On the Challenges of Cosmic-Ray Proton Shock Acceleration in the Intracluster Medium
OOn the Challenges of Cosmic-Ray Proton ShockAcceleration in the Intracluster Medium
Denis Wittor
Hamburger Sternwarte, Gojenbergsweg 112, D-21029 Hamburg, Germany
Abstract
Galaxy clusters host the largest particle accelerators in the Universe: Shock waves inthe intracluster medium (ICM), a hot and ionised plasma, that accelerate particles tohigh energies. Radio observations pick up synchrotron emission in the ICM, provingthe existence of accelerated cosmic-ray electrons. However, a sign of cosmic-ray pro-tons, in form of γ -rays. remains undetected. This is know as the missing γ -ray problem and it directly challenges the shock acceleration mechanism at work in the ICM.Over the last decade, theoretical and numerical studies focused on improving ourknowledge on the microphysics that govern the shock acceleration process in the ICM.These new models are able to predict a γ -ray signal, produced by shock acceleratedcosmic-ray protons, below the detection limits set modern γ -ray observatories. In thisreview, we summarise the latest advances in solving the missing γ -ray problem. Keywords: intracluster medium; shock waves; cosmic-ray acceleration
1. Introduction
Shock waves in the intracluster medium (ICM), the hot and ionised high β plasmainside galaxy clusters, belong to the largest sites of cosmic-ray acceleration. Galaxyclusters are gravitationally bound systems with masses of ∼ M (cid:12) . However, onlya small portion of their mass, (cid:46) few % , consists of galaxies and the main mass con-tributor is dark matter, ∼
85 % . The remaining mass, ∼ −
15 % , resides in ICM.During the processes of hierarchical structure formation, galaxy clusters form through Here, β is the plasma β that gives the ratio between gas pressure and magnetic pressure. Preprint submitted to Journal of L A TEX Templates February 17, 2021 a r X i v : . [ a s t r o - ph . H E ] F e b igure 1: Results from Wittor et al. (2020): the two panels show the expected γ -ray emission (color) of twosimulated galaxy clusters at redshift z = 0 . The underlying cosmic-ray model is the most restricted one, i.e.only supercritical and quasi-parallel shocks are allowed to accelerate cosmic-ray protons (see Sec. 3). Thewhite contours five the baryonic density of the ICM at (cid:2) − . , − , − . & 10 − . (cid:3) g / cm .The red circle marks the r of each cluster, i.e. the radius that encloses 200 times the critical density. the accretion of matter and the merging with other clusters (e.g. Planelles et al., 2015).These highly energetic processes drive both shock waves and turbulence in the ICM,giving rise to large acceleration sites for cosmic-rays. Radio observations of diffusesynchrotron emission proof the acceleration of cosmic-ray electrons in the ICM aswell as the existence of large-scale magnetic fields (e.g. Ferrari et al., 2008; Ferettiet al., 2012; van Weeren et al., 2019). Following the definition by (van Weeren et al.,2019), diffuse radio sources in the ICM are classified into three groups: 1) giant ra-dio halos and mini-haloes, 2) radio relics (or radio shocks) and 3) revived AGN fossilplasma sources. While the origin of the latter sources is still debated, it is commonlyassumed that halos and relics are produced by cosmic-ray electrons that have been(re-)accelerated by ICM turbulence and shocks, respectively. On the other hand, anysignatures of cosmic-ray protons in the ICM are still waiting for their detection. Inthe case of radio halos, the non-detection of cosmic-ray protons favors the turbulentre-acceleration model (e.g. Brunetti et al., 2001; Petrosian, 2001; Donnert et al., 2010,2013; Donnert and Brunetti, 2014; Pinzke et al., 2017, and references therein) over thehadronic (or secondary) models (e.g. Dennison, 1980; Blasi and Colafrancesco, 1999;2frommer et al., 2008; Enßlin et al., 2011, and references therein) for the origin of radiohaloes (e.g. Jeltema and Profumo, 2011; Brunetti et al., 2017). On the other hand, inthe case of radio relics, the non-detection of cosmic-ray protons directly challenges theshock acceleration mechanisms in the ICM (e.g. Vazza and Br¨uggen, 2014). Here, wefocus on the lack of shock-accelerated of cosmic-ray protons and, therefore, we pointthe reader to the references given above for further information on the other topics.The collocation of shock waves, observed in X-rays, and radio relics (e.g. Ogreanand Br¨uggen, 2013; Akamatsu and Kawahara, 2013; Botteon et al., 2016a,b; Hoanget al., 2018) gave rise to the idea that ICM electrons are shock accelerated to highenergies, where they emit synchrotron radiation (Ensslin et al., 1998). The originallyproposed acceleration mechanism is diffusive shock acceleration (DSA, see Blandfordand Ostriker, 1978; Bell, 1978a,b; Drury, 1983; Blandford and Eichler, 1987; Brunettiand Jones, 2014; Bykov et al., 2019; Marcowith et al., 2020, for physical details).While this picture is commonly accepted, some open questions remain. In severalcases, the Mach number estimates from X-ray and radio observations yield differentshock strengths (Stroe et al., 2014; Hong et al., 2015; van Weeren et al., 2016; Hoanget al., 2017). In addition, while the radio power of some relics is explained by theshock acceleration from the thermal pool (Locatelli et al., 2020), several other relicsrequire shock re-acceleration (e.g. Bonafede et al., 2014; van Weeren et al., 2017; Stu-ardi et al., 2019). In this review, we concentrate on the missing γ -ray problem , whichwe summarise in the following.In principle, the protons in the ICM should be accelerated by the same shock wavesthat produce radio relics. Eventually, the accelerated cosmic-ray protons collide withthe thermal protons of the ICM. These inelastic collisions would produce charged andneutral pions that further decay into electrons, positrons, neutrinos and γ -rays (Blasi3nd Colafrancesco, 1999): p + p → π + / − + π + anything (1) π + / − → µ + / − + ν µ (2) µ + / − → e + / − + ¯ ν µ ( ν µ ) + ν e ( ν e ) (3) π → γ. (4)Hence, if the shock waves in the ICM are efficient proton accelerators, galaxy clustersshould be observable in γ -rays , (see Fig. 1). However, such signal has not beendetected so far (e.g. Ackermann et al., 2014, 2015, 2016; Huber et al., 2013a). Thisnon-detection poses a fundamental challenge on the shock acceleration mechanismsin the ICM. Several solutions have been proposed to explain the non-detection of γ -rays: modifying the cosmic-ray proton distribution in clusters (e.g. Enßlin et al., 2011;Wiener et al., 2013; Zandanel et al., 2014; Pinzke et al., 2017), revising the particleacceleration efficiencies (e.g. Vazza and Br¨uggen, 2014; Vazza et al., 2015; Kang et al.,2012; Pinzke et al., 2013) and a better understanding of the microphysical processes ofthe shock acceleration mechanism (e.g. Caprioli and Spitkovsky, 2014a,b,c; Guo et al.,2014a,b; Ha et al., 2018b; Ryu et al., 2019; Kang, 2017; Kang and Ryu, 2018; Kang,2020).Reviewing all of these topics is beyond the scope of this work. Hence, we focuson the advances and implications of the latter, as new theoretical models on the micro-physics at work during the shock acceleration are able to explain the lack of γ -rays.This paper is structured as follows: in Sec. 2, we give an overview of the observationallimits obtained with modern γ -ray observatories. In Sec. 3, we put forward the latestadvances of theoretical and numerical modelling of proton shock acceleration and theireffect on the associated γ -ray emission. Before the conclusion, we briefly give two al-ternative perspectives on the missing γ -ray problem in Sec. 4. In Sec. 5, we concludethe review and give an outlook on the future of γ -ray observations in the ICM. We note that this kind of γ -ray signal is also expected from radio halos, if they are produced by theso-called secondary models (Dennison, 1980). . Observations of γ -rays in the ICM In the search for γ -rays, both space-based and ground-based telescope are used(e.g. Ackermann et al., 2014; Griffin et al., 2014; Zandanel and Ando, 2014; Aharonianet al., 2009; Aleksi´c et al., 2010, 2012, 2014; Ahnen et al., 2016). While ground-basedobservations mostly probe the energy range above (cid:38) , space-based observationsare mostly looking at energies between ∼
100 MeV and ∼ . Theoretical esti-mates yield that the γ -ray emission in the ICM is expected to be the strongest around ∼ (Brunetti et al., 2017; Pinzke and Pfrommer, 2010), hence, favoring space-based observations to detect γ -rays. Though, ground- and space-based observations donot compete but very well complement each other (e.g. Arlen et al., 2012). However,none of them has ever reported the detection of a γ -ray signal from the ICM. Hence,they all provide upper limits on the γ -ray flux expected from cosmic-ray protons in theICM. The deepest upper limits have been set by the Large Area Telescope on board ofthe Fermi Satellite (from here on Fermi-LAT). In the following, we will briefly reviewthese limits and we point to the literature for other works.Since its launch in 2008, the Fermi-LAT has been surveying the γ -ray sky between
20 MeV and
300 GeV (e.g. Atwood et al., 2009; Ackermann et al., 2012, for technicaldetails) and, hence, it perfectly covers the desired energy range around ∼ (e.g.Brunetti et al., 2017; Pinzke and Pfrommer, 2010). Using Fermi-LAT’s 4 year all-sky data, Ackermann et al. (2014) searched for spatially extended γ -ray emission ina sample of 50 X-ray bright galaxy clusters that included both cool-core and non-cool-core clusters. Using the cosmic-ray model by Pinzke and Pfrommer (2010), theythoroughly analysed the sample. Yet, they did not report any γ -ray emission attributedto the ICM. Hence, they presented upper limits for the γ -ray flux above
500 MeV thatare between . and . · − ph cm − s − .More extended searches targeted the γ -ray emission coming from individual clus-ters, namely the Virgo cluster (Ackermann et al., 2015) and the Coma cluster (Ack-ermann et al., 2016). Albeit, these observations used a larger energy range, i.e. ≥ https://fermi.gsfc.nasa.gov/
00 MeV , and, in the case of Coma, 6 years of data, they did not report any γ -ray signal.Hence, the γ -ray flux limits are . · − ph cm − s − and . · − ph cm − s − for Coma and Virgo, respectively. In an independent work, Zandanel and Ando (2014)used 63 months of Fermi-LAT observations to analyse the Coma cluster in the energy
100 MeV to
100 GeV . For different physical scenarios, they found upper limits in therange of − to − ph cm − s − .Huber et al. (2013b) used the stacking of γ -ray counts maps (also see Huber et al.,2012) to estimate statistical upper limits on a sample of 53 galaxy clusters taken fromthe HIFLUGS catalog (Reiprich and B¨ohringer, 2002). For energies between and
300 GeV , they found an upper limit of a few − ph cm − s − for their wholesample . Furthermore, they performed separate stacking analyses for cool-core andnon-cool-core clusters, that yielded upper limits of . and . · − ph cm − s − ,respectively. Griffin et al. (2014) performed an independent stacking analysis using 78nearby clusters. In the . to
100 GeV energy band, they recorded an upper limit of . · − ph cm − s − .Overall, these non-detections of γ -rays produced by inelastic collisions betweencosmic-ray protons and thermal protons in the ICM constrain the total content ofcosmic-ray protons in the ICM. In general, the ratio between the cosmic-ray protonpressure and the total gas pressure is limited to below (Vazza et al., 2016).
3. Advances from theory and simulations
A challenge for theoretical and numerical works is to explain the non-detection of γ -rays. In a different perspective, the above quoted limits on the γ -ray flux provide anideal testcase for shock acceleration models in the ICM. The major challenge of anynew model is to explain the non-detection cosmic-ray protons while still acceleratingenough cosmic-ray electrons to produce visible radio relics.The γ -ray signal strongly depends on the amount of energy injected into the cosmic-ray protons by the shock wave. Hence, by reducing the injected energy, it is possibleto decrease the corresponding γ -ray signal. One possibility to do this is to lower theshock acceleration efficiencies, η . For shock with Mach number M , the acceleration6 igure 2: Results given in Ryu et al. (2019): cosmic-ray acceleration efficiencies as a function of Machnumber. The red and black lines give the efficiencies for injection moments of p inj = 3 . · p th (red) and p inj = 3 . · p th (red), where p th is the momentum of the thermal post-shock electrons. The circles showthe results, if the injection momentum is assumed as a lower bound for computing the injection fraction,while the triangles assume the threshold energy for the pion-production reaction,
780 MeV /c , as a lowerbound. The points at M s = 2 and M s = 1 . are not able to accelerate protons. However, they are plottedfor completeness. efficiency is defined as the ratio of the injected cosmic-ray energy flux, f CR , and thetotal kinetic energy flux dissipated by the shock, f tot , (e.g. Ryu et al., 2003): η ( M ) = f CR ( M ) f tot ( M ) . (5)Using cosmological simulations, Vazza et al. (2016) tested a variety of accelerationefficiencies (that we discuss further below). They found that the efficiencies, availableat that time, were too large and produced visible γ -rays. Furthermore, they showedthat an overall constant acceleration efficiency of η = 10 − would be required to makegalaxy clusters undetectable to current instruments. A word of caution: simply lower-ing the acceleration efficiencies is difficult to be brought in line with the observations7f radio relics. Especially as low acceleration efficiencies already have difficulties inexplaining the radio power of several observed relics (e.g. Bonafede et al., 2014; vanWeeren et al., 2017; Stuardi et al., 2019). Figure 3: Results given in Wittor et al. (2017) and Banfi et al. (2020). Top left: pre-(blue) and post-shock(red) obliquity distribution of the relic simulated in Wittor et al. (2017) compared to the distribution of anglesfor a random distribution. Top right: obliquity distribution measured for different magnetic field seedingmodels (colored lines, see Banfi et al., 2020, for details) compared to the distribution of angles betweentwo random vectors in a three dimensional space (black dotted lines). Bottom panel: spacial distributionof obliquities in two galaxy clusters. The gray color gives the ICM density. The blue squares mark quasi-perpendicular shocks, while the red squares highlight quasi-parallel shocks.
Pioneering works by Kang and Jones (2007) used 1D diffusion convection equa-tions of shocks to estimate the shock acceleration efficiencies. Using a similar model,Enßlin et al. (2007) derived acceleration efficiencies for different post-shock temper-8tures. However, both of these are above the η = 10 − limit given by Vazza et al.(2016). By including more microphysical effects, Kang and Ryu (2013); Caprioli andSpitkovsky (2014a); Ryu et al. (2019) updated the efficiencies given in Kang and Jones(2007), always yielding lower efficiencies. Kang and Ryu (2013) included energy dis-sipation due to Alfven waves that are amplified at the shock. Caprioli and Spitkovsky(2014a) took into account the effect of the shock obliquity on the acceleration process(see discussion below), which gives efficiencies that are half the efficiencies derived byKang and Ryu (2013). Finally, Ryu et al. (2019) also included the dynamical feedbackof cosmic-ray pressure on the shock. This model yields acceleration efficiencies, seeFig. 2, that are lower than the ones of its predecessors and that fall below the η = 10 − limit given by Vazza et al. (2016). Yet, it was shown that these efficiency reduce the γ -ray emission significantly, but not enough to make galaxy clusters invisible in γ -rays(e.g. Wittor et al., 2020, and below).In the recent years, several kinetic simulations found that not all type of shocks areable to efficiently accelerate cosmic-ray protons (Caprioli and Spitkovsky, 2014a; Haet al., 2018b). Shocks can be characterized by two properties: their strength, given bythe Mach number, and their orientation to the underlying magnetic field, given by theshock obliquity. The latter is defined as the angle between shock normal, n shock , andmagnetic field, B : θ = arccos (cid:18) n shock · B | n shock | | B | (cid:19) (6)Using the obliquity, shocks are classified as either quasi-parallel, i.e. θ (cid:46) ◦ , orquasi-perpendicular, i.e. θ (cid:38) ◦ . The Mach number states if a shock is supercritical ornot (e.g. Marshall, 1955). Following the definition given in Ha et al. (2018b), a shockis supercritical if its Mach number is above (cid:38) . .Caprioli and Spitkovsky (2014a) showed that the cosmic-ray proton injection isonly efficient in quasi-parallel shocks. In the picture of DSA, particles gain energywhile they are scattered of magnetic inhomogeneities back and forth across the shockfront. Hence, one of the keys for efficient DSA are the magnetic inhomogeneities in theshock upstream and downstream. However, only at quasi-parallel shocks, cosmic-rayprotons are able to induce magneto-hydrodynamical instabilities that again cause such9nhomogeneities to grow. At quasi-perpendicular shocks, the growth of such instabil-ities is prevented for two reasons (also see the discussion in Caprioli and Spitkovsky,2014a). Most important, protons move only for one gyroradius into the upstream and,hence, the time available for any cosmic-ray driven perturbation to grow is significantlyreduced. Secondly, the particle anisotropies are mostly along the flow, which prohibitsthe growth of most instabilities. However, quasi-perpendicular shocks might still beable to re-accelerate an existing cosmic-ray proton population by DSA.Several works measured the distribution of shock obliquities in cosmological sim-ulations (Wittor et al., 2016, 2017; Roh et al., 2019; Banfi et al., 2020). These worksshowed that, to first order, the distribution of shock obliquities in the ICM follows thedistribution of angles between two random vectors in a three dimensional space, thatis ∝ sin ( θ ) , see in Fig. 3. Hence, only ∼
33 % of all shocks are expected to be quasi-parallel. Additionally, the excess of quasi-perpendicular shocks over quasi-parallelshocks is somewhat larger than for a pure random distribution, as shock waves amplifythe perpendicular component of the magnetic field leading to more quasi-perpendicularshocks (Wittor et al., 2017). Using a larger set of cosmological simulations, Banfi et al.(2020) measured the obliquity distribution in clusters, filaments and voids. They foundthat locally this excess can be up to factors of ∼ larger than for the random case.Furthermore, the excess of quasi-perpendicular shocks is maximized for Mach num-bers with ≤ M ≤ (also see Wittor et al., 2016) but it typically depends on thepre-shock temperature. Furthermore, the excess is maximized around filaments, wherethe shock is perpendicular to the filaments, see Fig. 3. Here, the magnetic field alignswith the filaments because of an equilibrium process that aligns local magnetic fieldwith the density gradient as found by Soler and Hennebelle (2017).However inside clusters, turbulent motions randomize the obliquity distributionagain. Hence, to first order, about ∼
33 % of all shocks are quasi-parallel. Restrict-ing the cosmic-ray proton acceleration to quasi-parallel shocks reduces the associated γ -ray emission by factor of about ∼ . Yet, this drop is not enough to explain thenon-detection of γ -rays (Wittor et al., 2016, 2017, 2020). It is worth to notice, thatthe obliquity requirement does not affect the emission from radio relics. In a comple-mentary work, Guo et al. (2014a) found that cosmic-ray electrons are only efficiently10ccelerated by quasi-perpendicular shocks. Therefore, most shocks in the ICM, i.e. ∼
66 % , are able to accelerate cosmic-ray electrons which leaves the power of relicsalmost unaltered (Wittor et al., 2016, 2017).
Figure 4: Results given in Ha et al. (2019): The various panels compare the γ -ray luminosities, as a functionof cluster mass, of the simulated sample (black dots) with the upper limits produces by the Fermi-LAT(red crosses Ackermann et al., 2014). The luminosities have been computed/measured in the energy band [0 . , . The different panels give the results for different spatial distributions of the cosmic-rayprotons, i.e. n CR ( r ) ∝ n gas ( r ) δ . The first three panels, (a)-(c), show the luminosities with re-accelerationincluded, while in the last panel, (d), re-acceleration was neglected. The blues lines give the mass-luminosityrelation, L γ ∝ M / . Using PIC simulations, Ha et al. (2018b) studied on the proton acceleration byquasi-parallel low Mach number shocks in high β plasmas. In their set-up, they as-sumed a shock obliquity of θ = 13 ◦ and a proton to electron mass ration of 100.They showed that under such conditions only supercritical shocks are able to acceler-ate cosmic-ray protons, i.e. the acceleration is quenched for shocks with Mach numbersbelow M (cid:46) . . However, the critical Mach number depends on the upstream plasmaparameters, such as the obliquity or the plasma β (e.g. Edmiston and Kennel, 1984).Furthermore, hybrid simulations of weak shocks in high β plasmas indicated that theion distribution and, hence, the critical Mach number are sensitive to the cosmic heliumabundance (Bykov et al., 2019). Yet, constraining the exact helium abundance remainsas a task for the next generation of X-ray observatories (e.g. Barret et al., 2018). Hence,despite the importance of the measurement by Ha et al. (2018b), the exact value of the11ritical Mach number might differ, if more realistic values for the proton to electronmass ration and the helium abundance are taken into account.Most shocks in the ICM are weak (Ryu et al., 2003; Vazza et al., 2009; Skill-man et al., 2013; Ha et al., 2018a; Wittor et al., 2019) and, hence, the supercritical-requirement reduces the number of shocks, that are able to accelerate protons, signif-icantly. However, Wittor et al. (2020) found that the supercritical criteria alone doesnot reduce the γ -ray emission enough to become invisible. Despite the small numberof supercritical shocks, they argue that most of the cosmic-ray energy flux is processedby these few supercritical shocks, on average ∼
53 % , and, hence, the associated γ -rayemission is only reduced by a factor of about ∼ . This reduction is not enough toexplain the non-detection of γ -rays.In a pioneering work, Ha et al. (2019) tested if shock accelerated cosmic-ray pro-tons become invisible to the Fermi-LAT if: a) the shock efficiencies are the ones de-rived by Ryu et al. (2019), and b) only supercritical and quasi-parallel shocks are ableto accelerated cosmic-ray protons. Using a particle-mesh/Eulerian cosmological hy-drodynamic (Ryu et al., 1993), they simulated 58 galaxy clusters that cover a massrange of − · M (cid:12) . In their simulation, they evolved the magnetic fieldspassively using the Biermann Battery mechanism (Biermann, 1950). As they did notself-consistently follow the transport of cosmic-ray protons, they computed the γ -rayemission in post-processing. Therefore, they identified all shocks in their simulationat redshift z = 0 and measured their properties such as strength, obliquity and en-ergy flux. By assuming a radial distribution of cosmic-ray protons, they computed thevolume-integrated momentum distribution of cosmic-ray protons that is produced bythe detected shocks. Finally, they estimated the associated γ -ray emission by applyingthe formalisms of Pfrommer and Enßlin (2004); Kelner et al. (2006). They find thatonly ∼ of the kinetic energy flux is processed by supercritical and quasi-parallelshocks. In addition, the shocks that are able to accelerate cosmic-ray protons have anaverage strength of M ≈ . − . . The γ -ray emission in the simulation was com-pared to the upper limits of the cluster sample given in Ackermann et al. (2014), seeFig. 4. Indeed, the γ -ray emission of the simulated clusters falls below the upper limitsgiven by the Fermi-LAT. 12 igure 5: Results given in Wittor et al. (2020): The plot compares the total γ -ray flux the simulated sampleof Wittor et al. (2020) with the upper limits given by the Fermi-LAT. Specifically, the grey crosses givethe limites in the energy range [0 . , Ackermann et al. (2014). The red solid cross marks thedeeper upper limit of the COMA cluster Ackermann et al. (2016), while the red dashed cross is the expectedupper limit after ten more years of Fermi-LAT observations of COMA. The different colored symbols givethe γ -ray fluxes computed for the simulated sample. The model that combines the acceleration efficienciesfrom Ryu et al. (2019) with acceleration by supercritical and quasi-parallel shocks only is given by the redsquares. The other models are: all shocks (black asterisks), only supercritical shocks (blue diamonds) or onlyquasi-parallel shocks (green triangles) are able to accelerate cosmic-ray protons. The black crosses displaythe γ -ray flux obtained for the “old” acceleration efficiencies calculated by Kang and Ryu (2013). (The plothas been modified from its original version to fit the style of this paper). In a complementary work, Wittor et al. (2020) combined cosmological simulationsand Lagrangian tracer particles to study the cosmic-ray proton acceleration in four mas-sive galaxy clusters, i.e. . · − . · M (cid:12) (see Fig. 1). In their modelling,they applied the acceleration efficiencies derived by Ryu et al. (2019). In addition,they measured the reduction of the γ -ray flux if a) only supercritical shocks, b) onlyquasi-parallel shocks and c) only both supercritical and quasi-parallel shocks are able toaccelerate cosmic-ray protons. In agreement with the results of Ha et al. (2019), theyfound that, using the acceleration efficiencies from Ryu et al. (2019), the associated γ -ray emissions is invisible to the Fermi-LAT, if only supercritical and quasi-parallelshocks are able to accelerate cosmic-ray protons, see Fig. 5. Furthermore, they foundthat most of the energy flux is processed by strong shocks and, hence, the key ingredi-ent, to explain the non-detection of γ -rays, is the obliquity cut.13n principle, shocks can also re-accelerate cosmic-ray protons. For weak shocks,the shock acceleration efficiencies are larger (e.g. Kang and Ryu, 2013) and, hence,they can inject a larger fraction of cosmic-rays. Furthermore, it is still uncertainwhether re-acceleration works at every type of shock or if it is restricted to super-critical (Ha et al., 2018b) and quasi-parallel (Caprioli and Spitkovsky, 2014a), as well.Hence, re-accelerated cosmic-ray protons might produce a γ -ray signal that is abovethe Fermi-limits.Ha et al. (2019) also tested their model for cosmic-ray re-acceleration. In theirmodel, cosmic-protons are shock accelerated at three fixed periods. While the first ac-celeration event is pure thermal acceleration, the second and third event also includere-acceleration. Their estimation yields that about ∼ − of pre-existing cosmic-ray protons undergo re-acceleration. Assuming that only quasi-parallel and supercrit-ical shocks are able to re-accelerate cosmic-ray protons, they found that the energystored in cosmic-ray protons increases about ∼
60 % on average. Yet, the associated γ -ray emission still remains below the Fermi-limits, see Fig. 4. For completeness,they tested the effect of re-accelerating when neglecting the supercritical or obliquitycriteria. Their estimates showed that, when relaxing the supercritical criteria, the totalcosmic-ray proton energy grows by ∼
90 % on average. On the other hand, if there-acceleration is independent of the shock obliquity, the cosmic-ray proton energy in-creases significantly and becomes too large to be compatible with upper limits set bythe Fermi-LAT.
4. A different perspective
Before concluding this work, we want to shortly highlight two different perspec-tives on the missing γ -ray problem. First, we present an other possibility to reduce theamount of cosmic-ray protons inside clusters and, hence, to lower the associated γ -raysignal. Second, we want to briefly point towards neutrinos which are an other tracer ofcosmic-ray protons in the ICM.As the γ -ray signal is produced by collisions between cosmic-ray protons and ther-mal protons, it strongly depends on the amount of cosmic-ray protons inside the clus-14er volume. More specifically, it is proportional to the number density squared, ∝ n .Hence, if the amount of cosmic-ray protons living in the clusters’ central regions is re-duced, the associated γ -ray signal drops significantly. Enßlin et al. (2011) were the firstto propose that cosmic-ray protons could stream out of the central regions and, hence,flattening the cosmic-ray protons profiles. Though, it is difficult to explain the lackof γ -ray emission in galaxy clusters that host radio halos with cosmic-ray streaming(e.g. van Weeren et al., 2019, and references therein). However, cosmic-ray stream-ing is complex topic as it depends on the interactions between cosmic-rays and plasmawaves and turbulence. Hence, we point the interested reader to sophisticated works andthe references therein: Pinzke et al. (2017); Enßlin et al. (2011); Wiener et al. (2013);Zandanel et al. (2014); Wiener et al. (2018).The pions, that are produced by the inelastic collisions of cosmic-ray protons andprotons, do not only decay into γ -rays but also into neutrinos (see Eq. 2 and 3). Hence,these neutrinos provide an other observable that could reveal the presence of cosmic-ray protons in the ICM. Several works have focused on estimating the neutrino fluxcoming from galaxy clusters (e.g. Murase et al., 2008, 2013; Murase and Waxman,2016; Zandanel et al., 2015). However, these works also showed that shock acceleratedcosmic-ray protons do not produce significant fraction of the neutrino flux measuredby IceCube . Nevertheless, Ha et al. (2019) computed the neutrino flux in their sampleof simulated galaxy clusters (see Sec. 3). They estimated that the neutrino fluxesin their sample are significantly below both the atmospheric neutrino fluxes at E ≤ and the IceCube flux at E = 1 PeV (Richard et al., 2016; Aartsen et al., 2014,respectively). Hence, they concluded that the detection of neutrinos, coming from thecollisions of cosmic-ray protons and protons, is more than unlikely with the currentand future generation of ground based neutrino telescopes, i.e. IceCube (Halzen andKlein, 2010), Super-Kamiokande (Hagiwara et al., 2019) or future Hyper-Kamiokande(Abe et al., 2011). https://icecube.wisc.edu/ . Will there be a detection in the future? In this review, we have reviewed the latest advances in understanding the missing γ -ray problem , which fundamentally challenges the shock acceleration mechanisms atwork in the ICM. Thanks to a collaborative effort of theoretical predictions, numericalsimulations as well as space-based and ground-based observations, it is now possibleto explain the non-detection of γ -rays associated to shock accelerated cosmic-ray pro-tons: Recent studies showed that if both the energy dissipation due to Alfven wavesand the dynamical feedback of the cosmic-ray pressure on the shock are included toderive shock acceleration efficiencies (Ryu et al., 2019), then they efficiencies becomesignificantly smaller than previously estimated. If these new efficiencies are pairedwith the fact that only supercritical (Ha et al., 2018b) and quasi-parallel (Caprioli andSpitkovsky, 2014a) shocks are able accelerated cosmic-ray protons, then the expected γ -ray signal from the ICM drops below the upper limits given by the Fermi-LAT (Haet al., 2019; Wittor et al., 2020).Yet, only a direct detection of a γ -ray signal associated with cosmic-ray protonsin the ICM will allow to pin-point the exact shock acceleration mechanism. Though,estimates by Wittor et al. (2020) showed that, even after ten more years of operation,the Fermi-LAT will most-likely not detect such a signal, see Fig. 5. Hence, all hopesto detect γ -rays in the future lie with the next generation of γ -ray telescopes.The Cherenkov Telescope Array (CTA) is the next ground-based γ -ray that iscurrently being build in Paranal, Chile, and on La Palma, Spain, and its construction isplanned to be completed in 2025. Yet, in the desired energy range, i.e. − MeV ,the CTA is expected to be less sensitive than the Fermi-LAT (Cherenkov TelescopeArray Consortium et al., 2019). Therefore, it will most-likely not be able to detect themissing γ -rays.However, the proposed space mission: All-sky Medium Energy Gamma-ray Ob-servatory (AMEGO McEnery et al., 2019) will have the capabilities to detect themissing γ -rays. If accepted, AMEGO will observe in the energy range . to https://asd.gsfc.nasa.gov/amego/ and it sensitivity will be factors of ∼ − below Fermi-LAT’s sensitiv-ity. This could potentially be enough to finally detect the missing γ -rays. Yet, also anon-detection with AMEGO would provide relevant information about the cosmic-rayproton spectrum. Some models of cosmic-ray proton re-acceleration by weak shock en-sembles with long-wavelength magneto-hydrodynamical waves produce a soft asymp-totic cosmic-ray proton spectrum (e.g. Bykov et al., 2019). Such a soft spectrum canhelp to understand the non-detection by the Fermi-LAT, yet it will be difficult to bedetected even with AMEGO. Acknowledgements
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