Constraining annihilating dark matter mass by the radio continuum spectral data of NGC4214 galaxy
aa r X i v : . [ a s t r o - ph . H E ] S e p Constraining annihilating dark matter mass by the radio continuum spectral data ofNGC4214 galaxy
Man Ho Chan, Chak Man Lee
The Education University of Hong Kong, Tai Po, Hong Kong, China (Dated: September 22, 2020)Recent gamma-ray and radio observations provide stringent constraints for annihilating darkmatter. The current 2 σ lower limits of dark matter mass can be constrained to ∼
100 GeV forthermal relic annihilation cross section. In this article, we use the radio continuum spectral dataof a nearby galaxy NGC4214 and differentiate the thermal contribution, dark matter annihilationcontribution and cosmic-ray contribution. We can get more stringent constraints of dark mattermass and annihilation cross sections. The 5 σ lower limits of thermal relic annihilating dark mattermass obtained are 300 GeV, 220 GeV, 220 GeV, 500 GeV and 600 GeV for e + e − , µ + µ − , τ + τ − , W + W − and b ¯ b channels respectively. These limits challenge the dark matter interpretation of thegamma-ray, positron and antiproton excess in our Milky Way. INTRODUCTION
Recent observations of gamma-ray, positrons and an-tiprotons indicate some excess emissions of these particlesin our Milky Way. These excess emissions could be ex-plained by dark matter (DM) annihilation (i.e. the DMinterpretations). For example, the gamma-ray excess canbe explained by DM annihilating via b ¯ b channel with DMmass m ∼ −
80 GeV [1–3]. For positron excess andantiproton excess, the suggested mass is m ∼ − m ≈ −
94 GeV (via b ¯ b channel) [5] re-spectively. Except for the positron excess interpretation,the mass ranges coincide with each other and the anni-hilation cross sections predicted are close to the thermalrelic annihilation cross section < σv > = 2 . × − cm s − [6].However, recent analyses of gamma-ray observationsgive very stringent constraints for annihilating DM. IfDM particles are thermal relic particles (the simplestmodel in cosmology), the latest Fermi-LAT gamma-rayobservations of Milky Way dwarf spheroidal satellite(MW dSphs) galaxies and two nearby galaxy clusters givethe lower limits of DM mass m ∼
100 GeV for b ¯ b quarkand τ + τ − channels [7–9]. For leptophilic channels like e + e − and µ + µ − , analyses of AMS-02 data [10, 11] andradio data [12–15] also give the lower limits m ∼ − m ∼
300 GeV [16]. These limitsobtained give some tension to the DM interpretations ofthe gamma-ray, positron and antiproton excess. Never-theless, most of these limits are only 2 σ limits and it isstill too early to rule out the possibility of the DM inter-pretations.In this article, we use the radio continuum spectraldata of a nearby galaxy NGC4214 and differentiate thethermal contribution, dark matter annihilation contri-bution and cosmic-ray contribution. We show that the5 σ lower limits of DM mass with thermal relic anni-hilation cross section can be improved to ≥
220 GeVfor leptophilic channels and ≥
500 GeV for two popu- lar non-leptophilic channels. These results provide somechallenges to the DM interpretations of the gamma-ray,positron and antiproton excess in our Milky Way.
THE MODEL
Previous studies show that radio data can give strin-gent constraints for annihilating DM [12–18]. The high-energy electrons and positrons produced from DM anni-hilation emit strong synchrotron radiation S DM (in ra-dio waves) when there is a strong magnetic field. Theseanalyses assume that all radio fluxes emitted S total orig-inate from high-energy electrons and positrons producedfrom DM annihilation (i.e. S total = S DM ). This as-sumption overestimates the contribution of DM annihi-lation because high-energy electrons and positrons canalso be produced by normal astrophysical processes suchas supernovae and pulsars (normal cosmic rays). There-fore, the limits obtained for DM are somewhat underes-timated.If one can differentiate the contributions of radio fluxemitted from a galaxy due to DM annihilation S DM andnormal cosmic rays S CR , the constraints of DM can bemuch more stringent. Furthermore, we can also eliminatethe radio flux due to thermal contribution S th . The elec-tromagnetic emission of the thermal electrons in a galaxywould contribute a small part in the radio flux. This ther-mal contribution part could be calculated by standardthermal physics [19]. Including the thermal contribu-tion can give more stringent limits for DM mass. In thefollowing, we assume S total = S DM + S CR + S th . If themagnetic field of the galaxy is strong enough ( B ≥ µ G)and remains uniform in the outer region, the diffusion ofhigh-energy electrons and positrons would be insignifi-cant so that the radio flux (in mJy) contributed by DMemitted from a galaxy can be simply given by [20, 21] S DM ( ν ) ≈ πνD " √ < σv > m (1 + C ) E ( ν ) Y ( ν, m ) Z ρ DM dV , (1)where ν is the radio frequency, < σv > is the annihila-tion cross section, D is the distance to the galaxy, C isthe correction factor for inverse Compton scattering con-tribution, ρ DM is the DM density profile of the galaxy, E ( ν ) = 14 . ν/ GHz) / ( B/µ G) − / GeV, Y ( ν, m ) = R mE ( ν ) ( dN e /dE ′ ) dE ′ and dN e /dE ′ is the energy spectrumof the electrons or positrons produced from DM annihi-lation (it depends on annihilation channels) [22]. Here,we have used the ‘point-source approximation’ in Eq. (1).For angular region smaller than 1 ◦ , the ‘J-factor’ is ap-proximately equal to J ≈ R ρ DM dV /D [23]. As we willsee below, the angular region of our target galaxy is muchsmaller than 1 ◦ so that using the ‘point-source approxi-mation’ can be justified. Generally speaking, S DM ( ν ) is apower-law of the radio frequency ν ( S DM ( ν ) ∝ ν − α DM ).The spectral index of radio spectrum α DM depends ondifferent annihilation channels. For example, α DM ≈ . e + e − channel while α DM > b ¯ b channel.On the other hand, numerical simulations show thatthe spectral index for GeV cosmic rays is very close toa constant (for ν ∼ GHz) [24]. This is also true for ourGalaxy based on observations [25]. In fact, many galaxiesshow nearly constant spectral index for a wide range offrequencies (e.g. NGC 4449, NGC 891) [26, 27], and evenfor galaxy clusters [28]. We will see later that the spectralindex of our target galaxy is also constant. Therefore, as-suming a constant spectral index for cosmic rays wouldgive a better fit rather than the non-constant spectral in-dex models. Otherwise, fine-tuning of the S DM and S CR is required to give a resultant constant spectral index fora wide range of frequencies. Also, the constant spectralindex model is the simplest model for cosmic-ray emis-sion (only two parameters are involved). Based on theabove arguments and minimizing the involved parame-ters, we can write S CR = S CR, ν − α CR . Our previousstudy using this model gives better constraints of DM inthe Ophiuchus galaxy cluster [29]. For thermal contribu-tion, we can write S th = S th , ν − . [26]. The total radioflux emitted by a galaxy is given by S total ( ν ) = S DM ( ν ) + S CR, ν − α CR + S th , ν − . . (2)The parameter S th , can be obtained from observationaldata. Unfortunately, it is very difficult to predict thetheoretical values of S CR, and α CR for a galaxy. Thesetwo values are free parameters when we apply this modelto fit the observational data. THE RADIO CONTINUUM SPECTRAL DATA
There are some criteria to follow for choosing the besttarget galaxy for analysis. First of all, the galaxy chosenshould have a large uniform magnetic field strength B .It is because a large uniform magnetic field strength cangreatly facilitate the cooling of high-energy electrons andpositrons produced from DM annihilation. Due to thehigh cooling rate, the diffusion of high-energy electronsand positrons would be insignificant and the resultantradio flux contributed by DM would be maximized. Sec-ond, the galaxy should be nearby and rich in DM content.Also, the radio data should have small uncertainties inboth large and small frequency regimes.We have examined some archival galactic radio con-tinuum spectral data and we have found a very goodcandidate - NGC 4214 galaxy - to do the analysis. Theradio continuum data can be found in [26]. The thermalcontribution can be modeled by S th = 20( ν/ . − . mJy [26]. Therefore, we can obtain the non-thermal ra-dio flux data S nth and their uncertainties (see Table 1).The non-thermal radio spectral index is very close to aconstant so that it is very good for analysis. Note thatthe data in [26] have included several observations fromdifferent telescopes. In particular, the data at ν = 1 . . ± . . ± . . ± . ±
25 mJy) and ν = 4 . − .
86 GHz (30 . ± . . ± . . ± . ν = 1 . ν = 4 . − .
86 GHz respectively as 62 . ± . . ± . D = 2 .
94 Mpc [26] andthe average uniform magnetic field strength is B ≈ µ G [30]. The angular size of the galaxy is smaller than0 . ◦ . Note that the radio flux data we considered areintegrated flux which represent the total emissions of thegalaxy. Since we do not have the information about theradio flux profile, we assume all radio signals come froma single halo with size smaller than 0 . ◦ . As the sizeis smaller than 1 ◦ , the ‘point-source approximation’ inEq. (1) is still a very good approximation.Generally speaking, the magnetic field of a galaxy usu-ally trace the matter distribution and an exponentialfunction is commonly assumed to model the magneticfield. However, this assumption requires two extra pa-rameters (the central magnetic field and scale radius)and the functional form also contributes systematic un-certainties. In modeling radio emission of DM annihi-lation, a larger magnetic field would give a larger radioflux (except for the e + e − channel) [13, 14]. Observationsindicate that the central magnetic field of the NGC 4214galaxy can be as high as 30 µ G and the magnetic fieldstrength in the outer region (even for the outskirt region)is close to a uniform strength 8 µ G [30]. Therefore, theradio emission near the center is much larger. However,the actual central magnetic strength and the magneticscale radius are quite uncertain. To avoid extra uncer-tain parameters involved, we adopt a ‘uniform field ap-proximation’ and use B = 8 µ G to model the magneticfield strength. This would underestimate the stronger ra-dio emission due to DM annihilation. Using the constantmagnetic field strength can give conservative limits of m ,except for the e + e − channel. Nevertheless, the effect ofthe assumption for the e + e − channel is not very large(the limit of m is larger by less than 40%).Note that the magnetic field strength of NGC 4214 issomewhat higher than that in normal dwarf galaxies (e.g.Local Group dwarf galaxies: B = 4 . ± . µ G [31]).Nevertheless, study in [31] point out that some higherstar-formation rate and starburst galaxies may have veryhigh average magnetic field strength. For example, themagnetic field strengths of the NGC 2976 galaxy, NGC1569 galaxy and NGC 4449 galaxy are B = 6 . ± . µ G, B = 14 ± µ G and B = 9 ± µ G respectively [31,32]. Since the star-formation rate of NGC 4214 galaxy(SFR= 0 . M ⊙ /yr) is close to that of NGC 1569 (SFR=0 . M ⊙ /yr) [33], the large magnetic field strength of theNGC 4214 galaxy ( B ≈ µ G) is not unexpected.The effect of the inverse Compton scattering is alsovery important. Since the cooling rate of the inverseCompton scattering also depends on E , this effect can besimply characterized by a correction factor C in Eq. (1).Assuming a conservative optical-infrared radiation en-ergy density ω opt = 0 . for NGC 4214 (same asour Galaxy) [34], the energy density for inverse Comp-ton scattering is about 0.75 eV/cm , which correspondsto C ≈ .
49. The synchrotron and the inverse Comptonscattering are the dominated cooling processes.As mentioned above, since the cooling rate is very high,the diffusion is not important in the NGC 4214 galaxy.This can be examined with the diffusion scale length [35] λ ∼ .
79 kpc (cid:18) D cm (cid:19) / (cid:16) ω − (cid:17) − / (cid:18) E (cid:19) − / , (3)which represents the approximated length traveled by anelectron with initial energy E . Here, D is the diffusioncoefficient and ω is the total radiation energy density.For B = 8 µ G and C = 0 .
49, we have ω = 2 . − .The value of the diffusion coefficient is scale-dependent.The standard diffusion coefficient used in our Milky Waygalaxy is D = 3 . × cm s − [36]. However, fora smaller NGC4214 galaxy, the smallest scale on whichthe magnetic field is homogeneous is somewhat smaller.The diffusion coefficient for a dwarf galaxy is of the order10 cm s − [37]. For the injection spectrum, most ofthe electrons and positrons having E ∼ −
100 GeV for m ≥
100 GeV, which correspond to λ ∼ . − E = 1 −
100 GeV would losemost of its energy by traveling a distance of 0 . − v and the rotational ve-locity contributed by baryonic matter v b . By subtract-ing the baryonic matter contribution and assuming astandard value of mass-to-luminosity ratio for galaxiesΥ = 0 . M ⊙ /L ⊙ [38], we can obtain the rotational veloc-ity contributed by DM v DM = v − v b . The DM densitycan be calculated by ρ DM = (4 πr ) − ( d/dr )( rv DM /G ).In Fig. 1, we can see that the resulting DM density canbe well-fitted by a power-law form ρ DM ∝ r − . for r > .
67 kpc. For the central density within r ≤ .
67 kpc,the uncertainties are quite large and we assume a con-stant density profile which can give a conservative pre-diction of DM annihilation signal. Therefore, we modelthe DM density as ρ DM = ρ r ≤ .
67 kpc ρ (cid:16) r .
67 kpc (cid:17) − . ± . .
67 kpc < r ≤ .
63 kpc , (4)where ρ = (3 . ± . × − g cm − . The uncertaintyof fitting is very small. Here, we do not assume anyparticular forms of DM density profile (e.g. Navarro-Frenk-White profile or Burkert profile). The DM densityprofile is just directly probed from the observed rotationcurve data. The systematic uncertainties involved wouldbe smaller than assuming any particular forms of DMdensity profile.Simulations show that the DM annihilation signalwould be enhanced due to the substructure contributions.These contributions can be quantified by considering theboost factor B f , which can be modeled by the followingempirical expression [39]:log B f = X i =0 b i (cid:18) log MM ⊙ (cid:19) i , (5)where M is the virial mass of the structure and b i isthe fitted coefficients [39]. Following the DM profile inEq. (4), the virial mass is M = 9 . × M ⊙ . Using themost conservative model in [39], the corresponding boostfactor is B f = 4 . < σv > = 2 . × − cm s − [6]. Therefore, we canget S DM ( ν ) for different annihilation channels and dif-ferent DM mass m . For each annihilation channel and m , we can fit the predicted S DM ( ν ) + S CR ( ν ) with thenon-thermal radio continuum spectral data of NGC4214 S nth obtained in [26]. We minimize the reduced χ value( χ ) by changing the values of two free parameters, S CR, and α CR .In Table 2, we present the corresponding χ valuesfor some DM mass and annihilation channels. The 5 σ lower limits of m are 300 GeV, 220 GeV, 220 GeV, 500GeV and 600 GeV for e + e − , µ + µ − , τ + τ − , W + W − and b ¯ b channels respectively, which are determined by the re-lation between χ and m in Fig. 2. We also plot thespectra for m just ruled out at 5 σ and just satisfied the2 σ lower limits respectively in Fig. 3. The correspondingcomponents of the thermal contribution, DM contribu-tion and the cosmic-ray contribution are shown in Fig. 4.In particular, we notice that for the e + e − , µ + µ − and τ + τ − channels, the best-fit scenarios do not have thecontributions of cosmic rays ( S CR = 0, see Fig. 4 andTable 2). It means that the spectral index of dark mat-ter annihilation for these three channels are already veryclose to the observed non-thermal radio spectrum so thatno cosmic-ray component is needed to give better fits. Inother words, dark matter contribution alone plus ther-mal component is sufficient to give the best-fit spectrafor these three channels.In fact, this is the first time that we can rule out m ≤
220 GeV at 5 σ for thermal relic annihilating DM.Generally speaking, the χ will decrease further andfinally approach to a constant if we increase the valueof m (see Fig. 2). It is because the non-thermal radiocontinuum spectrum of NGC 4214 is very close to a con-stant spectral index α nth = − . ± .
04 [26]. Increasingthe value of m would suppress the contribution of S DM so that S CR ≈ S nth . Therefore, we can only obtain thelower limits of m using this method.In the above analysis, we take the value of the ther-mal relic annihilation cross section < σv > = 2 . × − cm s − . Nevertheless, DM particles may not be thermalrelic particles and the annihilation cross section may belarger or smaller than the thermal relic annihilation crosssection. If we release the annihilation cross section as afree parameter, we can obtain its upper limit for eachannihilation channel. However, the values of the annihi-lation cross section and the two free parameters, S CR, and α CR , are quite degenerate for a particular value of m .Therefore, we fix the values of S CR and α CR as their con-vergent limits (the best-fit values when m is very large)and obtain the 5 σ upper limits of the annihilation crosssection as a function of m (see Fig. 5). We also show the2 σ upper limits of the annihilation cross section obtainedby the Fermi-LAT gamma-ray observations of the MWdSphs galaxies [7, 8] in Fig. 5. We can see that our 5 σ up-per limits are tighter than the 2 σ Fermi-LAT gamma-raylimits.We also examine the lower limits of DM if there isno cosmic-ray contribution. Generally speaking, if the cosmic-ray contribution is zero ( S CR = 0), the lowerlimits of m would be smaller because the DM contri-bution has to be larger to account for the radio spec-trum (smaller m gives larger S DM ). Therefore, setting S CR = 0 would give the most conservative lower limits of m . However, the thermal component S th is determinedby the H α emission measurement [26], which is indepen-dent of the radio observations. Therefore, the thermalcomponent cannot be set to zero arbitrarily. In Fig. 6, weshow the χ values for the 5 channels without cosmic-ray contributions. We can see that the 5 σ DM massranges for the e + e − , µ + µ − and τ + τ − channels are 300-540 GeV (best-fit: 360 GeV), 220-400 GeV (best-fit: 280GeV) and 220-430 GeV (best-fit: 280 GeV) respectively.For the b ¯ b and W + W − channels, the 5 σ ranges of m are 600-1050 GeV (best-fit: 800 GeV) and 490-830 GeV(best-fit: 600 GeV) respectively. The resulting 5 σ lowerlimits of m are nearly the same as the results includ-ing cosmic-ray contributions (compare with the resultsin Table 2). That means the dark matter contribution isstill dominating at these lower limits of m . However, thebest-fit χ for the b ¯ b and W + W − channels are largerthan 4, which are excluded at more than 3 . σ . There-fore, excluding the cosmic-ray contribution gives poorerfits (larger χ values) for these two channels and bet-ter fits will be obtained if cosmic-ray contributions areincluded (compare the χ in Table 2).Note that the χ values in Fig. 2 approach to a smallconstant value while the χ increase with m in Fig. 6 inthe large DM mass regime. It is because the calculationsof χ in Fig. 2 have included the cosmic-ray compo-nent. The cosmic-ray component would dominate thecontribution in the large DM mass regime and make the χ values small. The best-fit χ value is about 1.5without DM contribution (only cosmic-ray and thermalcontributions), which is a very good fit indeed. In otherwords, the cosmic-ray and thermal contributions alonecan give a very good explanation for the radio contin-uum data of the NGC 4214. A large contribution of theDM component (when m is sufficiently small) would givea large value of χ . That’s why we can obtain the lowerlimits of m by using the radio continuum data of NGC4214. DISCUSSION
In this article, we use the radio continuum spectrumof a galaxy (NGC 4214) to obtain the lower limits of DMmass m for five popular annihilation channels. Usingradio data is a very good option for constraining DM be-cause current radio telescopes can give observations withvery high resolution and sensitivity. For the NGC 4214radio data we used, the radio beam size and flux densitylevel detected can be as small as 5” and 1 mJy respec-tively [26]. Therefore, the radio data obtained may be r (kpc) D a r k m a tt e r d e n s it y ( g c m - ) FIG. 1. The DM density profile probed from the rotationcurve data in [38]. The red solid line is the density model inEq. (4) and the red dotted lines indicate the 1 σ uncertainty.
200 400 600 800 1000 1200 m (GeV) R e du ce d χ v a l u e e channel µ channel τ channelb channelW channel σ limit3 σ limit4 σ limit5 σ limit FIG. 2. The relation between the reduced χ values and theDM mass m for various channels. more effective in constraining annihilating DM than theMilky Way gamma-ray or positron data used in previousstudies. Furthermore, we have differentiated the contri-butions of the thermal emissions, DM annihilation andthe normal cosmic rays so that we can obtain a betterlower limit of m for each of the annihilation channels. Infact, many recent studies of gamma rays and positronshave included the astrophysical background components[40–42]. Nevertheless, using appropriate radio contin- S n t h ( m J y ) e channel (m = 360 GeV)e channel (m = 300 GeV) 0.1 1 101101001000 τ channel (m = 280 GeV) τ channel (m = 220 GeV)0.1 1 10 ν (GHz) S n t h ( m J y ) b channel (m = 1000 GeV)b channel (m = 600 GeV) 0.1 1 10 ν (GHz) FIG. 3. The best-fit spectra of S nth for m just ruled out at5 σ (dotted lines) and just satisfied the 2 σ lower limits (solidlines). The corresponding best-fit parameters and the reduced χ values are shown in Table 2.TABLE I. The radio continuum spectral data of NGC 4214[26]. The four data for ν = 1 . ν =4 .
855 GHz shown in [26] have been combined correspondinglyto allow for the largest possible uncertainties. ν (GHz) S total (mJy) S nth (mJy) Uncertainties (mJy)0.15 104.1 84.9 15.60.325 192.5 174.7 43.90.61 74.6 57.9 11.21.4 62.8 47.4 32.21.6 65 49.8 252.38 36 21.4 34.855 31.9 18.3 8.98.46 24.2 11.4 4.8 uum spectral data with the consideration of the back-ground cosmic-ray and thermal components seem to getbetter constraints. The limits obtained are the currentmost stringent radio limits for thermal relic annihilat-ing DM, which challenge the DM interpretations of thegamma-ray excess [1–3] and antiproton excess [5]. Forthe positron excess, it requires a much larger annihila-tion cross section ( ≥ − cm s − ) [4]. Our resultsalso rule out the proposed DM interpretation if we as-sume < σv > ≥ − cm s − (see Fig. 5). If we do notconsider the boost factor, the lower limits of m wouldapproximately decrease by a factor of 2.3. Therefore, theminimum 5 σ limits of m is still larger than 90 GeV for allpopular channels. This still challenges the DM interpre-tations of the positron and gamma-ray excess. In fact, S t o t a l ( m J y ) ν (GHz) S t o t a l ( m J y ) ν (GHz) e channel (m = 360 GeV) τ channel (m = 280 GeV)b channel (m = 1000 GeV) W channel (m = 800 GeV) FIG. 4. The best-fit spectra of S total and the correspondingcomponents (for m just satisfied the 2 σ lower limits). Theblack lines, red lines, green lines and blue lines indicate thetotal radio flux S total , thermal radio flux contribution S th ,DM flux contribution S DM and cosmic-ray flux contribution S CR respectively. The corresponding best-fit parameters areshown in Table 2.TABLE II. The best fit parameters for some DM mass m andannihilation channels.Channel m (GeV) χ S CR, (mJy) α CR Remark300 8.53 0 0 Ruled out at 5 σe + e −
330 3.45 0 0 Ruled out at 3 σ
360 1.71 0 0 Within 2 σ range220 7.56 0 0 Ruled out at 5 σµ + µ −
240 3.37 0 0 Ruled out at 3 σ
260 1.77 0 0 Within 2 σ range220 9.02 0 0 Ruled out at 5 στ + τ −
260 2.45 0 0 Ruled out at 2 σ
280 1.73 0 0 Within 2 σ range400 14.9 0 0 Ruled out at 5 σW + W −
600 3.11 9 0 Ruled out at 2 σ
800 1.80 16 0.19 Within 2 σ range600 8.10 4 0 Ruled out at 5 σb ¯ b
800 2.66 9 0 Ruled out at 3 σ σ range the DM interpretations of the gamma-ray and positronexcess are controversial. The ranges of DM mass andannihilation cross sections predicted are close to our ex-pected values while some other studies point out that theexcess emissions might originate from pulsars or molec-ular clouds [43–45]. Our results may provide some hintsfor settling this controversy.
100 1000 m (GeV) < σ v > ( c m s - ) e channel µ channel τ channelb channelW channel Cholis et al. (2019)Abazajian & Keeley (2016)Daylan et al. (2016)Calore et al. (2015) Boudaud et al. (2015)
FIG. 5. The colored solid lines are the 5 σ upper limits ofthe annihilation cross section in our analysis. The coloreddotted lines are the 2 σ upper limits of the annihilation crosssection obtained by the Fermi-LAT gamma-ray observationsof the MW dSphs galaxies [7, 8]. The orange dashed lineindicates the thermal relic annihilation cross section < σv > =2 . × − cm s − [6]. The data with error bars shown arethe predicted ranges of m and < σv > based on the darkmatter interpretations of gamma-ray excess [1–3], positronexcess [4] and antiproton excess [5]. The blue and red colorsof the data points correspond to the ranges for the b ¯ b and µ + µ − channels respectively.
200 400 600 800 1000 1200 m (GeV) R e du ce d χ v a l u e e channel µ channel τ channelb channelW channel FIG. 6. The relation between the reduced χ values and theDM mass m for various channels without cosmic-ray contri-butions. Note that the above results are solely based on thedata of a single galaxy. In fact, the diffusion processesand cosmic-ray emissions in a small galaxy are not verywell known. For instance, if the diffusion length of thehigh-energy electrons and positrons is much longer thanour expected, the radio emission due to the DM contri-bution would be suppressed and the resulting lower lim-its of DM mass would be smaller. Therefore, our resultsmay be affected by the systematic uncertainties involved.More observations and analysis using a larger sample ofgalaxies are definitely required to examine and verify ourclaims.The advantage of using the radio continuum spectraldata is that the spectral index is close to a constant.This is true for many galaxies and galaxy clusters [25–28]. Therefore, this method can be applied in many goodtargets (nearby DM-rich galaxies) to constrain DM. Moreradio continuum observations for these galaxies are defi-nitely helpful. This method can also be applied in ana-lyzing galaxy clusters [29, 46]. Using appropriate targetobjects, the 5 σ limits of DM mass could be improved to500 − ∼ m ∼ − σ constraints to ∼ − ACKNOWLEDGEMENTS
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