High-Energy Neutrino Production in Clusters of Galaxies
Saqib Hussain, Rafael Alves Batista, Elisabete M. de Gouveia Dal Pino, Klaus Dolag
MMNRAS , 1– ?? (2020) Preprint 8 February 2021 Compiled using MNRAS L A TEX style file v3.0
High-Energy Neutrino Production in Clusters of Galaxies
Saqib Hussain, ★ Rafael Alves Batista, † Elisabete M. de Gouveia Dal Pino, ‡ and Klaus Dolag, , , Institute of Astronomy, Geophysics and Atmospheric Sciences (IAG), University of São Paulo (USP), São Paulo, Brazil Radboud University Nijmegen, Department of Astrophysics/IMAPP, 6500 GL Nijmegen, The Netherlands University Observatory Munich, Scheinerstr. 1, 81679 Munchen, Germay Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str 1, 85741 Garching, Germany
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Clusters of galaxies can potentially produce cosmic rays (CRs) up to very-high energies via large-scale shocks and turbulentacceleration. Due to their unique magnetic-field configuration, CRs with energy ≤ eV can be trapped within these structuresover cosmological time scales, and generate secondary particles, including neutrinos and gamma rays, through interactions withthe background gas and photons. In this work we compute the contribution from clusters of galaxies to the diffuse neutrinobackground. We employ three-dimensional cosmological magnetohydrodynamical simulations of structure formation to modelthe turbulent intergalactic medium. We use the distribution of clusters within this cosmological volume to extract the propertiesof this population, including mass, magnetic field, temperature, and density. We propagate CRs in this environment using multi-dimensional Monte Carlo simulations across different redshifts (from 𝑧 ∼ 𝑧 = 𝛼 = . − . 𝐸 max = − × eV, clusters contribute to a sizeable fraction to the diffuse flux observed by the IceCubeNeutrino Observatory, but most of the contribution comes from clusters with 𝑀 (cid:38) 𝑀 (cid:12) and redshift 𝑧 (cid:46) .
3. If we includethe cosmological evolution of the CR sources, this flux can be even higher.
Key words: galaxies: clusters: intracluster medium, neutrinos, magnetic fields
The IceCube Neutrino Observatory reported evidence of an isotropicdistribution of neutrinos with ∼ PeV energies (Aartsen et al. 2017,2020). Their origin is not known yet, but the isotropy of the distri-bution suggests that they are predominantly of extragalactic origin.They might come from various types of sources, such as galaxy clus-ters (Murase et al. 2013; Hussain et al. 2019), starbursts galaxies,galaxy mergers, AGNs (Murase et al. 2013; Kashiyama & Mészáros2014; Anchordoqui et al. 2014; Khiali & de Gouveia Dal Pino 2016;Fang & Murase 2018), supernova remnants (Chakraborty & Izaguirre2015; Senno et al. 2015), gamma-ray bursts (Hümmer et al. 2012;Liu & Wang 2013). Since neutrinos can reach the Earth withoutbeing deflected by magnetic fields or attenuated due to any sort ofinteraction, they can help to unveil the sources of ultra-high-energycosmic rays (UHECRs) that produce them.Their origin and that of the diffuse gamma-ray emission are amongthe major mysteries in astroparticle physics. The fact that the observedenergy fluxes of UHECRs, high-energy neutrinos, and gamma raysare all comparable suggests that these messengers may have someconnection with each other (Ahlers & Halzen 2018; Alves Batista ★ E-mail: [email protected] (SH) † E-mail: [email protected] ‡ E-mail: [email protected] et al. 2019a; Ackermann et al. 2019). The three fluxes could, inprinciple, be explained by a single class of sources (Fang & Murase2018), like starburst galaxies or galaxy clusters (e.g., Murase et al.2008; Kotera et al. 2009; Alves Batista et al. 2019a, for reviews ).Clusters of galaxies form in the universe possibly through violentprocesses, like accretion and merging of smaller structures into largerones (Voit 2005). These processes release large amounts of energy,of the order of the gravitational binding energy of the clusters ( ∼ − erg). Part of this energy is depleted via shock waves andturbulence through the intracluster medium (ICM), which accelerateCRs to relativistic energies. These can be also re-accelerated bysimilar processes in more diffuse regions of the ICM, includingrelics, halos, filaments, and cluster mergers (e.g., Brunetti & Jones2014; Brunetti & Vazza 2020, for reviews). Furthermore, clustersof galaxies are attractive candidates for UHECR production due totheir extended sizes ( (cid:39) Mpc) and suitable magnetic field strength( ∼ 𝜇 G) (e.g., Fang & Murase 2018; Kim et al. 2019). Those withenergies
𝐸 > × eV have most likely an extragalactic origin(e.g., Aab et al. 2018; Alves Batista et al. 2019b), and those with 𝐸 (cid:46) eV are believed to have Galactic origin (see e.g., Blasi2013; Amato & Blasi 2018), although the exact transition betweengalactic and extragalactic CRs is not clear yet (see e. g., Aloisioet al. 2012; Parizot 2014; Giacinti et al. 2015; Thoudam et al. 2016;Kachelriess 2019).CRs with 𝐸 (cid:46) eV can be confined within clusters for a time © a r X i v : . [ a s t r o - ph . H E ] F e b S. Hussain et al. comparable to the age of the universe (e.g. Hussain et al. 2019).This confinement makes clusters efficient sites for the productionof secondary particles including, electron-positron pairs, neutrinosand gamma rays due to their interaction with the thermal protonsand photon fields (e.g. Berezinsky et al. 1997; Rordorf et al. 2004;Kotera et al. 2009). Non-thermal radio to gamma-ray and neutrinoobservations are, therefore, the most direct ways of constraining theproperties of CRs in clusters (Berezinsky et al. 1997; Wolfe & Melia2008; Yoast-Hull et al. 2013; Zandanel et al. 2015). Conversely, thediffuse flux of gamma rays and neutrinos depend on the energy budgetof CR protons in the ICM. Clusters also naturally can introduce aspectral softening due to the fast escape of high-energy CRs from themagnetized environment which might explain the second knee thatappears around ∼ eV, in the CR spectrum (Apel et al. 2013).To calculate the fluxes of CRs and secondary particles from clus-ters, there are many analytical and semi-analytical works (Berezinskyet al. 1997; Wolfe & Melia 2008; Murase et al. 2013), but in most ofthe approaches, the ICM model is overly simplified by assuming, forinstance, uniform magnetic field and gas distribution. There are morerealistic numerical approaches in Rordorf et al. (2004) and Koteraet al. (2009) exploring the three-dimensional (3D) magnetic fieldsof clusters. More recently, Fang & Olinto (2016) estimated the fluxof neutrinos from these objects assuming an injected CR spectrum ∝ 𝐸 − . , an isothermal gas distribution, a radial profile for the totalmatter (baryonic and dark) density profile, and a Kolmogorov tur-bulent magnetic field with coherence length ∼
100 kpc. They foundthese estimates to be comparable to IceCube measurements. Herewe revisit these analyses by employing a more rigorous numericalapproach. We take into account the non-uniformity of the gas densityand magnetic field distributions in clusters, as obtained from MHDsimulations. We consider additional factors such as the location ofCR sources within a given cluster, and the obvious mass depen-dence of the physical properties of clusters. This last considerationis important because massive clusters ( (cid:38) 𝑀 (cid:12) ) are much lesscommon than lower-mass ones ( (cid:46) 𝑀 (cid:12) ). Consequently, clustersthat can confine CRs of energy above PeV for longer are probablymore relevant for detection of high-energy neutrinos.Our main goal is to derive the contribution of clusters to the diffuseflux of high-energy neutrinos. To this end, we follow the propagationand cascading of CRs and their by-products in the cosmologicalbackground simulations by Dolag et al. (2005). We use the MonteCarlo code CRPropa (Alves Batista et al. 2016) that accounts for allrelevant photohadronic, photonuclear, and hadronuclear interactionprocesses. Ultimately, we obtain the CR and neutrino fluxes thatemerge from the clusters.This paper is organized as follows: in section 2 we describe thenumerical setup for both the cosmological background simulationsand for CR propagation through this environment; in section 3 wecharacterize the 3D-MHD simulations and present our results forthe fluxes of CRs and neutrinos; in section 4 we discuss our results;finally, in section 5 we draw our conclusions. To study the propagation of CRs in the ICM we consider the largescale cosmological 3D-MHD simulations performed by Dolag et al.(2005), who employed the Lagrangian smoothed particle hydrody-namics (SPH) code GADGET (Springel et al. 2001; Springel 2005).These simulations capture the essential features of the mass, tem-
T (K)
B (Gauss)
Figure 1.
This figure shows the temperature (upper panel) and magneticfield (lower panel) for one of the eight regions of our background 3D-MHDcosmological simulation at redshift 𝑧 = .
01, with dimension 240 Mpc ,performed by Dolag et al. (2005). perature, density, and magnetic field distributions in galaxy clusters,filaments and voids.We consider here seven snapshots of these simulations with red-shifts 𝑧 = .
01; 0 .
05; 0 .
2; 0 .
5; 0 .
9; 1 .
5; 5 .
0, each having the samevolume (
240 Mpc ) . We have divided the domain of each snapshotinto eight regions. Fig. 1 shows the temperature and magnetic-fielddistributions for one of the regions, at redshift 𝑧 = . ∼
50 Mpc , while the voids have dimensions of the sameorder, which are compatible with observations (e.g., Govoni et al.2019; Gouin et al. 2020). In this simulation, the comoving intensity ofthe seed magnetic field was chosen to be 𝐵 = × − G, which leadsto a quite reasonable match with the field strength observed in differ-ent clusters of galaxies today. Feedback and star formation were notincluded in these cosmological simulations. The background cosmo-logical parameters assumed are ℎ ≡ 𝐻 /(
100 km s − Mpc − ) = . Ω 𝑚 = . Ω Λ = .
7, and the baryonic fraction Ω 𝑏 / Ω 𝑚 =
14 %.
MNRAS , 1– ????
MNRAS , 1– ???? (2020) igh-Energy Neutrinos in Galaxy Clusters The simulations described in the previous section provide the back-ground magnetic field, gas density and temperature distributions ofthe ICM. In order to study the CR propagation in this environment,we employ the CRPropa 3 code (Alves Batista et al. 2016), withstochastic differential equations (Merten et al. 2017).In these simulations, we assume that CRs are composed only byprotons. We consider all relevant interactions during their propaga-tion including photohadronic, photonuclear, and hadronuclear pro-cesses, namely photopion production, photodisintegration, nucleardecay, proton-proton (pp) interactions, and adiabatic losses due tothe expansion of the universe. The cosmic microwave backgroundradiation (CMB) and the extragalactic background light (EBL) aretwo essential ingredients, but other contributions comes from the hotgas component of the ICM, of temperatures between ∼ − K,that produces bremsstrahlung radiation (Rybicki & Lightman 2008)and serves as target for pp-interactions. This is calculated in sec-tion 3.1.
To investigate the flux of different particle species and the change oftheir energy spectrum, we use the Parker transport equation, whichis a simplified version of the Fokker-Planck equation. It gives a gooddescription of the transport of CRs for an isotropic distribution in thediffuse regime. It is given by: 𝜕𝑛𝜕𝑡 +(cid:174) 𝑢. ∇ 𝑛 = ∇ . ( ˆ 𝜅 ∇ 𝑛 )+ 𝑝 𝜕𝜕 𝑝 (cid:18) 𝑝 𝜅 𝑝 𝑝 𝜕𝑛𝜕 𝑝 (cid:19) + (∇(cid:174) 𝑢 ) 𝜕𝑛𝜕 ln 𝑝 + 𝑆 ((cid:174) 𝑥, 𝑝, 𝑡 ) . (1)Here (cid:174) 𝑢 is the advection speed, ˆ 𝜅 is the spatial diffusion tensor, 𝑝 is theabsolute momentum, 𝜅 𝑝 𝑝 is the diffusion coefficient of momentumused to describe the reacceleration, n is the particle density, (cid:174) 𝑥 givesposition and 𝑆 ((cid:174) 𝑥, 𝑝, 𝑡 ) is the source of CRs (distribution of CRs atthe source).Propagation of CRs can be diffusive or semi-diffusive, dependingon the Larmor radius ( 𝑟 L = . 𝐸 / 𝐵 𝜇 G pc) of the particles and themagnetic field of the ICM. The diffusive regime corresponds to 𝑟 L (cid:28) 𝑅 cluster , and the semi-diffusive is for 𝑟 L (cid:38) 𝑅 cluster , wherein 𝑅 cluster isthe radius of the cluster, typically ∼ 𝐵 ∼ 𝜇 G, for theenergy range of interest (10 − eV), 𝑟 L (cid:28) 𝑅 cluster , so we arein the diffusive regime. CRs in this energy range would be confinedcompletely by the magnetic field of the clusters for a time longerthan the Hubble time ( 𝑡 H ∼
14 Gyr) (e.g. Fang & Murase 2018). Forinstance, a CR with energy ∼ eV in a cluster of mass ∼ 𝑀 (cid:12) with central magnetic field strength ∼ − 𝜇 G has 𝑟 L ∼ . ∼ ∼ Mpc. The confinementtime for this CR can be calculated as 𝑡 con ∼ / 𝑐 ∼ 𝑡 H (e.g. Hussain et al. 2019). Hence, CRs with energy 𝐸 > eVhave more chances to escape the magnetized cluster environment.The flux of CRs that can escape a cluster depends on its mass andmagnetic-field profile, with the latter directly correlated with thedensity distribution, being larger in denser regions. Our background simulation includes seven snapshots in the redshiftrange 0 . < 𝑧 < .
0. We have identified clusters in the densest regions of the isocontour maps of the whole volume, in each snap-shot (see Fig. 1). We then selected five clusters with distinct massesranging from 10 to 10 𝑀 (cid:12) , which we assumed to be represen-tative of all the clusters in the corresponding snapshot. Finally, weinjected CRs in each of these clusters to study their propagation andproduction of secondary particles. As an example, Fig. 2 illustratesrelevant properties for two of these clusters with masses ∼ 𝑀 (cid:12) (left panel) and ∼ 𝑀 (cid:12) (right panel) at redshift 𝑧 = .
01. To es-timate the total mass of a cluster from the simulations, we integratedthe baryonic and dark matter densities within a volume of 2 Mpc,assuming an approximate spherical volume. We note that this spe-cific evaluation is not much affected by the deviations from sphericalsymmetry that we detect in Fig. 2.To illustrate general average properties of the simulated clusters,we converted the Cartesian into spherical coordinates and dividedthe cluster in 10 concentric spherical shells of different radii ( 𝑅 shell ).Starting from the center of the cluster, the shells were first divided inintervals of 100 kpc, then between 300 kpc and 1500 kpc, they weredivided in intervals of 200 kpc, and the last shell in the outskirts wastaken between 1500 kpc < 𝑟 < ∼ 𝑀 (cid:12) at fourdifferent redshifts. The overdensity in Fig. 3 (bottom-right panel) isdefined as Δ = 𝜌 ( 𝑟 )/ 𝜌 bary , where 𝜌 ( 𝑟 ) is the total density at a givenpoint and 𝜌 bary is the mean baryonic density, 𝜌 bary = Ω bary × 𝜌 crit , 𝜌 crit = 𝐻 / 𝜋 G. We see that, in general, these radial profiles arevery similar across the cosmological time, except for the temperaturethat varies non-linearly with time by about four orders of magnitudein the inner regions of the cluster. Fig. 4 shows profiles for thetemperature, gas density, magnetic field and overdensity for a clusterof mass ∼ 𝑀 (cid:12) , as a function of the azimuthal ( 𝜙 ) angle fordifferent latitudes ( 𝜃 ), within a radial distance of 𝑅 =
300 kpc,at a redshift 𝑧 = .
01. We see that there are substantial variationsin the angular distributions of all the quantities. These variationscharacterize a deviation from spherical symmetry that may affect theemission pattern of the CRs and consequently secondary gamma raysand neutrinos.We also found that the magnetic field strength of a cluster dependson its mass: the heavier the cluster, the stronger the average magneticfield is, due to the larger extension of denser regions (see middlecolumn of Fig. 2 and Fig. 5). Inside all clusters, magnetic fields varyin the range 10 − < 𝐵 / G < − (see also Dolag et al. 2005; Ferrariet al. 2008; Xu et al. 2009; Brunetti & Jones 2014; Brunetti et al.2017; Brunetti & Vazza 2020).In the upper panel of Fig. 6, we compare the radial density profileof our simulated cluster of mass 10 𝑀 (cid:12) with the model used byFang & Olinto (2016). We see that both profiles look similar up to ∼ kpc. Above this scale, the density distribution of our simulatedclusters decays much faster than the assumed distribution in Fang &Olinto (2016).To estimate the total flux of CRs and neutrinos, we need to eval-uate the total number of clusters in our background simulationsas a function of their mass, at different redshifts. From the en-tire simulated volume, (240 Mpc ) , we selected 20 sub-samples of (
20 Mpc ) from different regions, as representative of the wholebackground. We then calculated the average number of clustersper mass interval in each of these sub-samples ( 𝑑𝑁 clusters, avg / 𝑑𝑀 ),between 10 𝑀 (cid:12) and 10 𝑀 (cid:12) . To obtain the total numberof clusters per mas interval we multiplied this quantity by thenumber of intervals 𝑁 = (
240 Mpc ) /(
20 Mpc ) in which thewhole volume was divided. So, the total number of clusters per MNRAS , 1– ?? (2020) S. Hussain et al. ρ ( gcm − ) B ( Gauss ) T ( K ) ρ ( gcm − ) B ( Gauss ) T ( K ) Figure 2.
Maps of gas density (left column), magnetic field (middle column), temperature (right column) of two clusters of masses ∼ 𝑀 (cid:12) (upper panels)and ∼ 𝑀 (cid:12) (bottom panels), at redshift 𝑧 = . M D M [ M ] z =0.01 z =0.05 z =0.5 z =0.9 M g a s [ M ] n [ c m ] R [kpc] B [ G a u ss ] R [kpc] T [ K ] R [kpc] [ g c m ] Figure 3.
Volume-averaged profiles as a function of the radial distance fromthe center for a cluster of mass 𝑀 ∼ 𝑀 (cid:12) , at four different redshifts.The quantities shown are: dark-matter mass (top left); gas number density(top center); gas mass (top right); magnetic field (bottom left); temperature(bottom-center) and overdensity (bottom right). mass interval was calculated as ( 𝑑𝑁 clusters, avg / 𝑑𝑀 ) × 𝑁 . Since wehave seven redshifts in our cosmological background simulations, 𝑧 = . , . , . , . , . , . , .
0, we then have repeated thecalculation above for each snapshot to obtain the number of clustersper mass interval at different redshifts. This is shown in the lowerpanel of Fig. 6 for different redshifts.To calculate the photon field of the ICM, we assume that the clus-ters are filled with photons from Bremsstrahlung radiation of the hot,rarefied ICM gas (see Figs. 1 to 4). For typical temperatures and den-sities, we can further assume an optically thin gas. Taking a photondensity ( 𝑛 ph ) distribution with approximately spherical symmetric T [ K ] o o < < 90 o o < < 135 o o < < 180 o n [ c m ] [degree] B [ G a u ss ] [degree] [ g c m ] Figure 4.
Volume-averaged profiles as a function of the azimuthal ( 𝜙 ) anglefor different latitudes ( 𝜃 ), within a radial distance 𝑅 =
300 kpc from thecenter, for a cluster of mass 𝑀 ∼ 𝑀 (cid:12) . From top left to bottom rightclockwise, temperature, gas number density, overdensity and magnetic field. within the cluster, we have the following relations for an opticallythin gas (Rybicki & Lightman 2008): 𝑑𝑛 ph 𝑑𝜖 = 𝜋𝐼 𝜈 𝑐ℎ𝜖 , 𝐼 𝜈 = 𝑅 shell 𝐽 ff 𝜈 . (2)where 𝐼 𝜈 is the specific intensity of the emission, 𝑐 is the speedof light, ℎ is the Planck constant, 𝜖 is the photon energy, 𝑅 shell isthe radius of concentric spherical shells, and 𝐽 ff 𝜈 is related with the MNRAS , 1– ????
300 kpc from thecenter, for a cluster of mass 𝑀 ∼ 𝑀 (cid:12) . From top left to bottom rightclockwise, temperature, gas number density, overdensity and magnetic field. within the cluster, we have the following relations for an opticallythin gas (Rybicki & Lightman 2008): 𝑑𝑛 ph 𝑑𝜖 = 𝜋𝐼 𝜈 𝑐ℎ𝜖 , 𝐼 𝜈 = 𝑅 shell 𝐽 ff 𝜈 . (2)where 𝐼 𝜈 is the specific intensity of the emission, 𝑐 is the speedof light, ℎ is the Planck constant, 𝜖 is the photon energy, 𝑅 shell isthe radius of concentric spherical shells, and 𝐽 ff 𝜈 is related with the MNRAS , 1– ???? (2020) igh-Energy Neutrinos in Galaxy Clusters M cluster [ M ]10 B [ G a u ss ] R cluster [kpc]10 B [ G a u ss ] M MM MM M Figure 5.
Upper panel shows the whole volume-averaged value of the mag-netic field as a function of the cluster mass. Lower panel compares the volume-averaged magnetic field as a function of the radial distance for clusters ofdifferent masses. R [ kpc ]10 n g a s [ g / c m ] MHD SimulationFang & Olinto (2016)10 M cluster [ M ] d N / d M z = 0.01 z = 0.2 z = 0.9 z = 1.5 z = 5.0 z Figure 6.
Comparison of the density profile of a cluster of mass 10 𝑀 (cid:12) ,from our simulation with the model used by Fang & Olinto (2016), givenin the upper panel. The lower panel shows the number of clusters per massinterval in our background simulation for different redshifts. [eV] d n / d [ e V / m ] Cluster 1: R <100 kpcCluster 1: 700< R / kpc <900Cluster 2: R <100 kpcCluster 2: 700< R / kpc <900EBL: Domínguez et al. 2011EBL: Gilmore et al. 2012 Figure 7.
Comparison of EBL with the Bremsstrahlung radiation of the ICMas a function of the photon energy. The Bremsstrahlung is calculated for twoclusters at different radial distance intervals. Cluster 1 has mass 10 𝑀 (cid:12) ,and Cluster 2 , 10 𝑀 (cid:12) . Bremsstrahlung emission coefficient:4 𝜋𝐽 ff 𝜈 = 𝜖 ff 𝜈 ( 𝜈, 𝑛, 𝑇 ) = . × − 𝑍 𝑛 𝑒 𝑛 𝑖 𝑇 − / 𝑒 − ℎ𝜈 / 𝑘 𝐵 𝑇 , (3)which is given in units of erg cm − s − Hz − .In Fig. 7 we compare the radiation fields for two EBL mod-els (Gilmore et al. 2012; Dominguez et al. 2011) with theBremsstrahlung photon fields of two clusters of masses ∼ 𝑀 (cid:12) (cluster 1) and ∼ 𝑀 (cid:12) (cluster 2). For both clusters, we calcu-lated the internal photon field at the center ( 𝑅 <
100 kpc) and for the(700 < 𝑅 / kpc < 𝜆 for these interactions is larger than the Hubble horizon.Thus deviations from spherical symmetry for this photon field willnot be relevant in this study.We have also implemented the proton-proton (pp) interactions us-ing the spatial dependent density field extracted directly from thebackground cosmological simulations, using the same procedure de-scribed by Rodríguez-Ramírez et al. (2019). We further notice that,for the computation of the CR fluxes, the magnetic field distributionhas been also extracted directly from the background simulations,without considering any kind of space symmetry. CRPropa 3 employs a Monte Carlo method for particle propagationand previously loaded tables of the interaction rates in order to cal-culate the interaction of CRs with photons along their trajectories.We implemented the spatially-dependent interaction rates into thecode, based on the gas and photon density distributions for the clus-ters of different masses. The mean free paths ( 𝜆 ) for the differentinteractions of CRs are described in appendix A.The values of 𝜆 for all the interactions of CRs with the backgroundphoton fields and the gas, are plotted in the upper panel of Fig. 8.For photopion production, we compare 𝜆 due to interactions with the MNRAS , 1– ?? (2020) S. Hussain et al. photon fields (i.e., the Bremsstrahlung radiation, red solid line) of acluster of mass 10 𝑀 (cid:12) with the EBL (red dotted line) and the CMB(red dashed line). For the Bremsstrahlung radiation, we consideredonly the photons within a sphere of radius 100 kpc around the centerof the cluster (i.e., the densest region, which is shown in Figs. 2& 4). High-energy CR interactions with CMB photons is a well-understood process that limits the distance from which CRs can reachEarth leading to the GZK cutoff. The upper panel of Fig. 8 showsthat 𝜆 for this interaction is much smaller than that for the EBL andBremsstrahlung. So, CR interactions with CMB photons dominateat energies 𝐸 (cid:38) eV. We also see that 𝜆 for Bremsstrahlungis greater than the size of the universe ( ∼ Mpc), and for EBL,it is ∼ Mpc. The 𝜆 for pp-interactions (green line) is muchless than the Hubble horizon. Therefore, this kind of interaction ismore likely to occur than photopion production specially at energies < eV. Upper panel of Fig. 8 also shows that we can neglect theCR interactions with the local Bremsstrahlung photon field, as wellas the interaction of high-energy gamma rays with the local gas ofthe ICM (yellow) in photopion production.The lower panel of Fig. 8 shows the distribution of the trajectorylengths (total distance travelled by a CR inside the cluster up tothe observation time), for different energy bins of CRs. There isa substantial number of events with trajectory length greater than 𝐷 (cid:38) Mpc for each energy bin. Thus, the trajectory lengthsof CRs are comparable to the mean free paths of pp-interactionsand photopion production in the CMB and EBL case, so that theseinteractions can produce secondary particles including gamma raysand neutrinos.
To study the propagation of CRs in the diffuse ICM, we used thetransport equation as implemented in CRPropa 3 by (Merten et al.2017, see also equation 1). There are three possible scenarios inCRPropa3 for each particle until its detection: the particle reachesthe detector within a Hubble time; the energy of the particle becomessmaller than a given threshold; or the trajectory length of a CRexceeds the maximum propagation distance allowed.We inject CRs isotropically with a power-law energy distributionwith spectral index 𝛼 and exponential cut-off energy 𝐸 max whichfollows the relation 𝑑𝑁 CR ,𝐸 / 𝑑𝐸 ∝ 𝐸 − 𝛼𝑖 exp (− 𝐸 𝑖 / 𝐸 max ) (see Ap-pendix B). We take different values for 𝛼 (cid:39) . − .
7, and for 𝐸 max = × − eV (e.g. Brunetti & Jones 2014; Fang& Olinto 2016; Brunetti et al. 2017; Hussain et al. 2019, for review).As stressed, the lower and upper limits of the mass of the galaxyclusters are taken to be 10 𝑀 (cid:12) and 10 𝑀 (cid:12) , respectively. This isbecause for 10 (cid:46) 𝐸 / eV (cid:46) , clusters with mass 𝑀 < 𝑀 (cid:12) barely contribute to the total flux of neutrino, due to low gas density,while there are few clusters with 𝑀 (cid:38) 𝑀 (cid:12) at high redshifts( 𝑧 > .
5) (Komatsu et al. 2009; Ade et al. 2014). The closest galaxyclusters are located at 𝑧 ∼ .
01, so we consider the redshift range0 . ≤ 𝑧 ≤ . . − E [ eV ] [ M p c ] Hubble Horizonpp-interactionphotopion: EBLphotopion: CMBphotopion: cluster 1 Inverse photopion: cluster 1 D [Mpc] N < E / eV < 10 < E / eV < 10 < E / eV < 10 < E / eV < 10 < E / eV < 10 Figure 8.
The upper panel shows the mean-free path 𝜆 for CR interactionswhich produce neutrinos. It is shown 𝜆 for photopion production in thebremsstrahung photon field (red solid line), CMB (red dashed line) and EBL(red dotted line). Also shown is 𝜆 for pp-interactions (green) calculated withina sphere of radius 𝑟 =
100 kpc around the center of a massive cluster (withmass 10 𝑀 (cid:12) and shown in Fig. 2). The 𝜆 for the interaction of high-energygamma rays with the local gas of the ICM (yellow) is also depicted. The thickblack line represents the Hubble horizon in the upper panel. The lower panelshows the distribution of the total trajectory length of CRs inside the clusteras a function of their energy bins. ies contain many supernova remnants that can also accelerate CRsup to very-high energies ( 𝐸 (cid:38)
100 PeV) (He et al. 2013). AGNare more powerful and more numerous at higher redshifts (Hasingeret al. 2005; Khiali & de Gouveia Dal Pino 2016; D’Amato et al.2020), and their luminosity density evolves more strongly for 𝑧 (cid:38) 𝜓 SFR ( 𝑧 ) = 𝐵 ( + 𝑧 ) . if 𝑧 < ( + 𝑧 ) − . if 1 < 𝑧 < ( + 𝑧 ) − . if 𝑧 > 𝜓 AGN ( 𝑧 ) = ( + 𝑧 ) 𝑚 𝐴 ( + 𝑧 ) . if 𝑧 < . . ( + 𝑧 ) − . if 0 . < 𝑧 < . . ( + 𝑧 ) − . if 𝑧 > .
48 (5)where 𝐴 = . 𝐵 = .
66 are normalization constants inequations (5) and (4), respectively. For AGN evolution 𝜓 𝐴𝐺𝑁 ( 𝑧 ) ∝ MNRAS , 1– ????
66 are normalization constants inequations (5) and (4), respectively. For AGN evolution 𝜓 𝐴𝐺𝑁 ( 𝑧 ) ∝ MNRAS , 1– ???? (2020) igh-Energy Neutrinos in Galaxy Clusters ( + 𝑧 ) , for low redshift 𝑧 < 𝑚 > . 𝑚 = . to 10 erg/sand their evolution depends on their luminosities. The AGNs withluminosities ∼ − erg/s are more important as they are morenumerous and believed to be able to accelerate particles to ultra-highenergies (e.g. Waxman 2004; Khiali & de Gouveia Dal Pino 2016).AGNs with luminosities greater than 10 erg/s are less numerous(Hasinger et al. 2005) and their evolution function ( 𝜓 AGN ( 𝑧 ) ) isdifferent from equation (5). For no source evolution, 𝜓 ( 𝑧 ) = 𝑑𝑁 / 𝑑𝑀 at redshift 𝑧 is given in the lower panel of Fig. 6, which was obtained from ourcosmological simulations. It is related to the flux through: 𝐸 Φ ( 𝐸 ) = 𝑧 max ∫ 𝑧 min 𝑑𝑧 𝑀 max ∫ 𝑀 min 𝑑𝑀 𝑑𝑁𝑑𝑀 𝐸 𝑑 (cid:164) 𝑁 ( 𝐸 /( + 𝑧 ) , 𝑀, 𝑧 ) 𝑑𝐸 (cid:32) 𝜓 ev ( 𝑧 ) 𝜋𝑑 𝐿 ( 𝑧 ) (cid:33) (6)where 𝜓 ev ( 𝑧 ) stands for, 𝜓 SFR ( 𝑧 ) and 𝜓 AGN ( 𝑧 ) , (cid:164) 𝑁 is the numberof CRs per time interval 𝑑𝑡 with energies between 𝐸 and 𝐸 + 𝑑𝐸 that reaches the observer. The quantity 𝐸 𝑑 (cid:164) 𝑁 / 𝑑𝐸 in equation (6) isthe power of CRs calculated from our propagation simulation and isseveral orders of magnitude smaller than the luminosity of observedclusters (e.g., Brunetti & Jones 2014).In order to convert the code units of the CR simulation to physicalunits, we have used a normalization factor (Norm). To calculateNorm, we first evaluate the X-ray luminosity of the cluster usingthe empirical relation 𝐿 X ∝ 𝑓 𝑔 𝑀 vir (Schneider 2014), where 𝑓 𝑔 = 𝑀 𝑔 / 𝑀 vir denotes the gas mass ( 𝑀 𝑔 ) fraction with respect to thetotal mass of the cluster within the Virial radius ( 𝑀 vir ) and then,since we are assuming that ( . − ) % of this luminosity goes intoCRs, this implies that Norm ∼ ( . − ) % 𝐿 X / 𝐿 CRsim and 𝐿 CRsim is the luminosity of the simulated CRs. Therefore, the CR powerthat reaches the observer (at the Earth) is ∼ 𝐸 𝑑 (cid:164) 𝑁 / 𝑑𝐸 × Norm. Inequation (6) 𝑑 𝐿 is the luminosity distance, given by: 𝑑 𝐿 = ( + 𝑧 ) 𝑐𝐻 𝑧 ∫ 𝑑𝑧 (cid:48) 𝐸 ( 𝑧 (cid:48) ) , (7)with 𝐸 ( 𝑧 ) = √︃ Ω 𝑚 ( + 𝑧 ) + Ω Λ = 𝐻 ( 𝑧 ) 𝐻 , (8)where the Hubble constant, as well as the matter ( Ω 𝑚 ) and dark-energy ( Ω Λ ) densities are defined in section 2.1, assuming a flat Λ CDM universe.We selected different injection points inside the clusters of differentmasses in order to study the spectral dependence with the position,which may correspond to different scenarios of acceleration of CRs.For instance, the larger concentration of galaxies near the center mustfavor more efficient acceleration, but compressed regions by shocksin the outskirts may also accelerate CRs. The schematic diagram ofthe simulation of CRs propagation is shown in Fig. 9. CRs are injectedat three different positions within each selected cluster denoted by 𝑅 Offset . The spectra of CRs have been collected by an observer in asphere of 2 Mpc radius ( 𝑅 Obs ), centred at the cluster, with a redshiftwindow ( − . ≤ 𝑧 ≤ .
1) for all the injection points of CRs. All-flavour neutrino fluxes are also computed at the same observer (seeSection-3.4 below).The spectrum of CRs obtained from our simulations is shown in R Cluster (kpc)R obs = 2MpcR
Offset
Cluster
Figure 9.
Scheme of the CR simulation geometry. They are injected at threedifferent positions inside each cluster represented by 𝑅 Offset , and 𝑅 Obs is theradius of the observer. E [eV] E d N / d E [ e V c m s s t ] M MM MM M Figure 10.
This figure shows the CR flux of individual clusters of distinctmasses, 𝑀 ∼ (red); 10 (green); and 𝑀 ∼ 𝑀 (cid:12) (blue color).This diagram shows the flux of CRs, for sources located at the center of thecluster (solid), at 300 kpc (dashed), and at 1 Mpc (dash-dotted lines) awayfrom the centre. The flux is computed at the edge of the clusters. The spectralparameters are 𝛼 = 𝐸 max = × eV, and it is assumed that 2% ofthe luminosity of the clusters is converted into CRs. Figs. 10 & 11. Its dependence on the position where the CR sourceis located within the cluster for 𝑧 = .
01 is shown for three clustersof different masses in Fig. 10. Particles injected at 1 Mpc distanceaway from the clusters center can leave them in short time, withalmost no interaction, as both the magnetic field and the gas numberdensity are very low compared to the central regions. On the otherhand, CRs injected at the center or at 300 kpc away from the clustercenter can be easily deflected by the magnetic field and trapped indense regions. This explains the higher CR flux for the injection pointat 1 Mpc in Fig. 10. Also, because the confinement of CRs in thecentral regions of the clusters is comparable to a Hubble time, andbecause of the value of 𝜆 for the relevant interactions, the productionof secondary particles including neutrinos and gamma rays in theclusters is substantial, as we will see in section 3.4.In Fig. 11 we show the CR spectrum of all the clusters at differentredshifts integrated up to the Earth. Although the spectra in this di-agram have been integrated up to the Earth, we have not consideredany interactions of the CRs with the background photon and mag-netic fields during their propagation from the edge of the clusters MNRAS , 1– ?? (2020) S. Hussain et al. E [eV] E d N / d E [ e V c m s s r ] z = 0.01 z = 0.05 z = 0.2z < 0.3 Figure 11.
This figure shows the total CR flux (at the Earth distance) fromall the clusters distributed in different redshifts: 𝑧 = .
01 (blue); 𝑧 = . 𝑧 = . . ≤ 𝑧 ≤ . to the Earth. Though not quantitatively realistic, it provides impor-tant qualitative information. One obvious result is that most of thecontribution in the CR flux comes from clusters at low redshifts.Moreover there is a significant suppression in the flux of CRs at (cid:38) eV, which indicates the trapping of lower-energy CRs withinthe clusters (Alves Batista et al. 2018). To calculate the neutrino flux, the CRPropa 3 code integrates a rela-tion similar to equation (6) for neutrino species, and the procedure isthe same as described in Section 3.3 .In general, neutrino production occurs mainly due to photopionproduction and pp-interactions. In Fig. 8, where we show 𝜆 for dif-ferent interactions, we see that protons with energies 𝐸 < eV produce neutrinos principally due to pp-interactions, while for 𝐸 > eV, they produce neutrinos both, by pp-interactions andphotopion process. We have also seen in Fig. 8 (lower panel) thatthe total trajectory length of CRs inside a cluster is comparable orlarger than 𝜆 for these interactions and thus, neutrino production isinevitable.In Fig. 12 we show the dependence of the neutrino flux with theposition of the corresponding CR source within clusters of differentmasses. As in the case of the CR flux, it can be seen that there isless neutrino production for the injection position at 1 Mpc awayfrom the center of the cluster. Furthermore, massive clusters producemore neutrinos than the light ones. In Fig. 13 we present the redshiftdistribution of neutrinos as a function of their energy, as observedat a distance of 2 Mpc from the center of individual clusters withdifferent masses.In Fig. 14 & 15, we present the total flux of neutrinos from thewhole population of clusters, as measured at Earth, integrated overthe entire redshift range within the Hubble time (solid brown curvein the panels). In the left panel of Fig. 14 and in Fig. 15, the injectedCR spectrum is assumed to follow 𝐸 − . , with an exponential cut-off 𝐸 max = × eV. Also, we assumed in these cases that 0 .
5% ofthe kinetic energy of the clusters is converted to the CRs. Besidesthe total flux, this panel also shows the flux of neutrinos for several E [eV] E d N / d E [ G e V c m s s t ] M MM MM M Figure 12.
This figure shows the neutrino flux of individual clusters of distinctmasses: 𝑀 ∼ (red); 10 (green) and 10 𝑀 (cid:12) (blue color). The CRsources are located at the center of the cluster (solid lines), at 300 kpc (dashedlines), and at 1 Mpc away from the center (dash-dotted lines). The flux iscomputed at the edge of clusters. The CR injection follows 𝑑𝑁 / 𝑑𝐸 ∝ 𝐸 − , 𝐸 max = × eV, and it is assumed that 2 % of the luminosity of theclusters is converted to CRs. E [eV]10 z M MM MM MM M Figure 13.
Redshift distribution of the neutrinos as a function of their energy,as observed at 2 Mpc away from the center of clusters with different masses. cluster mass intervals. The softening effect at higher energies is dueto the shorter diffusion time of the CRs, and to the mass distributionof the clusters, as higher flux reflects lower population of massiveclusters. In Fig. 15 we present the integrated flux in different redshiftintervals and it can also be seen that the clusters at high redshiftcontribute less to the total flux of neutrinos. Those at 𝑧 >
𝐸 >
20 TeV. In right panelof Fig. 14, instead, we have assumed that 2 % of the kinetic energyof the clusters is converted into CRs, with a CR energy power-lawspectrum 𝐸 − , with 𝐸 max following the dependence below with thecluster mass and magnetic field: 𝐸 max = . × (cid:18) 𝑀 cluster 𝑀 (cid:12) (cid:19) / (cid:18) 𝐵 cluster G10 − G (cid:19) eV , (9)which is similar to Fang & Olinto (2016). In this scenario we find that MNRAS , 1– ????
20 TeV. In right panelof Fig. 14, instead, we have assumed that 2 % of the kinetic energyof the clusters is converted into CRs, with a CR energy power-lawspectrum 𝐸 − , with 𝐸 max following the dependence below with thecluster mass and magnetic field: 𝐸 max = . × (cid:18) 𝑀 cluster 𝑀 (cid:12) (cid:19) / (cid:18) 𝐵 cluster G10 − G (cid:19) eV , (9)which is similar to Fang & Olinto (2016). In this scenario we find that MNRAS , 1– ???? (2020) igh-Energy Neutrinos in Galaxy Clusters the clusters contribution to the neutrino flux is smaller than IceCubemeasurements.For all diagrams of Fig. 14 & 15, we also compare our resultswith those of Fang & Olinto (2016)) (blue lines). The total fluxes inboth are similar, in general.Moreover, we see that in both cases, thelargest contribution to the flux of neutrinos comes from the clustermass group 10 𝑀 (cid:12) < 𝑀 < 𝑀 (cid:12) . However, the contributionfrom the mass group 10 𝑀 (cid:12) < 𝑀 < 𝑀 (cid:12) in our results is afactor twice larger than that of Fang & Olinto (2016), and smallerby the same factor for the mass group 𝑀 > 𝑀 (cid:12) , at energies 𝐸 > .
01 PeV (left panel of Fig. 14).A striking difference between the two results is that, accordingto Fang & Olinto (2016), the redshift range 0 . ≤ 𝑧 ≤ . ≤ 𝑧 ≤ . ∼ ∼ 𝑀 (cid:12) < 𝑀 < 𝑀 (cid:12) ) at high redshifts ( 𝑧 > 𝜓 ev ( 𝑧 ) = 𝑧 < . , . < 𝑧 < .
0, and1 . < 𝑧 < .
0. The flux is obtained for spectral index 𝛼 = 𝐸 max = × eV.Clusters can directly accelerate CRs through shocks, but any typeof astrophysical object that can produce HECRs can also contributeto the diffuse neutrino flux. In the former case, the sources evolveonly according to the background MHD simulations, dubbed here“no evolution”, whereas in the latter some assumptions have to bemade regarding the CR sources. In Fig. 18 we illustrate the impactof the source evolution. We consider, in addition to the case whereinsources do not evolve, SFR and AGN-like evolutions (see equations 5and 4 and accompanying discussion). Our results suggest that, whilethe neutrino fluxes for the AGN and the SFR evolutions are relativelyclose to each other, the case without evolution contributes slightlyless to the total flux. Moreover, at high redshifts (1 . < 𝑧 < . 𝑧 (cid:46) 𝛼 and 𝐸 𝑚𝑎𝑥 , with different source evolution assumptions as inFig. 18. In both panels all the combinations of 𝛼 and 𝐸 max are roughlymatching with IceCube data, except 𝛼 = .
5, and 𝐸 𝑚𝑎𝑥 = × eVin the upper panel as it overshoots the IceCube points. In our simulations, the central magnetic field strength and gas numberdensity of the ICM are ∼ 𝜇 G and ∼ − cm − , respectively, fora cluster with mass 10 𝑀 (cid:12) at 𝑧 = .
01, and both decrease towardthe outskirts of the cluster. These quantities depend on the mass ofthe clusters, being smaller for less massive clusters (see Fig. 2 & 5).Thus, high-energy CRs will escape with a higher probability without much interactions in the case of less massive clusters. Lower-energyCRs, on the other hand, contribute less to the production of high-energy neutrinos. Therefore, we have a lower neutrino flux from lessmassive clusters. In contrast, for massive clusters, higher magneticfield and gas density produce higher neutrino flux due to the longerconfinement time, as we see in Fig. 12.We tested several injection CRs spectral indices ( 𝛼 (cid:39) . − . 𝐸 max = × − eV), and source evolution(AGN, SFR, no evolution), in order to try to interpret the IceCube data(see Figs. 14, 15, 16, 17, 18 and 19). Overall, our results indicate thatgalaxy clusters can contribute to a considerable fraction of the diffuseneutrino flux measured by IceCube at energies between 100 TeV and10 PeV, or even all of it, provided that that protons compose most ofthe CRs.Our results also look, in principle, similar to those of Fang &Olinto (2016) with no source evolution, who considered essentiallythe same redshift interval, but employed semi-analytical profiles todescribe the cluster properties. In particular, in both cases, the largestcontribution to the flux of neutrinos comes from the cluster massgroup 10 < 𝑀 < 𝑀 (cid:12) . However, they did not consider theinteractions of CRs with CMB and EBL background as they consid-ered it subdominant compared to the hadronic background followingKotera et al. (2009). But, it can be seen from the upper panel of Fig. 8that 𝜆 for pp-interaction and photopion production in the CMB arecomparable for CRs of energy (cid:38) eV. Therefore, the neutrinoproduction due to CR interactions with the CMB is not negligible.Perhaps the most relevant difference between our results and theirsis that, in their case, the redshift range 0 . ≤ 𝑧 ≤ 𝑧 (cid:46) .
3, when considering no sourceevolution (see Fig 15).When including source evolution, there is also a dominance inthe neutrino flux from the redshift range 𝑧 (cid:46) .
3, though the con-tribution due to the evolution of star forming galaxies (SFR) fromredshifts 0 . ≤ 𝑧 ≤ ∼ ∼ 𝐸 (cid:38) eV and are in roughaccordance with (Murase 2017; Fang & Murase 2018). Neverthe-less, since there are uncertainties related to the choice of specificpopulations for the CR sources, obtaining a full picture of the diffusehigh-energy neutrino emission by clusters is not a straightforwardtask.It is also worth comparing our results with Zandanel et al. (2015),who evaluated the neutrino spectrum based on estimations of theradio to gamma-ray luminosities of the clusters in the universe. Al-though our work has assumed an entirely different approach, bothresults are consistent, especially for a CR spectral index 𝛼 (cid:39) 𝐸 > eV) CRs can escape easily from clusters,effectively leading to a spectral steepening that was not consideredby Zandanel et al. (2015). However, not all the clusters are expectedto produce hadronic emission (Zandanel et al. 2015, 2014). In fact,we observe less hadronic interactions in the case of low-mass clusters( 𝑀 (cid:46) 𝑀 (cid:12) ), which could further limit the neutrino contributionfrom clusters.The cluster scenario may get strong backing due to anisotropydetections above PeV energies. Recently, only a few sources of high-energy neutrinos have been observed (Aartsen et al. 2013, 2015;Albert et al. 2018; Ansoldi et al. 2018; Aartsen et al. 2020), but MNRAS , 1– ?? (2020) S. Hussain et al. E [eV] E d N / d E [ G e V c m s s t ] total (this work)total (Fang & Olinto) M > 10 M M < M < 10 M M < M < 10 M IceCube HESE (2015) E [eV] E d N / d E [ G e V c m s s t ] total (this work)total (Fang & Olinto) M > 10 M M < M < 10 M M < M < 10 M IceCube HESE (2015)
Figure 14.
Neutrino spectrum at Earth obtained using our simulations (brown lines), compared with the IceCube data (markers), and Fang & Olinto (2016)results (blue lines). The panels show the total flux integrated over all clusters and redshifts between 0 . ≤ 𝑧 ≤ 𝑀 (cid:12) < 𝑀 < 𝑀 (cid:12) (dash-dotted), 10 𝑀 (cid:12) < 𝑀 < 𝑀 (cid:12) (dashed), and 𝑀 > 𝑀 (cid:12) (dotted lines). The left panel corresponds to the case with 𝛼 = . 𝐸 max = × eV, whereas in the right panel 𝛼 = − 𝐸 max followsequation (9). These diagrams do not include the redshift evolution of the CR sources, 𝜓 𝑒𝑣 = E [eV] E d N / d E [ G e V c m s s t ] total (this work)total (Fang & Olinto)0.01 < z < 0.3 0.3 < z < 1.01.0 < z < 5.0IceCube HESE (2015) Figure 15.
This figure shows the neutrino spectrum for different redshiftranges: 𝑧 < . . < 𝑧 < . . < 𝑧 < . 𝑑𝑁 / 𝑑𝐸 ∝ 𝐸 − . , and 𝐸 max = × eV.This figure does not include the redshift evolution of the CR sources, 𝜓 𝑒𝑣 = there are also expectations to increase the observations with futureinstruments like IceCube-Gen2 (The IceCube-Gen2 Collaboration2020), KM3NeT (Adrián-Martínez et al. 2016), and the Giant RadioArray for Neutrino Detection (GRAND) (Álvarez-Muñiz et al. 2020).Specifically, neutrinos from clusters are more likely to be observedif the flux of cosmogenic neutrinos is low, which might contaminatethe signal, as discussed by Alves Batista et al. (2019b). E [eV] E d N / d E [ G e V c m s s t ] = 1.5= 1.9= 2.3= 2.7IceCube HESE (2015) Figure 16.
Total spectrum of neutrinos for different injected CR spectra, ∼ 𝐸 − 𝛼 , with 𝛼 = . . . . 𝐸 max = × eV. This figure does not include the redshift evolution of theCR sources, 𝜓 𝑒𝑣 = We considered a cosmological background based on 3D-MHD sim-ulations to model the cluster population of the entire universe, anda multidimensional Monte Carlo technique to study the propagationof CRs in this environment and obtain the flux of neutrinos theyproduce. Our results can be summarized as follows: • We found that CRs with energy 𝐸 (cid:46) eV cannot escapefrom the innermost regions of the clusters, due to interactions withthe background gas, thermal photons and magnetic fields. Massiveclusters ( 𝑀 (cid:38) 𝑀 (cid:12) ) have stronger magnetic fields which can MNRAS , 1– ????
Total spectrum of neutrinos for different injected CR spectra, ∼ 𝐸 − 𝛼 , with 𝛼 = . . . . 𝐸 max = × eV. This figure does not include the redshift evolution of theCR sources, 𝜓 𝑒𝑣 = We considered a cosmological background based on 3D-MHD sim-ulations to model the cluster population of the entire universe, anda multidimensional Monte Carlo technique to study the propagationof CRs in this environment and obtain the flux of neutrinos theyproduce. Our results can be summarized as follows: • We found that CRs with energy 𝐸 (cid:46) eV cannot escapefrom the innermost regions of the clusters, due to interactions withthe background gas, thermal photons and magnetic fields. Massiveclusters ( 𝑀 (cid:38) 𝑀 (cid:12) ) have stronger magnetic fields which can MNRAS , 1– ???? (2020) igh-Energy Neutrinos in Galaxy Clusters E [eV] E d N / d E [ G e V c m s s t ] E max = 5 × 10 eV E max = 10 eV E max = 10 eV E max = 5 × 10 eVIceCube HESE (2015) E [eV] E d N / d E [ G e V c m s s t ] E max = 5 × 10 eV E max = 10 eV E max = 10 eV E max = 5 × 10 eVIceCube HESE (2015) Figure 17.
Total neutrino spectrum for different cutoff energies i.e., 𝐸 𝑚𝑎𝑥 = × (red), 10 (green), 10 (orange), and 5 × eV (blue). In theupper panel the spectral index is 𝛼 =
2, and in lower panel 𝛼 = .
5. Thisfigure does not include the redshift evolution of the CR sources, 𝜓 𝑒𝑣 = confine these high-energy CRs for a time comparable to the age ofthe universe. • Our simulations predict that the neutrino flux above PeV ener-gies comes from the most massive clusters because the CR interac-tions with the gas of the ICM are rare for clusters with
𝑀 < 𝑀 (cid:12) . • Most of the neutrino flux comes from nearby clusters in theredshift range 𝑧 (cid:46) .
3. The high-redshif clusters contribute less tothe total flux of neutrinos compared to the low-redshift ones, as thepopulation of massive clusters at high redshifts is low. • The total integrated neutrino flux obtained from the interactionsof CRs with the ICM gas and CMB during their propagation inthe turbulent magnetic field can account for sizeable percentage ofthe IceCube observations, especially, between energy 100 TeV and10 PeV. • Our results also indicate that the redshift evolution of CR sourceslike AGN and SFR, enhance the flux of neutrinos.Finally, more realistic studies considering cosmological simula-tions that account for AGN and star formation feedback from galax- E [eV] E d N / d E [ G e V c m s s t ] total (no evolution)total (SFR)total (AGN) z < 0.3 0.3 < z < 1.01.0 < z < 5.0IceCube HESE (2015) Figure 18.
Neutrino spectrum for different assumptions on the evolution of theCR sources: SFR (blue), AGN (green), and no evolution (brown). The fluxesare shown for different redshift ranges: 𝑧 < . . < 𝑧 < . . < 𝑧 < . 𝛼 = 𝐸 max = × eV. ies ( e.g. Barai et al. 2016; Barai & de Gouveia Dal Pino 2019)will allow to constrain better the redshift evolution of the CR sourcesin the computation of the total neutrino flux from clusters. Fur-thermore, in the future, IceCube will have detected more events.Then, combined with diffuse gamma-ray searches by the forthcom-ing CTA (Cherenkov Telescope Array Consortium et al. 2019), itwill be possible to better assess the contribution of galaxy clusters tothe total extragalactic neutrino flux. ACKNOWLEDGEMENTS
Saqib Hussain acknowledges support from the Brazilian fundingagency CNPq. EMdGDP is also grateful for the support of theBrazilian agencies FAPESP (grant 2013/10559-5) and CNPq (grant308643/2017-8). RAB is currently funded by the Radboud Excel-lence Initiative, and received support from FAPESP in the early stagesof this work (grant 17/12828-4). KD acknowledges support by theDeutsche Forschungsgemeinschaft (DFG, German Research Founda-tion) under Germany’s Excellence Strategy – EXC-2094 – 39078331and by the funding for the COMPLEX project from the EuropeanResearch Council (ERC) under the European Union’s Horizon 2020research and innovation program grant agreement ERC-2019-AdG860744. The numerical simulations presented here were performedin the cluster of the Group of Plasmas and High-Energy Astrophysics(GAPAE), acquired with support from FAPESP (grant 2013/10559-5). This work also made use of the computing facilities of the Labora-tory of Astroinformatics (IAG/USP, NAT/Unicsul), whose purchasewas also made possible by a FAPESP (grant 2009/54006-4). We alsoacknowledge very useful comments from K. Murase on an earlierversion of this manuscript.
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Aab A., et al., 2018, The Astrophysical Journal, 868, 4Aartsen M. G., et al., 2013, Physical review letters, 111, 021103MNRAS , 1– ?? (2020) S. Hussain et al. E [eV] E d N / d E [ G e V c m s s t ] AGN + SFRAGNSFR no evolutionIceCube HESE (2015) E [eV] E d N / d E [ G e V c m s s t ] AGN + SFRAGNSFR no evolutionIceCube HESE (2015)
Figure 19.
Flux of neutrinos for different assumptions on the evolution ofthe CR sources: no evolution (solid lines), SFR (dashed lines), AGN (dottedlines) and AGN + SFR (dash-dotted lines). In upper panel green and redlines represent 𝛼 = . 𝐸 max = and 5 × eV respectively. Inlower panel orange and blue lines correspond to 𝛼 = 𝐸 max = and5 × eV, respectively.Aartsen M., et al., 2015, Physical Review D, 91, 022001Aartsen M., et al., 2017, The Astrophysical Journal, 835, 45Aartsen M., et al., 2020, Physical review letters, 124, 051103Ackermann M., et al., 2019, arXiv preprint arXiv:1903.04334Ade P. A., et al., 2014, Astronomy & Astrophysics, 571, A16Adrián-Martínez S., et al., 2016, Journal of Physics G Nuclear Physics, 43,084001Ahlers M., Halzen F., 2018, Progress in Particle and Nuclear Physics, 102,73Albert A., et al., 2018, The Astrophysical Journal Letters, 863, L30Aloisio R., Berezinsky V., Gazizov A., 2012, Astroparticle Physics, 39, 129Álvarez-Muñiz J., et al., 2020, Science China Physics, Mechanics & Astron-omy, 63, 219501Alves Batista R., et al., 2016, Journal of Cosmology and Astroparticle Physics,2016, 038Alves Batista R., Pino E., Dolag K., Hussain S., 2018, arXiv preprintarXiv:1811.03062Alves Batista R., et al., 2019a, Frontiers in Astronomy and Space Sciences,6, 23 Alves Batista R., de Almeida R. M., Lago B., Kotera K., 2019b, Journal ofCosmology and Astroparticle Physics, 2019, 002Amato E., Blasi P., 2018, Advances in Space Research, 62, 2731Anchordoqui L. A., Paul T. C., da Silva L. H., Torres D. F., Vlcek B. J., 2014,Physical Review D, 89, 127304Ansoldi S., et al., 2018, The Astrophysical Journal Letters, 863, L10Apel W., et al., 2013, Astroparticle Physics, 47, 54Barai P., de Gouveia Dal Pino E. M., 2019, MNRAS, 487, 5549Barai P., Murante G., Borgani S., Gaspari M., Granato G. L., Monaco P.,Ragone-Figueroa C., 2016, MNRAS, 461, 1548Berezinsky V. S., Blasi P., Ptuskin V., 1997, The Astrophysical Journal, 487,529Blasi P., 2013, The Astronomy and Astrophysics Review, 21, 70Brunetti G., Jones T. W., 2014, International Journal of Modern Physics D,23, 1430007Brunetti G., Vazza F., 2020, Physical Review Letters, 124, 051101Brunetti G., Zimmer S., Zandanel F., 2017, Monthly Notices of the RoyalAstronomical Society, 472, 1506Chakraborty S., Izaguirre I., 2015, Physics Letters B, 745, 35Cherenkov Telescope Array Consortium et al., 2019, Science with theCherenkov Telescope Array, doi:10.1142/10986.Dolag K., Grasso D., Springel V., Tkachev I., 2005, Journal of Cosmologyand Astroparticle Physics, 2005, 009Dominguez A., et al., 2011, Monthly Notices of the Royal AstronomicalSociety, 410, 2556D’Amato Q., et al., 2020, Astronomy & Astrophysics, 636, A37Fang K., Murase K., 2018, Nature Physics, 14, 396Fang K., Olinto A. V., 2016, The Astrophysical Journal, 828, 37Ferrari C., Govoni F., Schindler S., Bykov A., Rephaeli Y., 2008, in , Clustersof Galaxies. Springer, pp 93–118Gelmini G. B., Kalashev O., Semikoz D. V., 2012, Journal of Cosmology andAstroparticle Physics, 2012, 044Giacinti G., Kachelrieß M., Semikoz D., 2015, Physical Review D, 91, 083009Gilmore R., Somerville R., Primack J., Domínguez A., 2012, Not. Roy. As-tron. Soc, 422, 1104Gonzalez A. H., Sivanandam S., Zabludoff A. I., Zaritsky D., 2013, TheAstrophysical Journal, 778, 14Gouin C., Aghanim N., Bonjean V., Douspis M., 2020, Astronomy & Astro-physics, 635, A195Govoni F., et al., 2019, Science, 364, 981Hasinger G., Miyaji T., Schmidt M., 2005, Astronomy & Astrophysics, 441,417He H.-N., Wang T., Fan Y.-Z., Liu S.-M., Wei D.-M., 2013, Physical ReviewD, 87, 063011Heinze J., Boncioli D., Bustamante M., Winter W., 2016, The AstrophysicalJournal, 825, 122Hopkins A. M., Beacom J. F., 2006, The Astrophysical Journal, 651, 142Hümmer S., Baerwald P., Winter W., 2012, Physical Review Letters, 108,231101Hussain S., Alves Batista R., Dal Pino E. M. d. G., 2019, in ICRC. p. 81Kachelriess M., 2019, in EPJ Web of Conferences. p. 04003Kafexhiu E., Aharonian F., Taylor A. M., Vila G. S., 2014, Physical ReviewD, 90, 123014Kashiyama K., Mészáros P., 2014, The Astrophysical Journal Letters, 790,L14Khiali B., de Gouveia Dal Pino E. M., 2016, MNRAS, 455, 838Kim J., Ryu D., Kang H., Kim S., Rey S.-C., 2019, Science advances, 5,eaau8227Komatsu E., et al., 2009, The Astrophysical Journal Supplement Series, 180,330Kotera K., Allard D., Murase K., Aoi J., Dubois Y., Pierog T., Nagataki S.,2009, The Astrophysical Journal, 707, 370Liu R.-Y., Wang X.-Y., 2013, The Astrophysical Journal, 766, 73Merten L., Tjus J. B., Fichtner H., Eichmann B., Sigl G., 2017, Journal ofCosmology and Astroparticle Physics, 2017, 046Moriya T. J., et al., 2019, The Astrophysical Journal Supplement Series, 241,16MNRAS , 1– ????
Flux of neutrinos for different assumptions on the evolution ofthe CR sources: no evolution (solid lines), SFR (dashed lines), AGN (dottedlines) and AGN + SFR (dash-dotted lines). In upper panel green and redlines represent 𝛼 = . 𝐸 max = and 5 × eV respectively. Inlower panel orange and blue lines correspond to 𝛼 = 𝐸 max = and5 × eV, respectively.Aartsen M., et al., 2015, Physical Review D, 91, 022001Aartsen M., et al., 2017, The Astrophysical Journal, 835, 45Aartsen M., et al., 2020, Physical review letters, 124, 051103Ackermann M., et al., 2019, arXiv preprint arXiv:1903.04334Ade P. A., et al., 2014, Astronomy & Astrophysics, 571, A16Adrián-Martínez S., et al., 2016, Journal of Physics G Nuclear Physics, 43,084001Ahlers M., Halzen F., 2018, Progress in Particle and Nuclear Physics, 102,73Albert A., et al., 2018, The Astrophysical Journal Letters, 863, L30Aloisio R., Berezinsky V., Gazizov A., 2012, Astroparticle Physics, 39, 129Álvarez-Muñiz J., et al., 2020, Science China Physics, Mechanics & Astron-omy, 63, 219501Alves Batista R., et al., 2016, Journal of Cosmology and Astroparticle Physics,2016, 038Alves Batista R., Pino E., Dolag K., Hussain S., 2018, arXiv preprintarXiv:1811.03062Alves Batista R., et al., 2019a, Frontiers in Astronomy and Space Sciences,6, 23 Alves Batista R., de Almeida R. M., Lago B., Kotera K., 2019b, Journal ofCosmology and Astroparticle Physics, 2019, 002Amato E., Blasi P., 2018, Advances in Space Research, 62, 2731Anchordoqui L. A., Paul T. C., da Silva L. H., Torres D. F., Vlcek B. J., 2014,Physical Review D, 89, 127304Ansoldi S., et al., 2018, The Astrophysical Journal Letters, 863, L10Apel W., et al., 2013, Astroparticle Physics, 47, 54Barai P., de Gouveia Dal Pino E. M., 2019, MNRAS, 487, 5549Barai P., Murante G., Borgani S., Gaspari M., Granato G. L., Monaco P.,Ragone-Figueroa C., 2016, MNRAS, 461, 1548Berezinsky V. S., Blasi P., Ptuskin V., 1997, The Astrophysical Journal, 487,529Blasi P., 2013, The Astronomy and Astrophysics Review, 21, 70Brunetti G., Jones T. W., 2014, International Journal of Modern Physics D,23, 1430007Brunetti G., Vazza F., 2020, Physical Review Letters, 124, 051101Brunetti G., Zimmer S., Zandanel F., 2017, Monthly Notices of the RoyalAstronomical Society, 472, 1506Chakraborty S., Izaguirre I., 2015, Physics Letters B, 745, 35Cherenkov Telescope Array Consortium et al., 2019, Science with theCherenkov Telescope Array, doi:10.1142/10986.Dolag K., Grasso D., Springel V., Tkachev I., 2005, Journal of Cosmologyand Astroparticle Physics, 2005, 009Dominguez A., et al., 2011, Monthly Notices of the Royal AstronomicalSociety, 410, 2556D’Amato Q., et al., 2020, Astronomy & Astrophysics, 636, A37Fang K., Murase K., 2018, Nature Physics, 14, 396Fang K., Olinto A. V., 2016, The Astrophysical Journal, 828, 37Ferrari C., Govoni F., Schindler S., Bykov A., Rephaeli Y., 2008, in , Clustersof Galaxies. Springer, pp 93–118Gelmini G. B., Kalashev O., Semikoz D. V., 2012, Journal of Cosmology andAstroparticle Physics, 2012, 044Giacinti G., Kachelrieß M., Semikoz D., 2015, Physical Review D, 91, 083009Gilmore R., Somerville R., Primack J., Domínguez A., 2012, Not. Roy. As-tron. Soc, 422, 1104Gonzalez A. H., Sivanandam S., Zabludoff A. I., Zaritsky D., 2013, TheAstrophysical Journal, 778, 14Gouin C., Aghanim N., Bonjean V., Douspis M., 2020, Astronomy & Astro-physics, 635, A195Govoni F., et al., 2019, Science, 364, 981Hasinger G., Miyaji T., Schmidt M., 2005, Astronomy & Astrophysics, 441,417He H.-N., Wang T., Fan Y.-Z., Liu S.-M., Wei D.-M., 2013, Physical ReviewD, 87, 063011Heinze J., Boncioli D., Bustamante M., Winter W., 2016, The AstrophysicalJournal, 825, 122Hopkins A. M., Beacom J. F., 2006, The Astrophysical Journal, 651, 142Hümmer S., Baerwald P., Winter W., 2012, Physical Review Letters, 108,231101Hussain S., Alves Batista R., Dal Pino E. M. d. G., 2019, in ICRC. p. 81Kachelriess M., 2019, in EPJ Web of Conferences. p. 04003Kafexhiu E., Aharonian F., Taylor A. M., Vila G. S., 2014, Physical ReviewD, 90, 123014Kashiyama K., Mészáros P., 2014, The Astrophysical Journal Letters, 790,L14Khiali B., de Gouveia Dal Pino E. M., 2016, MNRAS, 455, 838Kim J., Ryu D., Kang H., Kim S., Rey S.-C., 2019, Science advances, 5,eaau8227Komatsu E., et al., 2009, The Astrophysical Journal Supplement Series, 180,330Kotera K., Allard D., Murase K., Aoi J., Dubois Y., Pierog T., Nagataki S.,2009, The Astrophysical Journal, 707, 370Liu R.-Y., Wang X.-Y., 2013, The Astrophysical Journal, 766, 73Merten L., Tjus J. B., Fichtner H., Eichmann B., Sigl G., 2017, Journal ofCosmology and Astroparticle Physics, 2017, 046Moriya T. J., et al., 2019, The Astrophysical Journal Supplement Series, 241,16MNRAS , 1– ???? 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APPENDIX A: MEAN FREE PATHS
The mean free path 𝜆 for different CR interactions in the ICM aredefined below.For a CR proton with Lorentz factor 𝛾 𝑝 traversing an isotropicphoton field, one obtains the rate 𝜆 − 𝑝𝛾 ( 𝐸 𝑝 ) (Schlickeiser 2002) 𝜆 − 𝑝𝛾 ( 𝐸 𝑝 ) = 𝛾 𝑝 ∞ ∫ 𝜖 th / 𝛾 𝑝 𝑑𝜖 𝑛 ph ( 𝜖, 𝑟 𝑖 ) 𝜖 𝛾 𝑝 𝜖 ∫ 𝜖 th 𝑑𝜖 (cid:48) 𝜖 (cid:48) 𝜎 𝑝𝛾 ( 𝜖 (cid:48) ) 𝐾 𝑝 ( 𝜖 (cid:48) ) , (A1) 𝜖 th = 𝐾𝑚 𝜋 𝑐 (cid:20) + 𝐾𝑚 𝜋 𝑚 𝑝 (cid:21) =
145 MeV . (A2)Where 𝑛 ph ( 𝜖, 𝑟 𝑖 ) denotes the number density of photons of energy 𝜖 at a given distance 𝑟 𝑖 from the center of the cluster and 𝜎 𝑝𝛾 is thecross section of the interaction of CRs with background photons. Thethreshold energy for the production of 𝐾 pions is given by equation(A2), so that for the production of a single ( 𝐾 =
1) pion the restsystem threshold energy is 𝜖 th =
145 MeV (Schlickeiser 2002).To calculate the rate for the interactions of high-energy photons(produced during the propagation of CRs inside a cluster) with thelocal protons in the ICM, we can use equation (A1) with the followingmodification in the center-of-mass (CM) energy. The energy 𝐸 and3-momentum p of a particle of mass 𝑚 form a 4-vector 𝑝 = ( 𝐸, 𝑝 ) whose square 𝑝 = ( 𝐸 / 𝑐 ) − p = 𝑚 𝑐 . The velocity of the particleis 𝛽𝑐 = v / 𝑐 = p / 𝐸 . In the collision of two particles of masses 𝑚 and 𝑚 , the total CM energy can be expressed in the Lorentz-invariantform as 𝜖 𝐶𝑀 = (cid:20) 𝑐 ( 𝐸 + 𝐸 ) − ( p + p ) 𝑐 (cid:21) / (A3) = (cid:20) 𝑚 𝑐 + 𝑚 𝑐 + 𝐸 𝐸 𝑐 ( − 𝛽 𝛽 cos 𝜃 ) (cid:21) / , (A4)where 𝜃 is the angle between the particles that we can consider zero.In the frame where one particle (of mass 𝑚 ) is at rest (lab frame)then, 𝜖 𝐶𝑀 = ( 𝑚 𝑐 + 𝑚 𝑐 + 𝐸 𝑚 𝑐 ) / . (A5)If we consider 𝑚 is proton and 𝑚 is photon, then the above relationbecomes 𝜖 𝐶𝑀 = ( 𝑚 𝑐 + 𝐸 𝑚 𝑐 ) / . (A6) 𝜆 − 𝛾 𝑝 ( 𝜖 ph ) = 𝜖 𝑝 𝜖 ph ∞ ∫ 𝜖 th /( 𝜖 ph / 𝜖 𝑝 ) 𝑑𝜖 𝑛 𝑝 ( 𝜖, 𝑟 𝑖 ) 𝜖 𝑝 ( 𝜖 ph / 𝜖 𝑝 ) 𝜖 ∫ 𝜖 th 𝑑𝜖 (cid:48) 𝜖 (cid:48) 𝜎 𝛾 𝑝 ( 𝜖 (cid:48) ) , (A7)so that the rest frame is in the local protons. We used equation (A6) forthe energy of the CM in equation (A7). In equation (A7), 𝑛 𝑝 ( 𝜖, 𝑟 𝑖 ) isnumber density of local protons with energy 𝜖 𝑝 = 𝑚 𝑝 𝑐 ∼ 𝑟 𝑖 from the center of a cluster and decreases toward theoutskirt, 𝜖 th ∼ . × eV is the threshold energy for this interactionand the cross section 𝜎 𝛾 𝑝 ( 𝜖 (cid:48) ) is of the order ∼ − ( cm ) . Withthese values used in equation (A7) we solve this integral to calculate 𝜆 for 𝛾 -proton interaction. We calculated 𝜆 from equations A1-A7with some modifications to include the information of the spatiallydependent Bremsstrahlung photon field of the clusters 𝑛 ph ( 𝜖, 𝑟 ) .For proton-proton (pp) interaction, the rate is given by 𝜆 − ( 𝐸 𝑝 , 𝑟 𝑖 ) = 𝐾 pp 𝜎 pp ( 𝐸 𝑝 ) 𝑛 𝑖 ( 𝑟 𝑖 ) (A8)Where 𝐾 pp = . 𝑛 𝑖 ( 𝑟 𝑖 ) denotes the numberdensity of proton at a given distance 𝑟 𝑖 from the center of the clusterand 𝐸 𝑝 is the energy of the protons.To obtain the proton number density, we consider that the back-ground plasma consists of electrons and protons in near balancing.Since the abundance is mostly of H and this is mostly ionized inthe hot ICM, this is a reasonable assumption. Thus 𝑛 𝑝 (cid:39) 𝑛 𝑒 , and 𝜌 gas = 𝑛 𝑝 𝑚 𝑝 + 𝑛 𝑒 𝑚 𝑒 ∼ 𝑛 𝑝 𝑚 𝑝 , so that 𝑛 𝑖 (cid:39) 𝑛 𝑒 (cid:39) 𝜌 gas / 𝑚 𝑝 , where 𝑚 𝑝 is the proton mass and 𝜌 gas is the gas mass density in the system.For 𝜎 pp =
70 mb (1barn = − m ), we have for the crosssection (Kafexhiu et al. 2014): 𝜎 pp = (cid:34) . − .
96 log (cid:32) 𝐸 𝑝 𝐸 th 𝑝 (cid:33) + .
18 log (cid:32) 𝐸 𝑝 𝐸 th 𝑝 (cid:33)(cid:35) − (cid:32) 𝐸 th 𝑝 𝐸 𝑝 (cid:33) . mb , (A9)where 𝐸 𝑝 is the energy of the proton and 𝐸 th 𝑝 is the threshold kineticenergy 𝐸 th 𝑝 = 𝑚 𝜋 + 𝑚 𝜋 / 𝑚 𝑝 ≈ . 𝜆 pp . MNRAS , 1– ?? (2020) S. Hussain et al.
APPENDIX B: SPECTRAL INDEX
To calculate the flux of neutrinos corresponding to injected CRswith an arbitrary power-law spectrum with power law index 𝛼 , 𝑑𝑁 CR , 𝐸 / 𝑑𝐸 ∝ 𝐸 − 𝛼𝑖 exp {− 𝐸 𝑖 / 𝐸 max } , we can normalize the spec-trum as follows: 𝐽 ( 𝛼 ) = ln ( 𝐸 CR , max / 𝐸 min ) 𝐸 CR, max ∫ 𝐸 min 𝐸 − 𝛼𝑖 exp (cid:16) − 𝐸 𝑖 𝐸 max (cid:17) 𝑑𝐸 𝐸 − 𝛼𝑖 exp (cid:18) − 𝐸 𝑖 𝐸 max (cid:19) (B1)Where, 𝐸 𝑖 is the injection energy of the simulated CRs, 𝐸 max is theexponential cut-off energy, and 𝐸 CR, max is the maximum injectionenergy of the CRs.
MNRAS , 1– ????