Astrophysical Limits on Massive Dark Matter G.Bertone1,2, G.Sigl1, J.Silk2 1. Institut d’Astrophysique, F-75014 Paris, France 1 0 2. Department of Astrophysics, University of Oxford, NAPL 0 Keble Road, Oxford OX13RH, United Kingdom 2 n a J Abstract 9 Annihilations of weakly interacting dark matter particles provide an 1 important signature for the possibility of indirect detection of dark mat- v ter in galaxy halos. These self-annihilations can be greatly enhanced in 4 thevicinityofamassiveblackhole. Weshowthatthemassiveblackhole 3 presentatthecentreofourgalaxyaccretesdarkmatterparticles,creating 1 a region of veryhigh particle density. Consequentlytheannihilation rate 1 is considerably increased, with a large number of e+e− pairs being pro- 0 duced either directly or by successive decays of mesons. We evaluate the 1 0 synchrotron emission (and self-absorption) associated with the propaga- / tionoftheseparticlesthroughthegalacticmagneticfield,andareableto h constrain the allowed values of masses and cross sections of dark matter p particles. - o r t 1 Introduction s a : v There is convincing evidence for the existence of an unseen non-baryonic com- i ponent in the energy-density of the universe. The most promising dark matter X candidates appear to be weakly interacting massive particles (WIMPs) and in r particularthe so-calledneutralinos,arising in supersymmetric scenarios as well a as much heavier particles (WIMPzillas) which could have been produced non- thermally in the early universe (for a review of particle candidates for dark matter see e.g. Ellis (1998)). TheannihilationsoftheseX-particleswouldgeneratequarks,leptons,gauge and Higgs bosons and gluons. Consequently e+e− pairs would be produced ei- ther directly or by successive decays of mesons, and they are expected to lose their energy through synchrotron radiation as they propagate in the galactic magnetic field. This radiation is expected to be greatly enhanced in the prox- imity of the galactic centre, where the existence of a massive black hole creates a region of very high dark matter particle density and consequently a great increase in the annihilation rate and synchrotron radiation. Wediscussinsection2thedistributionofdarkmatterparticlesinourgalaxy andinparticulararoundthecentralblackhole,followingGondolo&Silk(1999) 1 (fromnowon,PaperI).Insection3weevaluatetheannihilatione+e− spectrum and their synchrotron emission. Finally in section 4 we discuss our results and determine the regions of the mass-annihilation cross section plane which give predictions compatible with experimental data. 2 Dark matter distribution around the galactic centre There is strong evidence for the existence of a massive compact object lying withintheinner0.015pcofthegalacticcentre(seeYusef-Zadeh,Melia&Wardle (2000) and references therein). This object is a compelling candidate for a massive black hole, with mass M = (2.6 0.2) 106M⊙. The galactic halo ± × density profile is modified in the neighborood of the galactic centre from the adiabatic process of accretion towards the central black hole. If we consider an initially power-law type profile of index γ, similar to those predicted by high resolution N-body simulations (Navarro, Frenk & White 1997; Ghigna et al. 2000), the corresponding dark matter profile after accretion is, from paper I M 3−γ γsp−γ D γsp ρ′ = α ρ g(r) (1) γ ρ D3 D r " (cid:18) D (cid:19) # (cid:18) (cid:19) whereγ =(9 2γ)/(4 γ),Disthesolardistancefromthegalacticcentreand sp − − ρ =0.24GeV/c2/cm3isthedensityinthesolarneighbourhood. Thefactorsα D γ and g (r) cannot be determined analytically (for approximate expressions and γ numericalvaluesseepaperI).Theexpression(1)isonlyvalidinacentralregion ofsizeR =α D(M/ρ D3)1/(3−γ) wherethecentralblackholedominatesthe sp γ D gravitationalpotential. Ifwetakeintoaccountthe annihilationofdarkmatterparticles,thedensity cannot grow to arbitrarily high values, the maximal density being fixed by the value ρ =m/σvt (2) core BH where t 1010years is the age of the central black hole. The final profile, BH ≈ resulting from the adiabatic accretionof annihilating dark matter on a massive black hole is ρ′(r)ρ core ρ (r)= (3) dm ρ′(r)+ρ core following a power law for large values of r, and with a flat core of density ρ core and dimension ρ(R ) (1/γsp) sp R =R (4) core sp ρ (cid:18) core (cid:19) 2 Figure 1: Values of Y as a function of particle mass. e 3 Constraints from Synchrotron Emission Among the products ofannihilationof darkmatter particles, there will be elec- tronsandpositrons,whichareexpectedtoproducesynchrotronradiationinthe magnetic field around the galactic centre. The e+e− component produced by the hadronic jets has been computed using the MLLA approximation1. The galactic magnetic field can be described using the ’equipartition as- sumption’, where the magnetic, kinetic and gravitational energy of the matter accreting on the central black hole are in approximate equipartition (see Melia (1992)). In this case the magnetic field can be expressed as −5/4 r B(r)=1µG (5) pc (cid:18) (cid:19) Energy-losslengthscale. Asweshallsee,mostoftheannihilationshappen at very small distances from the centre, typically at min(R ,10Rs), i.e. core ≈ in a region with magnetic fields of the order of > 1G. Under these conditions, comparable to the size of the region where most of the annihilations occur, the electrons lose their energy almost in place. To see this, consider the critical synchrotron frequency ν (E) i.e. the frequency around which the synchrotron c 1Fordetails seeDokshitzer etal. (1991), Ellis,Stirling,& Webber (1996), Khoze & Ochs (1997). Forapplicationstoultra-highenergycosmicraysseeBhattacharjee &Sigl(2000). 3 Figure 2: Left panel: A as a function of frequency for m = 1TeV. The two ν X upper curves correspond to the cross section σv 10−28/m2 (GeV)cm3s−1, ≈ X close to the unitarity limit; the two lower curves correspond to σv 10−38/m2 (GeV)cm3s−1, a cross section more typical for wimps. Results fo≈r X two values of the density profile are shown in each case: γ = 1 (solid curves) andγ =1.5(dashedcurves). Rightpanel: A asafunctionoftheparticlemass ν for ν=408MHz, σv =10−10/m2 (in physical units) and two values of γ. X emission of an electron of energy E, in a magnetic field of strength B, peaks, namely 2 3 eB E ν (E)= . (6) c 4πm c m c2 e (cid:18) e (cid:19) Inverting this relation, we find the energy of the electrons which give the maxi- mum contribution at that frequency, 4πm3c5 ν 1/2 ν 1/2 r 5/8 E (ν)= e =0.25 GeV (7) m 3 e B MHz pc (cid:18) (cid:19) (cid:16) (cid:17) (cid:18) (cid:19) The typical synchrotronloss length for an energy corresponding to E (ν) can m be expressed as a function of the distance r from the central black hole for the magnetic field profile in Eq. (5): 27 1/2 m5c11 1/2 1 1 l (E (ν))= e e m 16π e7 ν1/2B3/2 (cid:18) (cid:19) (cid:18) (cid:19) 4 ν −1/2 r 15/8 =1.025 1010 pc (8) · MHz pc (cid:16) (cid:17) (cid:18) (cid:19) Forafrequencyof408MHz,whichproduces,asweshallsee,themoststringent upper bound on the Galactic Centre emission, we get 15/8 r l (E (ν))=5.074 108 pc (9) e m · pc (cid:18) (cid:19) The diffusion length D =D(B(r),E), which can be approximatedas a third of the radius of gyration of the electron, is given by E ν 1/2 r 15/8 D(E (ν))= 8.991 10−8 c pc (10) m 3eB ≈ · MHz pc (cid:16) (cid:17) (cid:18) (cid:19) Since l (E (ν)) >> D(E (ν)) for all practically relevant parameters, an elec- e m m tronwilldiffuse adistance√l D beforeitlosesmostofits energy. Numerically, e √l D r 7/8 e 3.035 101 (11) r ≈ · pc (cid:18) (cid:19) or, expressing the distance as a function of the Schwarzschildradius √l D r 7/8 e 7.059 10−5 (12) r ≈ · R (cid:18) s(cid:19) We can thus assume that the electrons lose their energy practically in place. Electron Production Spectrum. To compute the synchrotronluminosity produced by the propagation of e± in the galactic magnetic field, we need to evaluate their energy distribution in the magnetic field which is given by (see Gondolo (2000)) dn ΓY (>E) e = f (r) (13) e dE P(E) where Γ is the annihilation rate σv ∞ Γ= ρ2 4πr2 dr. (14) m2 sp x Z0 The function f (r) is given by e ρ2 sp f (r)= (15) e ∞ ρ2 4πr2 dr 0 sp and R 2e4B2E2 P(E)= (16) 3m4c7 e 5 Figure 3: ComparisonofSgr A* (see Narayanet al. (1998))observedspectrum withexpectedfluxes. Thevaluesofparticlemassandcrosssectionwerechoosen to fit the experimental data normalisation. is the total synchrotron power spectrum. Note that the general expression for f (r) would have to take into account spatial redistribution by diffusion (see e e.g. Gondolo (2000)), which we demonstrated to be negligible in our model. The quantityY (>E)isthenumberofe+e− withenergyaboveEproduced e per annihilation, which depends on the annihilation modes. Eq. (7) shows that for the frequencies we are interested in, E (ν) << m , and thus the energy m X dependence of Y (> E) can be neglected. We will estimate Y (> E) by the e e number ofchargedparticles producedin quarkfragmentation(see footnote); in figure 1 we show the values of Y as a function of the particle mass. Synchrotron Luminosity. For each electron the total power radiated in the frequency interval between ν and ν+dν is given by √3e3 ν ∞ P(ν,E)= B(r) K (y)dy m c2 ν (E) 5/3 e c Z0 √3e3 ν = B(r)F (17) m c2 ν (E) e (cid:18) c (cid:19) where we introduced ∞ ν ν F = K (y)dy (18) ν (E) ν (E) 5/3 (cid:18) c (cid:19) c Z0 6 Integrating this formula we obtain the total synchrotron luminosity ∞ mx dn L = dr 4πr2 dE eP(ν,E) (19) ν dE Z0 Zme which by substitution becomes √3e3Γ ∞ mx Y (>E) ν L = dr4πr2f (r)B(r) dE e F (20) ν m c2 e P(E) ν (E) e Z0 Zme (cid:18) c (cid:19) It is possible to simplify this formula by introducing the following approxi- mation for the function F ν (see Rybicki & Lightman (1979)) νc(E) (cid:16) (cid:17) ν F δ(ν/ν (E) 0.29). (21) c ν (E) ≈ − (cid:18) c (cid:19) The evaluation of the integral then gives 9 1 m3c5 1/2 ΓY (>E) L e e I (22) ν ≈ 8 0.29π e √ν (cid:18) (cid:19) where ∞ I = dr 4πr2f (r)B−1/2(r) (23) e Z0 Synchrotron Self-Absorption. The synchrotronself-absorptioncoefficent is by definition (see Rybicki & Lightman (1979)) 1 ∞ A = (1 e−τ(b))2πb db (24) ν a − ν Z0 where τ(b) is the optical depth as a function of the cylindrical coordinate b ∞ τ(b)=a f (b,z) dz (25) ν e Zd(b) and the coefficent a is given by ν e3ΓB(r) m d Y (>E) ν a = E2 e F dE (26) ν 9m ν2 dE E2P(E) ν e Zme (cid:20) (cid:21) (cid:18) c(cid:19) The final luminosity is obtained by multiplying Eq. (19) with A given by ν Eq. (24). It is evident that in the limit of small optical depths the coefficent A 1, as can be seen by expanding the exponential. ν → Furthermore the lower limit of integration of expression 25 is d(b)=0 for b2+z2 >(4Rs)2 (27) d(b)= (4Rs)2 b2 elsewhere − p 7 Figure 4: Exclusion plot based on the comparison between predicted flux and radioobservationsofthegalacticcentre.The3solidcurvesindicate,fordifferent values of the density profile power law index, the lower edge of the excluded regions. The dashed line shows, for comparison, the unitarity bound, σv ≃ 1/m2 . The shaded region is the portion of the parameter space occupied by X cosmologically interesting neutralinos (i.e. those leading to 0.025<Ω h2 < 1; X see, e.g. Bergstrom,Ullio & Buckley (1997)). Using the approximationintroducedin Eq.(21) we find fora the following ν expression ΓY c2 a = , (28) ν 4π ν3 which can in turn be used to evaluate τ(b) in Eq. (25) and A in Eq. (24). ν Note that the synchrotronlosstime-scaleis proportionalto B−2 r5/2.We ∝ comparethiswiththeannihilationtime,whichdetermineswheretheinnerden- sitydropsdrastically: t r(9−2γ)/(4−γ)+1/2.Inotherwords,theannihilation ann ∝ time goes to zero more rapidly than the synchrotron loss time as r goes to 0. Hence synchrotronlosses are important throughout the annihilation region. Extensiontothewimpzillamassrangeintroducesfarmoreuncertainphysics. The wimpzilla cross sections strongly depend on new physics beyond the elec- troweakscale. Similarlytothecaseofelectroweakgaugebosons,onecanexpect ageneralscalingσv α2/m2 fortheself-annihilationcrosssectionwhereα<1 ∼ X issomedimensionlessgaugecoupling. Infour dimensionalfieldtheory,the c∼on- servationofprobability(unitarity)roughlycorrespondstothisscalingforα 1 → 8 (see,e.g.,Weinberg(1995)). Inordertoretaingenerality,we willthereforecon- sider Wimpzilla annihilation cross sections in the range σv < 1/m2 . We note, X however, that beyond four dimensional field theory, such a∼s in theories with extra dimensions and in string theory, larger cross sections may be possible. Wefirstevaluatetheself-absorptioncoefficentforselectedvaluesofthemass, asafunctionoffrequency. Theresult,showninfig.2,isacoefficentwhichgrows from very low values (showing that absorption is important at 408 MHz) and thenreachesthevalue1,aroundafrequencywhichisstronglydependentonthe crosssectionandthe massm , but notsomuchonthe profilepowerlaw index X γ. On the left the above coefficent is evaluated for two different values of the crosssection,thefirstonecorrespondingtothemaximumpossiblevalue(the so called unitary bound, see Griest & Kamionkowski (1990)) and the second one correspondingtotypicalcrosssectionsonecanfindinsupersymmetricscenarios (see, e.g., Bergstrom, Ullio & Buckley (1997)). The right part of fig.2 shows the self-absorption coefficent at the fixed fre- quency of 408MHz as a function of the neutralino mass. The behaviour shown is qualitatively the same for any value of the cross-section and for different γ. Infigure3wecomparethepredictedspectrumwiththeobservations;wechoose a set of parameters m , γ and σv in order to reproduce the observed normal- X isation. It is remarkable that in that way one can reproduce the observe d spectrumoverasignificantrangeoffrequencies. The setofdarkmatterparam- etersforwhichthefluxespredictedinourmodelareconsistentwithobservation is shown in the exclusion plot of fig. 4. The boundaries of the excluded range represent the parameter values for which the observed flux is explained by the dark matter scenario discussed here. 4 Conclusions and perspectives We have shown that present data on the emission from Sgr A* are compatible withquiteawidesetofdarkmatterparameters. Theevaluationofsynchrotron self-absorption has enabled us to reach an alternative conclusion from an ear- lier claim of incompatibility of cuspy halos with the existence of annihilating wimp dark matter. Our final result is that the experimental data on Sgr A* spectrum at radio wavelengths could be explained by synchrotron emission of electrons produced in the annihilation of relatively massive dark matter parti- cles, extending from TeV masses to m > 108GeV. The former is relevant to X recentstudies ofcoannihilation(Boehm, Djouadi&Drees (2000)),thatsuggest that WIMPs with Ω h2 = 0.2 can extend up to several TeV, and the latter X is relevant for particles (wimpzillas) that are produced non-thermally in the primordial universe. We have found that with the current data situation, the synchrotron emis- sion tends to give somewhat sharper constraints on masses and cross sections than the gamma-ray fluxes (cf. Baltz et al. (2000)). This situation could be reversed by more sensitive gamma-ray observations anticipated from upcom- ing experiments. However the synchrotronpredictions are uncertainbecause of 9 our relative ignorance about the magnetic field strength near the central black hole, and gamma ray fluxes are subject to similarly uncertain amounts of self- absorption.The ANTARES neutrino experiment will eventually be able to set a relatively model-independent limit on the annihilation flux from the Galactic Centre. 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