sky coverage is quite incom- ±30◦ and ±60◦, reflects basically the structure of the mag- plete and the data quality is non-optimal. Observations neticfieldasseeninprojection fromtheSolarSystem,where inthisverylowfrequencyrangeshouldsubstantiallyim- theobserverislocated (comparealso withFig.2). Thegalac- ticcenterregionandthefirstfewmagneticarmsarevisibleas provewith the next generationradio arraysLOFARand regions of high magnetic field intensity and hence prominent SKA. WIMP masses below 1 GeV still would produce synchrotron emission. observable synchrotron radiation at the galactic center where the magnetic field is likely much higher than µG scale (possibly O(mG) ). This kind of analysis would be Notice that although the astrophysical background however quite model dependent and would face further which we compare with at 1 GHz is an interpolation, backgrounduncertainties. the derived constraints are still valid given the smooth behavior and the broad frequency extent of the DM sig- V. SINGLE CLUMPS DETECTABILITY nal, which does not exhibit narrow peaks at particular frequencies. However, effective measurements have been performed for example at 408 MHz and 1.4 GHz (see Tohaveareliableestimateofthesensitivitytoasingle [40]). Quoting our constraints at these exact frequencies clumpdetectiondiffusioneffectscannotbeneglected. Al- would change the results only slightly. though the integrated synchrotron clump signal is given by Eq.(23), the clumps appear extended rather than The DM signalhas thus a broadfrequency extent and pointlike with a dimension typically of several degrees. also below 1 GHz is still relevant. This is a potential As a reasonable approximation we can assume that the problem for the DM interpretation of the WMAP Haze signalis spreadover an area of radius equal to the diffu- given that, in the Haze extraction procedure, the ob- sionlengthoftheelectronsl = K(E )τ (E ),where served radio emission at 408 MHz is used as template D e loss e K is the diffusion coefficient and τ is the energy loss of the synchrotron background. In fact, naively, a DM loss p time given by Eq.(17). We use for K the Galpropmodel signal at 23 GHz should be relevant at 408 MHz as well, [49] unless either the DM mass or the magnetic field is so high to shift the DM contribution to higher frequencies δ E and making it negligible at 408 MHz. K =K e , (31) 0 E The second relevant point to notice is that the con- (cid:18) e0(cid:19) straint depends quite sensibly on the magnetic field as- with a reference energy E = 3 GeV, a Kolmogorov e0 sumptions. TheconstraintsweobtainwiththeTTmodel spectrum δ =1/3 and K =1028cm2/s. 0 aregenerallyalmosttwoordersofmagnitudeweakerwith Taking as reference the parameters of a very bright respect to the results reported in [21]. They are instead clump like the #3 in table II, we get (1pc=3×1018cm) moreinagreementwith[22]whereGalprophasbeenem- ployed to calculate the DM synchrotron signal. For a E −1 closercomparisonwith[21]wechoose,astheydo,acon- lD =sK(Ee) GeeV µ(~x)τs0yn ≈1kpc, (32) stant magnetic field of 10µG although still keeping the (cid:18) (cid:19) GalpropISRF model. Evenin this case our derivedcon- for E ≈ 10 GeV and for a radiation density U ≈ e rad straints are a factor of 5 weaker (despite the inclusion 5eV/cm3. The energy losses are basically dominated by of the contribution from substructures). The remaining ICSthustheresultisalmostindependentofthemagnetic factorof5canbefinallyrecoveredusingaconstantISRF fieldvalue. Moreoverthe dependence onthe electronen- with U = 5eV/cm3 as assumed in [21]. In this case, ergy and the radiation density itself is very weak. Of rad 10 10-21 10-21 23GHz Observedradioemission Observedradioemission at23GHz 10-22 10GHz 10-22 1GHz WMAPHaze -1Ls10-23 -1Ls10-23 at23GHz 3m 3m H>c 0.1GHz H>c v v <ΣA10-24 <ΣA10-24 10-25 allowedregion 10-25 allowedregion 3´10-26cm3s-1 3´10-26cm3s-1 10-26 10-26 12.517.7 25. 35.4 50 70.7 100 141 200 283 400 566 8001130 12.517.7 25. 35.4 50 70.7 100 141 200 283 400 566 8001130 mΧHGeVL mΧHGeVL FIG. 7: (Left) Constraints in the m -hσ vi plane for various frequencies without assuming synchrotron foreground removal. χ A (Right)ConstraintsfromtheWMAP23GHzforegroundmapand23GHzforeground–cleanedresidualmap(theWMAPHaze) for the TT model of magnetic field (filled regions) and for a uniform 10 µG field (dashed lines). course the clumps will have a certain profile peaked in VI. SUMMARY AND CONCLUSIONS the center and will not be perfectly smoothed all over lD. Howeverthe dilutionofthe signalinthemuchlarger Using conservative assumptions for the DM distribu- volume with respect to the region of emission makes it tioninourgalaxywederivetheexpectedsecondaryradi- quite hard to detect the clump. We can consider for ationduetosynchrotronemissionfromhighenergyelec- example the signal from a very bright clump at a dis- tronsproducedinDMannihilation. The signalfromsin- tance of 5 kpc with a flux of 10−22GeVcm−2s−1Hz−1 glebrightclumpsoffersonlypoorsensitivitiesbecauseof at 20 GHz, corresponding approximately to the char- diffusioneffects whichspreadthe electrons overlargear- acteristics of clump #3 in Fig.3. With a dilution over eas diluting the radio signal. The diffuse signalfrom the 1 kpc, the clump emission is seen under a steradian halo and the unresolved clumps is instead relevant and A = πα2 ≈ 0.1 sr with α ≈ 10◦, giving a diffuse clump can be compared to the radio astrophysical background flux of 10−21GeVcm−2s−1Hz−1sr−1. The WMAP sensi- to derive constraints on the DM mass and annihilation tivity of about 10µK translates into a flux sensitivity1 cross section. of ∼10−18GeVcm−2s−1Hz−1sr−1, meaning that the ex- Constraints in the radio band, in particular, are com- pected, optimistic signal is about 3 order of magnitude plementary to similar (less stringent but less model de- below the reach of the current sensitivity. The situa- pendent) constraints in the X-ray/gamma band [55, 56] tion is only slightly better at 150 MHz where the ex- and from neutrinos [57]. Radio data, in particular, are pected LOFAR sensitivity is 50 mK [54] i.e. ∼ 2 × more sensitive in the GeV-TeV region while neutrinos 10−19GeVcm−2s−1Hz−1sr−1. provide more stringent bounds for very high DM masses (> 10 TeV). Gammas, instead, are more constraining ∼ for m <1 GeV. The combination of the various obser- χ ∼ The chance of clump radio detection seems thus quite vations provides thus interesting constraints over a wide poor evenwith the next generationexperiments. Onthe rangeofmassespushingtheallowedwindowsignificantly other side, the fact that the signal is anyway extended near the thermal relic possibility. andnotpointlikemakestheclumpsignalnotreallycom- Moreintodetails,weobtainconservativeconstraintsat plementary to the diffuse component sharing the same the level of hσAvi∼10−23cm3s−1 for a DM mass mχ = systematics with a much fainter signal. It is likely thus 100GeVfromtheWMAPHazeat23GHz. However,de- that the a rolefor DM investigationsin the radio will be pending on the astrophysicaluncertainties, in particular played basically by the diffuse signal. on the assumption on the galactic magnetic field model, constraints as strong as hσ vi ∼ 10−25cm3s−1 can be A achieved. Complementary to other works which employ the WMAP Haze at 23 GHz, we also use the informa- 1 Atradiofrequencies theRayleigh-Jeans lawFν =2ν2/c2kBT is tion in a wide frequency bandin the range100 MHz-100 employedtotranslatefluxesintobrightness temperatures GHz. Adding this information the constraints become

Radio constraints on dark matter annihilation in the galactic halo and its substructures

Similar Radio constraints on dark matter annihilation in the galactic halo and its substructures