Contribution of dark matter annihilation to gamma-ray burst afterglows near massive galaxy centers
Bao-Quan Huang, Tong Liu, Feng Huang, Da-Bin Lin, Bing Zhang
aa r X i v : . [ a s t r o - ph . H E ] S e p Draft version September 22, 2020
Typeset using L A TEX twocolumn style in AASTeX63
Contribution of dark matter annihilation to gamma-ray burst afterglows near massive galaxy centers
Bao-Quan Huang, Tong Liu, Feng Huang, Da-Bin Lin, and Bing Zhang Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, China Laboratory for Relativistic Astrophysics and Department of Physics, Guangxi University, Nanning 530004, China Department of Physics and Astronomy, University of Nevada, Las Vegas, Las Vegas, NV 89154, USA
ABSTRACTGamma-ray bursts (GRBs) are believed to be powered by ultrarelativistic jets. If these jets encounterand accelerate excess electrons and positrons produced by particle dark matter (DM) annihilation, theobserved electromagnetic radiation would be enhanced. In this paper, we study GRB afterglow emis-sion with the presence of abundant DM under the weakly interacting massive particle annihilationconditions. We calculate the light curves and spectra of the GRB afterglows with different parame-ters, i.e., DM density, particle DM mass, annihilation channel, and electron density of the interstellarmedium. We find that the effect of DM may become noticeable in the afterglow spectra if the circum-burst has a low electron number density ( n . . − ) and if the DM has a high number density( ρ χ & GeV cm − ). According to the standard galaxy DM density profile, GRB afterglows withDM contribution might occur at distances of several to tens of parsecs from the centers of massivegalaxies. Keywords: dark matter - galaxies: general - gamma-ray burst: general - shock waves - relativisticprocesses INTRODUCTIONThe existence of dark matter (DM) is strongly sup-ported by convincing evidence in cosmology and as-trophysics (e.g., Bertone et al. 2005). Numerous as-tronomical processes and evolutions are partly or pre-dominantly affected by DM. However, the nature ofDM remains a mystery, and its direct detection has yetto be achieved. The mainstream DM model invokesDM particles, the most promising candidate being theweakly interacting massive particles (WIMP, see e.g.,Jungman et al. 1996; Bertone et al. 2005; Bergstr¨om2012). The WIMP model can be tested by detectingpotential signals from WIMP annihilations, which canultimately produce Standard Model particles, such asneutrinos, photons, electrons, and positrons. Search-ing for gamma-ray emissions produced by DM annihi-lation is one method to indirectly detect DM; for thistask, nearby galaxy centers and dwarf spheroidal galax-ies are appealing targets (e.g., Ackermann et al. 2015;Abdallah et al. 2018; Johnson et al. 2019). Moreover,
Corresponding author: Tong [email protected] the radio detection of synchrotron radiation induced byelectrons and positrons produced by DM annihilationin galaxies is considered a promising method to con-strain the particle nature of DM (e.g., Storm et al. 2013;Egorov & Pierpaoli 2013). However, such emissions aregenerally weak, and possible detections are limited togalaxies in the local universe.If significant signals can be powered by electrons pro-duced by DM annihilation, an accelerator should be es-tablished in a high-density DM halo. In addition, thenumber density of system-provided electrons should becomparable to that of DM electrons (DMEs). The jetsgenerated from gamma-ray bursts (GRBs) near galaxycenters are ideal accelerators.GRBs are the most luminous explosions in the uni-verse. The ultrarelativistic jets in the line of sightlaunched by newly born magnetars or hyperaccretingblack holes can trigger observable GRBs (see the re-views by M´esz´aros 2006; Zhang 2007, 2018; Liu et al.2017). The prompt gamma-ray emissions and multi-band afterglows of GRBs generally originate fromthe internal and external shock phases, respectively(e.g., M´esz´aros & Rees 1993; Rees & M´esz´aros 1994;M´esz´aros & Rees 1997; Sari et al. 1998; Piran 2004).Most GRB afterglows can be explained by synchrotron
Huang et al. and synchrotron self-Compton (SSC) processes underthe conditions of external shocks sweeping the interstel-lar medium (ISM, or interstellar wind) and acceleratingelectrons. This explanation has been thoroughly veri-fied by multi-wavelength observations of GRB 190114C(MAGIC Collaboration et al. 2019; Wang et al. 2019;Fraija et al. 2019). In the prompt emission phase, thenumber of DMEs is much less than that of electronsemitted from the central engine, so the electromagneticradiation of DMEs might be observable only in the af-terglow phase.In this paper, the lightest supersymmetric particle inthe WIMP model, namely, the neutralino, is chosen;this particle has four different annihilation channels, i.e., w + w − , b ¯ b , µ + µ − , and τ + τ − . We then study the accel-eration and radiation mechanisms of DMEs in externalshocks and subsequently predict and analyze the DMcontributions to the light curves and spectra of GRB af-terglows. In Section 2, we provide a detailed descriptionof the method employed to calculate the light curvesand spectra of GRB afterglows, including the effects ofDM. In Section 3, the results using different DM param-eters are shown. In Section 4, we analyze the locationsof GRBs in massive galaxies associated with the effectsof DM. The conclusions and discussion are presented inSection 5. METHOD2.1.
Dynamical evolution of the external forward shock
As the GRB ejecta launched from the central en-gine interact with the circumburst medium, a relativis-tic shock is generated which propagates through themedium. We adopt an approximate dynamical evolutionmodel to discuss the evolution of the external forwardshock (Huang et al. 1999, 2000) d Γ dm = − Γ − M ej + ǫm + 2(1 − ǫ )Γ m , (1)where Γ, M ej , m , ǫ are the Lorentz factor, the ejectamass, the swept mass from the external medium, and theradiation efficiency of the external shock, respectively.The problem can be solved by introducing two moredifferential equations dmdR = 2 πR (1 − cos θ j ) nm p (2)and dRdt = βc Γ(Γ + p Γ − , (3)where θ j is the half opening angle of the ejecta, n isthe number density of the circumburst medium, t is thetime measured in the observer frame, m p is the proton mass, and β = p − / Γ is the ejecta velocity. Gener-ally, the circumburst medium can be classified into twocases: an interstellar medium (ISM) and a stellar wind.The mass density of the circumburst medium is a con-stant in the former case and declines with ∼ R − in thelatter case. In this work, we consider only the ISM case.We also neglect the evolution of θ j since numerical simu-lations showed that sideways expansion is not important(Zhang & MacFadyen 2009).2.2. Electron distribution
With the contribution of DMEs considered, the evolu-tion of the shock-accelerated electrons can be expressedas a function of the radius R , i.e., ∂∂R (cid:18) dN ′ e dγ ′ e (cid:19) + ∂∂γ ′ e (cid:18) ˙ γ ′ e dt ′ dR dN ′ e dγ ′ e (cid:19) = ˆ Q ′ ISM + ˆ Q ′ DM , (4)where dN ′ e /dγ ′ e is the instantaneous electron energyspectrum, γ ′ e is the Lorentz factor of the shock-accelerated electrons in the comoving frame of the shock, dt ′ /dR = 1 / Γ c (with t ′ being the time in the shock co-moving frame and Γ being the Lorentz factor of the ex-ternal forward shock), ˙ γ ′ e is the cooling rate of electronswith the Lorentz factor γ ′ e , and ˆ Q ′ ISM dR and ˆ Q ′ DM dR represent the injection of electrons from the ISM andDM into the shock during its propagation from R to R + dR .Here, ˆ Q ′ ISM = ¯ Kγ ′− p e with ¯ K ≈ π ( p − R nγ ′ p − , min is adopted to describe the injection behavior of newlyshocked circumburst medium electrons (CMEs) in theISM, where p ( >
2) is the power law index, n is the num-ber density of the circumburst medium, and γ ′ e , min ≤ γ ′ e ≤ γ ′ e , max is adopted for γ ′ e .The shocked electrons and the magnetic fields sharethe fractions ǫ e and ǫ B of the thermal energy densityin the forward shock downstream. Since DMEs are in-volved in our work, the minimum Lorentz factor of theshock-accelerated electrons can be expressed as γ ′ e , min = ǫ e η (Γ − p − m p / ( p − m e , where η ( ≤
1) is the ra-tio of the number density of CMEs to the number den-sity of all electrons, including both CMEs and DMEs,and m e denotes the electron masses. The maximumLorentz factor of electrons is γ ′ e , max = p m c / B ′ e ,with B ′ = p π Γ(Γ − nm p ǫ B c being the magneticfield behind the shock (e.g., Kumar et al. 2012).The evolution of the DME fluid is described by the fol-lowing diffusion-loss equation (neglecting re-accelerationand advection effects, see e.g., Colafrancesco et al. 2006;Borriello et al. 2009), i.e., ∂∂t dn e ,χ dγ e ,χ = ~ ∇ · (cid:20) K ( γ e ,χ , ~r ) ~ ∇ dn e ,χ dγ e ,χ (cid:21) + ∂∂γ e ,χ (cid:20) b ( γ e ,χ , ~r ) dn e ,χ dγ e ,χ (cid:21) + Q ( γ e ,χ , ~r ) , (5) ark matter annihilation in gamma-ray burst afterglows dn e ,χ /dγ e ,χ is the DME equilibrium spectrum, K ( γ e ,χ , ~r ) is the diffusion coefficient, and b ( γ e ,χ , ~r )stands for the energy loss rate. Since GRBs are stellar-scale events, we reasonably assume that the values ofthe DM densities in these regions are constants. Thenthe gradients of the DME densities should be consid-ered as 0, so the first term on the r.h.s of the aboveequation can be neglected. Thus, the steady-state en-ergy spectrum for the DMEs in the interstellar mediumcan be expressed as (e.g., Colafrancesco et al. 2006;Borriello et al. 2009) dn e ,χ ( γ e ,χ ) dγ e ,χ = 1 b ( γ e ,χ , ~r ) Z γ e ,χ, max γ e ,χ Q ( ζ, ~r ) dζ, (6)where γ e ,χ, max = M χ /m e is the maximum Lorentz factorof the DMEs, M χ is the DM mass, and Q ( γ e ,χ , ~r ) is thesource item, which can be expressed as Q ( γ e ,χ , r ) = 12 (cid:18) ρ χ ( r ) M χ (cid:19) h σ A υ i dN inj dγ e ,χ , (7)where ρ χ ( r ) is the DM density profile, h σ A υ i is the an-nihilation cross section that has a typical value (3 . × − cm s − is used in this work), and dN inj /dγ e ,χ isthe DME injection spectrum, which can be obtainedby the Dark SUSY package (e.g., Gondolo et al. 2004;Bringmann et al. 2018). In addition, the energy lossterm, b ( γ e ,χ , ~r ), involving the energy loss of synchrotronradiation and the inverse Compton scattering of the cos-mic microwave background (CMB) and starlight pho-tons, which are the faster energy loss processes for driv-ing the equilibrium of DMEs, can be expressed by b ( γ e ,χ , ~r ) = b IC ( γ e ,χ ) + b Synch . ( γ e ,χ , ~r )= b u CMB E ,χ + b u SL E ,χ + b . B ( r ) E ,χ , (8)where the coefficients have the values b . = 0 . b = 0 .
76 in units of 10 − GeV − (e.g.,Colafrancesco et al. 2006; McDaniel et al. 2018), thephoton energy densities are u SL = 8 eV / cm forstarlight and u CMB = 0 .
25 eV / cm for CMB photons(e.g., Porter et al. 2008; Profumo & Ullio 2010), E e ,χ = γ e ,χ m e c is the energy of the DMEs in units of GeV,and B ISM ( r ) is the strength of the interstellar magneticfield, which is assumed to be a constant, ∼ µ G.When the external forward shock encounters the cir-cumburst medium, the DMEs are accelerated. Here,a simple scenario of Fermi-type shock acceleration isapplied in which the energy of the electrons can in-crease by a factor on the order of Γ during the firstshock crossing and increase by a factor of ¯ g ∼ . δ = ǫ e ηm p ( p − − /m e ( p − P ret ( ∼ .
4) to return to the upstream regionand that the probability of returning upstream is unity(e.g., Lemoine & Pelletier 2003). Hence, the Lorentzfactors and number density spectra of the escapingDMEs after the N th cycle are γ ′ N e ,χ = ¯ g N Γ( γ e ,χ + δ )and f N esc ( γ ′ N e ,χ ) = ¯ P N ret ¯ g − N (1 − ¯ P ret ) dn e ,χ ( γ e ,χ ) /k Γ dγ e ,χ ,respectively. Here, k = ( γ e ,χ + δ ) /γ e ,χ . By accumulat-ing the number density spectra of the accelerated DMEs, dn ′ e ,χ ( γ ′ e ,χ ) /dγ ′ e ,χ = P N = ∞ N =0 f N esc ( γ ′ e ,χ ) is obtained. Thefact that this method can obtain the initial spectrumof shock-accelerated CMEs that is generally assumed tohave a power-law form illustrates that simplifying theacceleration of the DMEs in this way is feasible.It should be noted that when the number density ofDMEs is much larger than that of CMEs, the value of η is much less than unity, resulting in γ ′ e , min ≪ Γ, whichconflicts with the increasing energy of DMEs during thefirst shock crossing. Thus, the shock-induced accelera-tion of DMEs in this case is inefficient. In other words,not all DMEs are efficiently accelerated. Considering thecontributions of nonaccelerated DMEs simultaneously,ˆ Q ′ DM ( γ ′ e ,χ , R ) can be derived asˆ Q ′ DM ( γ ′ e ,χ , R ) = 4 πR (cid:20) A dn ′ e ,χ ( γ ′ e ,χ ) dγ ′ e ,χ +(1 − A ) dn e ,χ ( γ e ,χ )Γ dγ e ,χ (cid:21) , (9)where A is set as 1 when δ &
1; otherwise, the valueof A equals the ratio of the number density of shock-accelerated DMEs to the number density of all DMEs.Here, δ ≃ Afterglow radiation
The synchrotron radiation power at the frequency ν ′ can be obtained by (Rybicki & Lightman 1979) P ′ syn ( ν ′ ) = √ e B ′ m e c Z γ ′ e , max γ ′ e , min (cid:18) dN ′ e dγ ′ e (cid:19) F (cid:18) ν ′ ν ′ c (cid:19) dγ ′ e , (10) Huang et al. -13 -11 -9 -7 -5 -3 t [s] n F n [ e r g c m - s - ] (a) r c = GeV cm -3 r c = GeV cm -3 R band0.3-10 KeV (*10)0.1-1 GeV 0.3-1 TeV (*100) -1 -14 -12 -10 -8 -6 n F n [ e r g c m - s - ] hv [eV] (b) r c = GeV cm -3 r c = GeV cm -3 t = s t = 10 s t = 10 s Figure 1. (a) Light curves of GRB afterglows including and excluding the effects of DM with n = 0 . − . DM annihilationoccurs in the b ¯ b channel, and the DM particle mass is M χ = 10 GeV. (b) Synchrotron + SSC spectra in the same situations of(a). where ν ′ c = 3 eB ′ γ ′ / πm e c and F ( ν ′ /ν ′ c ) =( ν ′ /ν ′ c ) R + ∞ ν ′ /ν ′ c K / ( x ) dx , where K / ( x ) is the modifiedBessel function of order 5/3. The number density of SSCseed photons can be written as (e.g., Fan et al. 2008) n ′ ph ( ν ′ ) ≈ πR chν ′ P ′ syn ( ν ′ ) , (11)and thus, the SSC power at the frequency ν ′ ic is (e.g.,Blumenthal & Gould 1970) P ′ SSC ( ν ′ ic ) = 3 σ T chν ′ ic Z ν ′ max ν ′ min n ′ ph ( ν ′ ) dν ′ ν ′ × Z γ ′ e , max γ ′ e , min y ( q, g ) γ ′ dN ′ e dγ ′ e dγ ′ e , (12)where y ( q, g ) = 2 q ln q + (1 + 2 q )(1 − q ) + 8 q g (1 − q ) / (1 + 4 qg ), q = w/ g (1 − w ), g = γ ′ e hν ′ /m e c , and w = hν ′ ic /γ ′ e m e c . The observed spectral flux is (e.g.,Granot et al. 1999) F ν obs = 1 + z πD ZZ (EATS) P ′ ( ν ′ ) D d Ω , (13)where “EATS” is the equal-arrival time surface corre-sponding to the same observer time, ν ′ = (1 + z ) ν obs /D (with D being the Doppler factor of the emitter), D L is the luminosity distance in the standard ΛCDM cos-mology model (Ω M = 0 .
27, Ω Λ = 0 .
73, and H =71 km s − Mpc − ), and z is the redshift of the burst. RESULTSFollowing the above method, we can calculate the lightcurves and spectra of GRB afterglows, including the effects of DM. The universal parameters of the exter-nal shock model are set as the isotropic kinetic energy E k , iso = 10 ergs, ǫ e = 10 − , ǫ B = 10 − , Γ = 200, p = 2 . θ j = 0 .
1, and z = 1.Figure 1(a) shows the light curves of GRB after-glows including and excluding the effects of DM with n = 0 . − . DM annihilation occurs in the b ¯ b channel,and the DM particle mass is M χ = 10 GeV. The solidand dashed lines represent the light curves and spectrawith ρ χ = 5 . × GeV cm − and 0, respectively. Themagenta and blue lines correspond to the synchrotronemission in the optical R band and 0 . −
10 KeV, re-spectively. The red lines correspond to the synchrotron+ SSC emissions in the range of 0 . − . − . −
10 KeVand 0 . − s, 10 sand 10 s, respectively. Once the effects of DM are con-sidered, in addition to fluence increasing, the shapes ofthe spectra also change.Figure 2 shows the synchrotron + SSC spectra of GRBafterglows with different parameters, i.e., DM density,particle DM mass, annihilation channel, and CME den-sity. Figure 2(a) displays the synchrotron + SSC spec-tra of GRB afterglows with different n at t = 10 safter the GRB is triggered. The DM density is set to5 × GeV cm − or 0 GeV cm − , and the DM par-ticle mass is taken as 10 GeV. The red, blue, green, ark matter annihilation in gamma-ray burst afterglows -2 -14 -13 -12 -11 -10 -9 b channel n= 1 cm -3 n= 10 -1 cm -3 n= 10 -2 cm -3 n= 10 -3 cm -3 M c = 10 GeV t = 10 sr c = 5·10 GeV cm -3 r c = 0 GeV cm -3 n F n [ e r g c m - s - ] hv [eV] (a) -2 -13 -12 -11 -10 -9 bb w + w - t + t - m + m - n F n [ e r g c m - s - ] hv [eV] r c = 5·10 GeV cm -3 M c = 10 GeVn= 10 -1 cm -3 t = 10 s (b) -2 -14 -12 -10 -8 -6 n F n [ e r g c m - s - ] hv [eV] r c = 0 GeV cm -3 n= 1 cm -3 t = 10 s r c = 5·10 GeV cm -3 , M c = 100 GeV r c = 5·10 GeV cm -3 , M c = 100 GeV r c = 5·10 GeV cm -3 , M c = 10 GeV r c = 5·10 GeV cm -3 , M c = 10 GeV (c) b channel -2 -14 -12 -10 -8 -6 n F n [ e r g c m - s - ] hv [eV] b channelr c = 0 GeV cm -3 t = 10 sn= 10 -3 cm -3 r c = 5·10 GeV cm -3 , M c = 100 GeV r c = 5·10 GeV cm -3 , M c = 100 GeV r c = 5·10 GeV cm -3 , M c = 10 GeV r c = 5·10 GeV cm -3 , M c = 10 GeV (d) Figure 2. (a) Synchrotron + SSC spectra of GRB afterglows for different CME number densities ( n = 1 cm − , 10 − cm − ,10 − cm − , and 10 − cm − ) at t = 10 s with ρ χ = 5 × GeV cm − and M χ = 10 GeV in the b ¯ b channels. The solid lines anddashed lines denote the cases including and excluding the contributions of DMEs, respectively. (b) Synchrotron + SSC spectraof GRB afterglows in four different annihilation channels ( b ¯ b , w + w − , τ + τ − , and µ + µ − ) at t = 10 s with n = 10 − cm − , ρ χ = 5 × GeV cm − , and M χ = 10 GeV. The black solid line denotes the spectrum excluding the contributions of DMEs. (c)Synchrotron + SSC spectra of GRB afterglows for different DM densities ( ρ χ = 5 × and 5 × GeV cm − ) and particle masses( M χ = 10 and 100 GeV) at t = 10 s. DM annihilation occurs in the b ¯ b channel, and the CME number density is n = 1 cm − .The black solid line denotes the spectrum excluding the contributions of DMEs. (d) Same as (c) except n = 10 − cm − and ρ χ = 5 × and 5 × GeV cm − . and magenta lines represent n = 1 . − , 10 − cm − ,10 − cm − , and 10 − cm − , respectively. The solid anddashed lines represent the cases including and excludingthe contribution of the DMEs, respectively. DifferentCME densities result in different spectral shapes, andthe inclusion of DMEs further makes the variety of thespectra. For a lower CME density, the DME effect ismore significant.The spectra in four different annihilation channels arepresented in Figure 2(b). It is obvious that the fluxes inthe w + w − and b ¯ b annihilation channels are higher thanthose in the τ + τ − and µ + µ − annihilation channels. Figure 2(c) shows the synchrotron + SSC spectraof GRB afterglows for different DM densities and DMmasses at t = 10 s. The values of the DM density isset to 5 × GeV cm − and 5 × GeV cm − , andthe values of DM particle mass are taken as 10 GeVand 100 GeV. The DM annihilation channel is assumedto be the b ¯ b channel, and the CME number density is n = 1 cm − . Figure 2(d) presents the afterglow spectrafor the same parameters of Figure 2(c), except the pa-rameters n = 10 − cm − , and ρ χ = 5 × GeV cm − and 5 × GeV cm − . The black solid lines in Figures2(c) and 2(d) denote the spectra excluding the contri- Huang et al. butions of the DMEs. The flux values of the black linesare slightly higher than those of the red solid lines (cor-responding to lower DM densities and high DM particlemasses) because γ e , min decreases slightly when the ef-fects of DM are considered. According to all the lines,a higher DM density and lower DM particle mass candramatically increase the fluxes. As shown by the bluedashed line in Figure 2(d), the flux exhibits prominentimprovements in the low- and high-energy parts due tothe contributions of inefficiently accelerated DMEs andthe increased number of seed photons, respectively.Since many parameters related to jets are uncertainand the ISM and DM can prominently influence the lightcurves and spectra of GRB afterglows, it is difficult todistinguish the distributions of DM unless the proper-ties of the circumburst medium can be constrained byobservations. For a full description of the medium prop-erties of GRB afterglows, the location of GRBs in thegalaxy should be considered, as discussed below. GRB LOCATION IN THE GALAXYTo simplify, we assume a DM halo with the virial mass M vir in the virial radius R vir . The mean density is equalto the virial overdensity ∆ vir times the mean backgrounddensity ρ u , given by (e.g., Bullock et al. 2001) M vir ≡ π vir ρ u R . (14)∆ vir = 200 is generally preferred, which is independentof cosmology. Thus, the corresponding virial radius andvirial mass are R and M , respectively. Further-more, the concentration parameter (e.g., Bullock et al.2001) can be defined as c ≡ R r c , (15)which is also related to the virial mass (e.g., Buote et al.2007), i.e., c = 6 . z ) (cid:18) M M ⊙ (cid:19) − . . (16)Moreover, the virial mass can be derived by integratingdifferent DM density profiles in the viral radius R , M = 4 π Z R ρ ( r ) r dr. (17)The DM density of the galaxies is a function of ra-dius. The general Navarro-Frenk-White (NFW) densityprofile reads (Navarro et al. 1997) ρ NFW ( r ) = ρ c ( r/r c ) γ c [1 + ( r/r c )] − γ c , (18) Figure 3.
DM density as a function of the distance to thecenter of a massive galaxy with the virial mass ∼ M ⊙ .The lines of different colors denote different DM density pro-files. In addition, the red solid, dashed, dashed-dotted, anddotted lines denote the results with the following values ofthe inner slope of the NFW DM density profile: 1 .
0, 1 .
1, 1 . .
3, respectively. where ρ c and r c are the characteristic density and ra-dius, respectively. Here, we consider four values of theslope: γ c = 1 .
0, 1 .
1, 1 .
2, and 1 .
3, respectively. Theprofile proposed by Moore et al. (1999) correspondingto the slope γ c = 1 . ρ Einasto ( r ) = ρ c exp[ − d N (( r/r c ) /N − , (19)where d N ≈ N − / . /N for N & . M = 10 M ⊙ . The resultsare shown in Figure 3. The blue and green solid linesrepresent the Einasto law and Moore law, respectively.The red lines represent the NFW formula, where thesolid, dashed, dotted, and dashed-dotted lines denotethe NFW profile with the following four values of theinner slope: 1 .
0, 1 .
1, 1 .
2, and 1 .
3, respectively. Thegray shaded region indicates the DM density range of ∼ − GeV cm − . The vertical black dashed linedenotes the location 1 pc from the center of the galaxy.According to the magnetic field model used in the galaxycenter (Aloisio et al. 2004), B ISM increases rapidly withdecreasing radius within 1 pc, so the value of B ISM as-sumed above would be inappropriate within 1 pc. Fi-nally, the brown shaded region represents the possiblelocations of GRBs with significant contributions from ark matter annihilation in gamma-ray burst afterglows ∼ M ⊙ .Moreover, the effects of DM spatial diffusion (e.g.,Colafrancesco et al. 2006), re-acceleration, and advec-tion (e.g., Strong et al. 2007) might be important at thegalaxy scale or beyond, which would decrease the DMEdensities around GRBs. These effects are neglected forour calculations. CONCLUSIONS AND DISCUSSIONIn this paper, we study the contributions of DM anni-hilation to GRB afterglows. If GRBs occur at distancesof several to tens of parsecs from the centers of mas-sive galaxies, the effects of DMEs should be significant.Moreover, the influences of different DM particle massesand annihilation channels on GRB afterglows are ade-quately reflected in the flux changes.To date, no GRB has been discovered at a distance oftens of parsecs from a galaxy center, but one may expectthat these events could be detected in the future becausethe density of stars near the galaxy center is generallyhigher than that in other regions within the host galaxy.Observations show that the majority of stars formed atdistances of a few parsecs from the center of the MilkyWay are older than 5 Gyr (e.g., Blum et al. 2003), andone pulsar, PSR J1745 − ∼ ⋆ (Eatough et al. 2013). Of course,the dense gas and bright galaxy center will trouble theobservations of those distant GRBs.In addition to DMEs, gamma-ray emission is anotherfinal product of DM annihilation due to π decay. Thegamma-ray flux produced by π decay from a galaxywith M vir = 10 M ⊙ at z = 1 is far below the obser-vational threshold and hence can be neglected. How- ever, if this galaxy is located near the Milky Way, thegamma-ray flux can be detected. Therefore, we useobservations of a neighboring galaxy, M31, which hasa virial mass of ∼ M ⊙ , to test our model un-der the assumption that the gamma-ray emission nearthe center of M31 originated from DM annihilation.In the observation, a gamma-ray excess with a flux of(5 . ± . × − erg cm − s − was detected in the en-ergy range from 0 . ∼ d Φ γ ( E γ ) dE γ = 1 D A h σ A υ i m χ dN γ ( E γ ) dE γ E γ × Z r max r min ρ χ ( r ) r dr, (20)where dN γ ( E γ ) /dE γ is the gamma-ray spectrum cor-responding to M χ = 100 GeV and the b ¯ b annihilationchannel and r max = 5 kpc, and D A is the distance ofM31. The NFW profile with γ c = 1 . . . × − erg cm − s − . Thisflux is less than that observed in M31, indicating thatour model is self-consistent.ACKNOWLEDGMENTSThis work was supported by the National NaturalScience Foundation of China under grants 11822304,11890692, 11773007, and U1531130, and the GuangxiScience Foundation under grant 2018GXNSFFA281010.REFERENCES Abdallah, H., Abramowski, A., Aharonian, F., et al. 2018,PhRvL, 120, 201101Achterberg, A., Gallant, Y. A., Kirk, J. G., et al. 2001,MNRAS, 328, 393Ackermann, M., Albert, A., Anderson, B., et al. 2015,PhRvL, 115, 231301Ackermann, M., Ajello, M., Albert, A., et al. 2017, ApJ,836, 208Aloisio, R., Blasi, P., & Olinto, A. V. 2004, JCAP, 2004,007Bergstr¨om, L. 2012, Annalen der Physik, 524, 479Bertone, G., Hooper, D., & Silk, J. 2005, PhR, 405, 279Blum, R. D., Ram´ırez, S. V., Sellgren, K., et al. 2003, ApJ,597, 323 Blumenthal, G. R., & Gould, R. J. 1970, Reviews ofModern Physics, 42, 237Borriello, E., Cuoco, A., & Miele, G. 2009, PhRvD, 79,023518Bringmann, T., Edsj¨o, J., Gondolo, P., et al. 2018, JCAP,2018, 033Bullock, J. S., Kolatt, T. S., Sigad, Y., et al. 2001,MNRAS, 321, 559Buote, D. A., Gastaldello, F., Humphrey, P. J., et al. 2007,ApJ, 664, 123Colafrancesco, S., Profumo, S., & Ullio, P. 2006, A&A, 455,21Eatough, R. P., Falcke, H., Karuppusamy, R., et al. 2013,Nature, 501, 391Egorov, A. E., & Pierpaoli, E. 2013, PhRvD, 88, 023504