The diffuse gamma-ray flux associated with sub-PeV/PeV neutrinos from starburst galaxies
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9, 2018
Preprint typeset using L A TEX style emulateapj v. 05/04/06
THE DIFFUSE GAMMA-RAY FLUX ASSOCIATED WITH SUB-PEV/PEV NEUTRINOS FROM STARBURSTGALAXIES X IAO -C HUAN C HANG , X
IANG -Y U W ANG
School of Astronomy and Space Science, Nanjing University, Nanjing, 210093, China; [email protected] Key laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China
Draft version September 9, 2018
ABSTRACTOne attractive scenario for the excess of sub-PeV/PeV neutrinos recently reported by IceCube is that theyare produced by cosmic rays in starburst galaxies colliding with the dense interstellar medium. These proton-proton ( pp ) collisions also produce high-energy gamma-rays, which finally contribute to the diffuse high-energy gamma-ray background. We calculate the diffuse gamma-ray flux with a semi-analytic approach andconsider that the very high energy gamma-rays will be absorbed in the galaxies and converted into electron-position pairs, which then lose almost all their energy through synchrotron radiation in the strong magneticfields in the starburst region. Since the synchrotron emission goes into energies below GeV, this synchrotronloss reduces the diffuse high-energy gamma-ray flux by a factor of about two, thus leaving more room forother sources to contribute to the gamma-ray background. For a E - ν neutrino spectrum, we find that the diffusegamma-ray flux contributes about 20% of the observed diffuse gamma-ray background in the 100 GeV range.However, for a steeper neutrino spectrum, this synchrotron loss effect is less important, since the energy fractionin absorbed gamma-rays becomes lower. Subject headings: neutrinos- cosmic rays INTRODUCTION
The IceCube Collaboration reported 37 events rangingfrom 60 TeV to 3 PeV within three years of operation,correspond to a 5 . σ excess over the background atmo-spheric neutrinos and muons(Aartsen et al. 2014). The ob-served events can be fitted by a hard spectra ( E - Γ ν ) with Γ = 2 and a cutoff above 2 PeV, implied by the non-detection of higher energy events, or alternatively by aslightly softer but unbroken power-law spectrum with in-dex Γ ≃ . - . ν + ¯ ν ) is E ν Φ ν = 0 . ± . × - GeVcm - s - sr - , assuminga E - ν spectrum. The sky distribution of the these eventsis consistent with isotropy (Aartsen et al. 2014), imply-ing an extragalactic origin or possibly Galactic halo origin(Taylor et al. 2014), although a fraction of them could comefrom Galactic sources (Fox et al. 2013; Ahlers & Murase2013; Neronov et al. 2014; Razzaque 2013; Lunardini et al.2013).The source of these IceCube neutrinos is, however, un-known. Jets and/or cores of active galactic nuclei (AGN),and gamma-ray burst (GRBs) have been suggested to be pos-sible sources (Stecker et al. 1991; Anchordoqui et al. 2008;Kalashev et al. 2013; Stecker 2013; Waxman & Bahcall1997; He et al. 2012; Murase & Ioka 2013; Liu & Wang2013), where the photo-hadronic (e.g., p γ ) reaction is typi-cally the main neutrino generation process (Winter 2013). Onthe other hand, starburst galaxies and galaxy clusters may alsoproduce PeV neutrinos, mainly via the hadronuclear (e.g., pp )reaction (Liu et al. 2014; He et al. 2013; Murase et al. 2013;Tamborra et al. 2014; Anchordoqui et al. 2014b). In this pa-per, we concern the pp process in starburst galaxies, follow-ing earlier works in which remnants of supernovae in gen-eral (Loeb & Waxman 2006) or a special class of supernova–hypernovae (Liu et al. 2014) in starburst galaxies acceleratecosmic rays (CRs), which in turn produce neutrinos by in- teracting with the dense surrounding medium during propa-gation in their host galaxies. Here we use starburst galaxiesto mean those galaxies that have much higher specific starformation rates (SSFR) than the "Main Sequence" of star-forming galaxies. To produce neutrinos with energy E ν , werequire a source located at redshift z to accelerate protonsto energies of 50 (1 + z )2 E ν (Kelner et al. 2006). It is suggestedthat semi-relativistic hypernova remnants in starburst galax-ies, by virtue of their fast ejecta, are able to accelerate pro-tons to EeV energies (Wang et al. 2007; Liu & Wang 2012).Alternatively, assuming sufficient amplification of the mag-netic fields in the starburst region, normal supernova remnantsmight also be able to accelerate protons to 10 eV and pro-duce PeV neutrinos (Murase et al. 2013).In pp collisions, charged and neutral pions are bothgenerated. Charged pions decay to neutrinos ( π + → ν µ ¯ ν µ ν e e + , π - → ¯ ν µ ν µ ¯ ν e e - ), while neutral pions decay togamma-rays ( π → γγ ). Thus, PeV neutrinos are accompa-nied by high-energy gamma-rays. For a E - ν neutrino spec-trum with a single-flavor flux of E ν Φ ν ≃ - GeVcm - s - sr - produced by π ± decay, the flux of gamma-rays from π -decay is 2 × - GeVcm - s - sr - . While low-energy gamma-rays may escape freely from the galaxy, very high energy(VHE) gamma-rays are absorbed by the soft photons instarbursts, as starburst galaxies are optically thick to VHEgamma rays (Inoue 2011; Lacki & Thompson 2013). Theabsorbed gamma-rays will convert to electron-positron ( e ± )pairs, which then initiate electromagnetic cascades on softbackground photons, reprocessing the energy of VHE pho-tons to low-energy emission and contributing to the dif-fuse gamma-ray background . Detailed calculations show A simple estimate of the contribution to the gamma-ray backgroundcan be given below if the synchrotron loss of e ± pairs formed duringthe cascades is not taken into account. Assuming a flat cascade gamma-ray spectrum (i.e., Φ γ ∝ E - α γ γ with α γ ∼
2) (Berezinskii & Smirnov that the diffuse gamma-ray emission associated with sub-PeV/PeV neutrinos contribute a fraction &
30% to thegamma-ray background when the source density follows thestar-formation history and contribute a fraction &
40% fornon-evolving source density case (Murase et al. 2013). An-alytical calculations in Liu et al. (2014) show that this frac-tion is about & e ± ) pairs formed during the cas-cades. As we argue in this paper, this synchrotron loss effectcould reduce the flux of cascade gamma-ray emission signifi-cantly.In this paper, we calculate the accumulated diffuse gamma-ray flux by considering the synchrotron loss of e ± producedby the absorbed gamma-rays in the magnetic fields of the star-burst region. For reasonable assumptions of the strength of themagnetic fields in the starburst region, we find that the syn-chrotron loss dominates over the inverse-Compton (IC) scat-tering loss. Since the synchrotron emission goes into energiesbelow GeV, the intensity of the cascade gamma-ray compo-nent is significantly reduced in the ∼
100 GeV energy range.In §2, we first obtain the normalization of the gamma-ray fluxof an individual starburst galaxy with the observed neutrinoflux. In §3, we calculate the gamma-ray flux after consideringthe synchrotron loss. Then in §4, we calculate the accumu-lated gamma-ray flux from the population of starburst galax-ies. Finally we give our conclusions and discussions in §5. NORMALIZATION
To calculate the accumulated diffuse gamma-ray flux, weneed to know the the gamma-ray flux from individual star-burst galaxies and then integrate over contributions by all thestarburst galaxies in the Universe. The gamma-ray flux ofan individual galaxy can be then calibrated with the observedneutrino flux, assuming that all the observed neutrinos by Ice-Cube are produced by starburst galaxies.As the neutrino flux is linearly proportionally to the cosmic-ray proton flux and the efficiency in transferring the proton en-ergy to secondary pions, the single-flavor ( ν + ¯ ν ) neutrino lu-minosity per unit energy from a starburst galaxy is expressedby L ν i = dN ν i dtdE ν ∝ f π L p E - p ν , (1)where i = ( e , µ, τ ), L p is the CR proton flux at some fixed en-ergy and f π is the pion production efficiency. p is the index of × - GeVcm - s - sr - . Another component that can contribute to the dif-fuse gamma-ray emission is e ± pairs produced simultaneously with theneutrinos in the π ± decay. The IC scatterings of these pairs with in-frared photons will produce very high energy photons, which may be ab-sorbed and finally contribute to the cascade gamma-ray component with aflux ∼ - GeVcm - s - sr - . Noting that the observed gamma-ray back-ground flux observed by Fermi/LAT is ∼ - GeVcm - s - sr - at ∼ pp scenario would thus contribute ∼
50% of the ob-served diffuse gamma-ray background in the ∼
100 GeV range. the proton spectrum ( dn / dE p ∝ E - pp ), and we assume p = 2, asexpected from Fermi acceleration, unless otherwise specified.Starbursts have very high star formation rates ( SFR ) of mas-sive stars. Massive stars produce supernova or hypernovae,which accelerate CRs in the remnant blast waves, so we ex-pect that L p ∝ SFR . On the other hand, the radiation of a largepopulation of hot young stars are absorbed by dense dust andreradiated in the infrared band, so L TIR ∝ SFR , where L TIR isthe total infrared luminosity of the starburst galaxy. Thus weexpect that L p ∝ L TIR , so L ν i ( E ν , L TIR ) = C f π L TIR L ⊙ (cid:18) E ν (cid:19) - p , (2)where C is the normalization factor and L ⊙ is the bolo-metric luminosity of the Sun. The pionic efficiency is f π =1 - exp( - t esc / t loss ), where t loss is the energy-loss time for pp collisions and t esc is the escape time of protons. t loss =(0 . n σ pp c ) - , where the factor 0.5 is inelasticity, n is the par-ticle density of gas and σ pp is the inelastic proton collisioncross section, which is insensitive to proton energy. Introduc-ing a parameter Σ g = m p nR as the surface mass density of thegas (where R is the length scale of the starburst region), theenergy loss time is t loss = 7 × yr R pc (cid:18) Σ g - (cid:19) - . (3)We use R = 500pc as the reference value because neutrinosare dominantly produced by high redshift starbursts, as im-plied by the star-formation history, and high redshift starbursttypically have R ∼ t adv = R / v w = 4 . × yr R pc (cid:16) v w - (cid:17) - , (4)where v w is the velocity of galactic wind. One can seethat protons lose almost all their energy in dense starburstgalaxies with Σ g & . v w / - )gcm - , if the advec-tive time is shorter than the diffusive escape time. Little isknown about the diffusive escape time in starbursts. The dif-fusive escape time can be estimated as t di f f = R / D , where D = D ( E / E ) δ is the diffusion coefficient, D and E are nor-malization factors, and δ = 0 - D . cm s - at E = 3GeV for starburst galaxies, becausethe magnetic fields in nearby starburst galaxies such as M82and NGC253 are observed to be 100 times stronger than inour Galaxy and the diffusion coefficient is expected to scalewith the CR Larmor radius in the case of Bohm diffusion. Thefact that no break in the GeV-TeV gamma-ray spectrum up toseveral TeV from HESS observations also suggests a smalldiffusion coefficient in starburst galaxies (Abramowski et al.2012). The diffusive escape time is thus t di f f = 10 yr( R pc ) ( D cm s - ) - ( E p - . (5)for δ = 0 . For comparison, we also consider R = 200pc and R = 1kpc in the follow-ing calculations. (2011) assume that CRs stream out the starbursts at the av-erage Alfvén speed v A = B / p π m p n , so t di f f = 2 . (cid:18) R pc (cid:19) / (cid:18) Σ g - (cid:19) / (cid:18) B (cid:19) - , (6)where B is the magnetic field in the starburst region. The es-cape time is the minimum of the advective time and the diffu-sive time, i.e. t esc = min( t adv , t di f f ).The accumulated neutrino flux can be estimated by sum-ming up the contribution by all starburst galaxies throughoutthe whole universe, i.e. E ν Φ accu ν i = E ν c π Z z max Z L TIR , max L TIR , min φ ( L TIR , z ) L ν i [(1 + z ) E ν , L TIR ] H p (1 + z ) Ω M + Ω Λ dL TIR dz , (7)where φ ( L TIR , z ) represents the luminosity function of star-burst, H = 71 km s - Mpc - , Ω M = 0 . Ω Λ = 0 .
73. Her-schel PEP/HerMES has recently provided an estimationof the IR-galaxy luminosity function for separate classes(Gruppioni et al. 2013), which is a modified-Schechter func-tion φ ( L TIR ) = φ ∗ (cid:18) L TIR L ∗ (cid:19) - α exp (cid:20) - σ log (cid:18) + L TIR L ∗ (cid:19)(cid:21) . (8)For starburst galaxies, α = 1 . ± . σ = 0 . ± .
10, log ( L ∗ / L ⊙ ) = 11 . ± .
16, log ( φ ∗ / Mpc - dex - ) = - . ± . L ∗ evolves with redshift as ∝ (1 + z ) k L with k L =1 . ± .
13. For φ ∗ , it evolves as ∝ (1 + z ) k ρ , when z < z b , ρ ,and as ∝ (1 + z ) k ρ , when z > z b , ρ , with k ρ , = 3 . ± . k ρ , = - . ± .
05 and z b , ρ = 1. We set z max = 4 and the lu-minosity ranges from 10 L ⊙ to 10 L ⊙ .We assume all the observed neutrinos are produced by star-burst galaxies, so we can normalize the accumulated diffuseneutrino flux in Eq. (7) with the observed flux at PeV energy,i.e. E ν Φ accu ν i | E ν =1PeV = 0 . × - GeV cm - s - sr - . (9)Using the central values of the above parameters in the lumi-nosity function (e.g. α = 1 . σ = 0 .
35 and log ( L ∗ / L ⊙ ) =11 . C = 4 . × eV - s - for p = 2. Varying thevalues of these parameters within their error, the value of C changes within a factor of a few. However, we note that theresult of the accumulated gamma-ray flux, as calculated be-low in §4, remains almost unchanged. We also find that thespectrum of the accumulated neutrino flux, as given by Eq.(7),is flat ( ∝ E - ν ) for either choice of the diffusive escape timesin Eq.(5) or Eq.(6).For pp collisions, the energy flux in π -decay gamma-rays is E γ L γ ∝ (1 / E p Q p , where Q p represents the differ-ential CR energy flux. Since the neutrino flux is E ν i L ν ∝ (1 / E p Q p , the gamma-ray luminosity per unit energy re-sulted from the π -decay is related to the neutrino luminosityby E γ L γ ≈ E ν L ν i | E ν =0 . E γ . (10)Thus the π -decay photon luminosity per unit energy of a sin-gle starburst should be L γ = 8 . × f π L TIR L ⊙ ( E γ - eV - s - (11)for p = 2. GAMMA-RAY FLUX FROM ONE SINGLE STARBURST GALAXY
The pp process produces neutral pions π and chargedpions π ± . The neutral pion decay produces gamma-rays.Besides, e ± pairs from the decay of π ± , can also producegamma-rays through IC scattering off the soft photons in thestarburst region, mainly the infrared photons (in competingwith the synchrotron radiation in the magnetic field). Whilethe low-energy gamma-rays can escape and contribute to thediffuse gamma-ray background, the VHE gamma-rays will beabsorbed due to the dense photon field in the starburst region.The absorbed gamma-rays will be converted into e ± pairs,which initiate cascades through interactions with the soft pho-tons. As the VHE gamma-rays cascade down to enough lowenergies, they can escape out of the starburst and also con-tribute to the diffuse gamma-ray background. Below we con-sider the separate contributions by the absorbed gamma-raysand the unabsorbed ones. Absorption of high-energy gamma-rays in starbursts
Rapid star formation process makes an intense in-frared photon field in the starburst. The VHE gamma-ray photons may be absorbed by background photonsand generate e ± pairs. The optical depth is τ γγ ( E γ ) = R σ γγ ( ε, E γ ) n ( ε ) R d ε , where σ γγ is the cross section forpair-production (Bonometto & Rees 1971; Schlickeiser et al.2012) and n ( ε ) is the number density the infrared back-ground photons. We adopt the optical-infrared spectral en-ergy models for the nuclei of starburst galaxies developed bySiebenmorgen & Krügel (2007) and take Arp 220 as a tem-plate for starbursts to calculate the absorption optical depth.Using Arp220 as a template is because its infrared luminosityis close to luminosities of those starburst galaxies that con-tribute dominantly to the diffuse neutrino background (see§4). We find that the absorption energy does not change toomuch if we use M82 as a template instead. For simplicity,we assume all starbursts have the same photon spectra as Arp220. With such simplifications, the optical depth τ γγ ( E γ ) ina starbusrt depends only on its total infrared luminosity L TIR .Fig. 1 shows the corresponding absorption energy E cut (where τ γγ = 1) as a function of the total infrared luminosity of star-burst galaxies for different choices of R = 200pc, R = 500pcand R = 1kpc. It can be seen that photons with energies above1 - Absorbed gamma-rays
The π decay gamma-rays above E cut will be absorbedin the soft photons in the starbursts and produce pairs,each carrying about half energy of the initial photons( γγ → e + e - ). These pairs, together with the pairs re-sulted from π ± decay, will cool through synchrotron andIC emission. The synchrotron photon energy is E syn =50( E e / ( B / E e is the energy of e ± pairs and B is the magnetic field in the starburst region. Un-der reasonable assumptions about the magnetic field in thestarburst region, the synchrotron photon energy is below GeVenergies, so their contribution to the diffuse gamma-ray fluxat ∼
100 GeV is negligible. Note that the most constrainingpoint of the gamma-ray background is around 100 GeV, sincethe observed gamma-ray background flux decreases with theenergy. The IC photon energy is E IC γ ∼ min[ γ e ε b , E e ], where γ e is the Lorentz factor of the pair and ε b is the energy ofthe soft background photons. Approximating the soft photonsas infrared photons with energy ε b = 0 . E IC γ = 5TeV( E e / ( ε b / . E cut will be absorbed and an electro-magneticcascade will be developed. This cascade process transfers theenergy of absorbed gamma-rays to lower and lower energies,until the secondary photons can escape from absorption. Theescaped gamma-ray photons can then contribute to the diffusegamma-ray background. In order to calculate the fraction ofthe energy loss of e ± pairs transferred to the cascade emis-sion, we need to know the energy loss fraction of e ± throughthe IC emission. Synchrotron vs IC loss
Energy loss timescales are crucial for assessing the fractionof energy loss in these two processes. Thompson et al. (2006)argued that in order for starburst galaxies to fall on the ob-served FIR-radio correlation, the synchrotron cooling time instarbursts must be shorter than the IC cooling time and theescape time for relativistic electrons. The reason is that, ifthis constraint is not satisfied, any variation in the ratio be-tween the magnetic field and photon energy density ( U B / U ph )would lead to large changes in the fraction of cosmic ray elec-tron energy radiated via synchrotron radiation. A linear FIR-radio correlation would then require significant fine tuning.The synchrotron timescale depends on the magnetic field inthe starburst region. The magnetic fields in starbursts are notwell-understood and we parameterize the strength of the mag-netic field in terms of the gas surface density Σ g with the fol-lowing three scalings :i) First, in the assumption that the magnetic energy densityequilibrates with the total hydrostatic pressure of the interstel-lar medium, B ≃ (8 π G) / Σ g (Thompson et al. 2006), whereG is the gravitational constant. Fields strength as large asthis equipartition are possible if the magnetic energy den-sity equilibrates with the turbulent energy density of the ISM.The measurements of Zeeman splitting associated with OHmegamasers for eight galaxies suggest that the magnetic en-ergy density in the interstellar medium of starburst galax-ies is indeed comparable to their hydrostatic gas pressure(McBride et al. 2014).ii) B ∝ Σ . g is sometimes assumed, motivated by setting themagnetic energy density equal to the pressure in the ISM pro-duced by star formation ( P SF ). Because of P SF ∝ Σ SFR (where Σ SFR is the star formation rate per unit area) and the Schmidtscaling law for star-formation Σ SFR ∝ Σ . g (Kennicutt 1998),the scaling B ∝ Σ . g follows.iii)The third is the minimum energy magnetic field case B ∝ Σ . g , which is obtained using the observed radio flux and as-suming comparable cosmic-ray and magnetic energy densities(Thompson et al. 2006). Although such a low magnetic fieldis not favored, as argued in various aspects (Thompson et al.2006; McBride et al. 2014), we keep this case in the calcula-tion just to illustrate the difference when a low magnetic fieldstrength is considerred.Thus, the strength of the magnetic fields can be summarizedas (Thompson et al. 2006; Lacki & Thompson 2010) B µ G = (cid:16) Σ g gcm - (cid:17) ( B ∝ Σ g )400 (cid:16) Σ g gcm - (cid:17) . ( B ∝ Σ . g )150 (cid:16) Σ g gcm - (cid:17) . ( B ∝ Σ . g ) (12)As the gas surface density Σ g scales with the surface density of star formation rate as Σ g ∝ Σ . SFR and the star formation rate( π R Σ SFR ) scales linearly with the total infrared luminosity L TIR , we get the relation (assuming a constant star-formationradius R ) Σ g g cm - ≃ . (cid:18) L TIR L ⊙ (cid:19) . . (13)The synchrotron energy loss timescale of e ± is t syn =6 π m e c / ( c σ T B γ e ) (Rybicki & Lightman 1979), while the ICenergy loss timescale of relativistic e ± , for a graybody ap-proximation for the soft background photons, is given by(Schlickeiser & Ruppel 2010) t IC ≈ m e c c σ T U ph γ K + γ e γ e γ K (14)where U ph ≈ L / (2 π R c ) is the energy density of the soft pho-tons, γ K ≈ . × ( T / K ) - , T is the temperature of thegraybody radiation field and γ e is the Lorentz factor of e ± .Note that Eq.(14) applies to both the Thomson scatterings andthe scatterings in the Klein-Nishina regime when γ e is veryhigh.The fraction of the energy loss through synchrotron radia-tion is t - syn / (cid:0) t - syn + t - IC (cid:1) for one electron or positron of a par-ticular energy. Since this fraction is a function of the energyof e ± , we integrate it over a proper energy range to estimatethe fraction of the total energy loss of e ± through synchrotronradiation in the energy range, which is r ≈ R E max / E cut / t - syn t - syn + t - IC E - p + e dE e R E max / E cut / E - p + e dE e , (15)where E max = 4PeV is used for the maximum energy of e ± ,corresponding to a maximum neutrino energy of 2PeV. Weshow, in Fig.2, this fraction for starburst galaxies with differ-ent L TIR . The black, red and blue lines represent, respectively,the B ∝ Σ g , B ∝ Σ . g and B ∝ Σ . g cases. It shows that for boththe B ∝ Σ g and B ∝ Σ . g cases, the synchrotron loss constitutea fraction &
90% of the total energy loss. There are two fac-tors that leads to such a large synchrotron loss fraction: 1) themagnetic energy density is larger than the photon energy den-sity in these cases; 2) the Klein-Nishina effect for e ± above ∼ T / - TeV (Lacki & Thompson 2013) causes the ICenergy loss time to increase with γ e , while the synchrotronloss time continues to fall as γ - e . Cascade gamma-rays
While the synchrotron loss energy goes into low-energyemission and thus does not contribute to the diffuse high-energy gamma-ray background, the IC loss energy will cas-cade down to the relevant energy range of the diffuse gamma-ray emission. If the cascade develops sufficiently, the spec-trum of the cascade emission has a nearly universal form of L cas ∝ (cid:26) E - . γ ( E γ < E γ , b ) E - α γ γ ( E γ , b < E γ < E cut ) (16)where E cut is the absorption cutoff energy, E γ , b ≈ (4 / E cut / m e c ) ε b is the break energy corresponding to E γ , cut , and α γ ≃ E cut de-creases rapidly as e - τ γγ . The normalization of the cascadeemission spectrum is determined by equating the total cas-cade energy with the total IC energy loss above E cut , i.e. Z E γ L cas dE γ ≃
32 (1 - r ) E abs , (17)where E abs = R E max / E cut / E γ L γ dE γ is the energy of all π -decaygamma-rays above E cut and the factor represents the sumcontributions by the π -decay gamma-rays and the gamma-rays from IC scattering of π ± -decay e ± , the latter of whichhas a flux about half of that of π -decay gamma-rays. Unabsorbed gamma-rays
The gamma-rays from π decay with energies below E cut will escape out of the starburst galaxies and contribute di-rectly to the diffuse gamma-ray background. This flux ismodel-independent and readily obtained from the observedneutrino flux by using Eq.(10). For a flat neutrino spectrum,the differential flux of π -decay gamma-rays below E cut is E γ Φ γ ≃ × - GeVcm - s - sr - .As we pointed out before, the electrons and positrons ac-companying with the neutrino production in the π ± decayalso contribute to the diffuse gamma-rays through IC scatter-ing of soft background photons. These e ± pairs cool by bothsynchrotron and IC emission. We denote the flux contributedby IC process with L π ± , IC , which is important only when themagnetic energy density in the starburst region is low. There-fore the flux of the unabsorbed gamma-rays is L γ , un = ( L γ + L π ± , IC ) e - τ γγ (18)Adding the luminosity of the unabsorbed gamma-rays andthat of the cascade gamma-rays, we get the total gamma-rayphoton luminosity emitted from one starburst galaxy L total = L γ , un + L cas . (19) THE ACCUMULATED DIFFUSE GAMMA-RAY FLUX
Once the gamma-rays escape out of the starburst galaxy,they will further interact with the extragalactic infrared andmicrowave background photons, and similar cascades areformed for VHE gamma-rays. In this case, however, E cut isdifferent and dependent on the redshift distribution of star-burst galaxies. The cascade emission spectrum has the sameform as equation (16). Taking such cosmic cascades intoconsideration, the accumulated gamma-ray flux includes twoparts, one is from the direct source contribution and the otheris from the intergalactic cascades, i.e. Φ accu γ = Φ sour γ + Φ cas γ , (20)where Φ sour γ represents the direct source contribution by star-burst galaxies and Φ cas γ represents the flux from the intergalac-tic cascades.The direct source contribution can be obtained by integrat-ing the contributions of all starburst galaxies in the universeover their redshift and luminosity range, i.e., E γ Φ sour γ = E γ c π Z z max Z L TIR , max L TIR , min φ ( L TIR , z ) L ′ total [(1 + z ) E γ ] H p (1 + z ) Ω M + Ω Λ dL TIR dz , (21)where L ′ total [(1 + z ) E γ ] = L total [(1 + z ) E γ ] e - τ ′ γγ ( E γ , z ) and τ ′ γγ isthe absorption optical depth due to the extragalactic infraredand microwave background photons.The accumulated diffuse gamma-ray flux are shown in Fig-ures 3-5 for the cases of R = 500pc, R = 200pc and R = 1kpc respectively. It can be seen that, the diffuse gamma-ray fluxin the case considering the synchrotron loss is less than halfof that in the case neglecting the synchrotron loss. This is be-cause that the cascade component resulted from the absorbedgamma-rays is strongly suppressed due to the synchrotronradiation of the e ± . The accumulated diffuse gamma-rayflux after considering the synchrotron loss effect contributes ∼
20% of the observed diffuse gamma-ray background byFermi/LAT at ∼ B ∝ Σ g and B ∝ Σ . g cases.For a steeper neutrino spectrum, since the energy fractionin absorbed gamma-rays above E cut is lower, this effect is ex-pected to be less important. As in Murase et al. (2013), westudy the allowed range of the spectral index by the observedgamma-ray background data. We find that the index must be p . .
18, as shown in Fig.6 , in order not to violate the ob-served gamma-ray background data by Fermi/LAT, which isin agreement with the results in Murase et al. (2013).We also study how much the starburst galaxies in differentluminosity ranges and redshift ranges contribute to the dif-fuse gamma-ray background. Fig.7 shows the contributionsin each luminosity range to the total flux. It can be seen thatmost flux is contributed by the starbursts in the luminosityrange from 10 to 10 L ⊙ . For starburst galaxies of suchhigh luminosity, the pion production efficiency f π is close to1 and these galaxies are proton calorimeters. Fig.8 shows thecontributions by starburst galaxies in different redshift rangesto the total flux. As expected, the dominant contribution is bystarburst galaxies in the redshift range of 1 < z < DISCUSSIONS AND CONCLUSIONS
Starburst galaxies, which have many supernova or hy-pernova explosions, are proposed to be a possible sourcefor the sub-PeV/PeV neutrinos recently detected by Ice-Cube (Murase et al. 2013; Liu et al. 2014; Anchordoqui et al.2014b). We have shown that the minimum diffuse gamma-ray flux associated with these sub-PeV/PeV neutrinos is about(2 - × - GeVcm - s - sr - , which is a factor of two lowerthan the case without considering the synchrotron loss of the e ± pairs resulted from the absorption of very high energy pho-tons. This minimum diffuse gamma-ray flux constitute a frac-tion of ∼
20% of the observed gamma-ray background flux at ∼
100 GeV energies, thus leaving a relatively large room forother sources to contribute to the gamma-ray background.It was proposed that hypernovae remnants in starburstgalaxies accelerate protons to > eV (Wang et al. 2007),which then produce PeV neutrinos via pp collisions with thedense surrounding medium (Liu et al. 2014). Remnants ofnormal supernovae may accelerate protons to PeV energies,so they may not contribute to the &
100 TeV neutrino flux,but they can produce <
100 TeV gamma rays, thus contribut-ing to the diffuse gamma-ray background as well. But as longas the diffuse gamma-ray flux contributed by these normal su-pernovae does not exceed that contributed by the hypernovaremnants too much, the total diffuse flux may still fall be-low the observed background. It may be also possible that,if the remnants of normal supernovae produce a gamma-rayflux 3-4 times higher, the total gamma-ray flux from starburstgalaxies can reach the level of the observed one. Interestingly,Tamborra et al. (2014) recently show that star-forming andstarburst galaxies can explain the whole diffuse gamma-raybackground in the 0.3-30 GeV range.We thank Peter Mészáros, Kohta Murase and Ruoyu Liufor useful discussions, and the referee for the valuable re-port. This work is supported by the 973 program undergrant 2014CB845800, the NSFC under grants 11273016 and 11033002, and the Excellent Youth Foundation of JiangsuProvince (BK2012011).
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R=200pc B~
R=200pc B~ g R=500pc B~
R=500pc B~
R=500pc B~ g R=1kpc B~
R=1kpc B~
R=1kpc F IG . 2.— The fraction of total energy loss of relativistic e ± through the synchrotron radiation in the starburst magnetic field. The solid, dashed and dottedlines represent the cases of R = 200pc, R = 500pc and R = 1kpc respectively. The black, red and blue lines show the cases of B ∝ Σ g , B ∝ Σ . g and B ∝ Σ . g ,respectively. Note that the B ∝ Σ g lines overlap with the horizontal line of r=1. -10 -9 -8 -7 -6 -5 E [ G e V c m - s - s r - ] E[GeV] g zero B Fermi 2010 atm IceCube 2014 F IG . 3.— The accumulated diffuse gamma-ray flux of starburst galaxies for different assumptions of the magnetic fields in the starburst region. R = 500pcand p = 2 are assumed. The black, red and blue lines show the cases of B ∝ Σ g , B ∝ Σ . g and B ∝ Σ . g , respectively. For illustration, the green line shows thecase of B = 0. The neutrino flux is obtained using Eq.(7). The extragalactic gamma-ray background data from Fermi/LAT are depicted as the black dots. Theatmospheric neutrino data and the IceCube data are also shown. -10 -9 -8 -7 -6 -5 E [ G e V c m - s - s r - ] E[GeV] g zero B Fermi 2010 atm IceCube 2014 F IG . 4.— The same as figure 3, but with R = 200pc. -10 -9 -8 -7 -6 -5 E [ G e V c m - s - s r - ] E[GeV] g zero B Fermi 2010 atm IceCube 2014 F IG . 5.— The same as figure 3, but with R = 1kpc. -10 -9 -8 -7 -6 -5 E [ G e V c m - s - s r - ] E[GeV] g zero B Fermi 2010 atm IceCube 2014 F IG . 6.— The same as figure 3, but assuming a steeper proton spectrum with p = 2 . -10 -9 -8 -7 -6 -5 E G e V c m - s - s r - E[GeV] -10 -10 -10 -10 Total Fermi 2010
IceCube 2014 F IG . 7.— The diffuse gamma-ray flux contributed by starburst galaxies in different luminosity ranges. The grey, red, blue and orange lines represent thecontributions by the starburst galaxies in the luminosity ranges of 10 - L ⊙ , 10 - L ⊙ , 10 - L ⊙ , and 10 - L ⊙ , respectively. The black linerepresents the sum of them. R = 500pc, B ∝ Σ g and p = 2 are assumed. -10 -9 -8 -7 -6 -5 E G e V c m - s - s r - E[GeV] z 0-1 z 1-2 z 2-3 z 3-4 Total Fermi 2010
IceCube 2014 F IG . 8.— The diffuse gamma-ray flux contributed by starburst galaxies in different redshift ranges. The grey, red, blue and orange lines represent thecontributions by the starburst galaxies in the redshift ranges of 0 -
1, 1 -
2, 2 - -
4, respectively. The black line represents the sum of them. R = 500pc, B ∝ Σ g and pp