COSMIC RAYS IN GALAXY CLUSTERS AND THEIR NON-THERMAL EMISSION

Cosmic Rays in Galaxy Clusters and Their Non-Thermal Emission

GIANFRANCO BRUNETTI    THOMAS W. JONES
Abstract

Radio observations prove the existence of relativistic particles and magnetic field associated with the intra-cluster-medium (ICM) through the presence of extended synchrotron emission in the form of radio halos and peripheral relics. This observational evidence has fundamental implications on the physics of the ICM. Non-thermal components in galaxy clusters are indeed unique probes of very energetic processes operating within clusters that drain gravitational and electromagnetic energy into cosmic rays and magnetic fields. These components strongly affect the (micro-)physical properties of the ICM, including viscosity and electrical conductivities, and have also potential consequences on the evolution of clusters themselves. The nature and properties of cosmic rays in galaxy clusters, including the origin of the observed radio emission on cluster-scales, have triggered an active theoretical debate in the last decade. Only recently we can start addressing some of the most important questions in this field, thanks to recent observational advances, both in the radio and at high energies. The properties of cosmic rays and of cluster non-thermal emissions depend on the dynamical state of the ICM, the efficiency of particle acceleration mechanisms in the ICM and on the dynamics of these cosmic rays. In this review we discuss in some detail the acceleration and transport of cosmic rays in galaxy clusters and the most relevant observational milestones that have provided important steps on our understanding of this physics. Finally, looking forward to the possibilities from new generations of observational tools, we focus on what appear to be the most important prospects for the near future from radio and high-energy observations.

Received Day Month Year

Revised Day Month Year

Keywords: Galaxies: clusters: general; Radiation mechanisms: non-thermal; Acceleration of particles.

PACS numbers: 95.30.Cq; 95.30.Gv; 95.30.Qd; 98.65.Cw; 98.65.Fz; 98.65.Hb

1 Introduction

Clusters of galaxies and the filaments that connect them are the largest structures in the present universe in which the gravitational force due to the matter overdensity overcomes the expansion of the universe. Massive clusters have typical total masses of the order of , mostly in the form of dark matter ( of the total mass), while baryonic matter is in the form of galaxies () and especially in the form of a hot () and tenuous () gas (), the intra-cluster-medium (ICM) (Figure 1). That ICM emits thermal X-rays, mostly via bremsstrahlung radiation, and also Compton-scatters the photons of the cosmic microwave background, leaving an imprint in the mm-wavelengths band that provides information complementary to the X-rays (Figure 1). In the current paradigm of structure formation, clusters are thought to form via a hierarchical sequence of mergers and accretion of smaller systems driven by dark matter that dominates the gravitational field. Mergers, the most energetic phenomena since the Big Bang, dissipate up to ergs during one cluster crossing time ( Gyr). This energy is dissipated primarily at shocks into heating of the gas to high temperature, but also through large-scale ICM motions. Galaxy clusters are therefore veritable crossroads of cosmology and astrophysics; on one hand they probe the physics that governs the dynamics of the large-scale structure in the Universe, while on the other hand they are laboratories to study the processes of dissipation of the gravitational energy at smaller scales. In particular, a fraction of the energy that is dissipated during the hierarchical sequence of matter accretion can be channeled into non-thermal plasma components, i.e., relativistic particles (cosmic rays, or “CRs”) and magnetic fields in the ICM. Relativistic particles in the ICM are the main subject of our review.

The evidence for non-thermal particles in the ICM is routinely obtained from a variety of radio observations that detect diffuse synchrotron radiation from the ICM (Figure 1). They also open fundamental questions of their origins as well as their impact on both the physics of the ICM and the evolution of galaxy clusters more broadly. Cosmological shock waves and turbulence driven in the ICM during the process of hierarchical cluster formation are obvious potential accelerators of cosmic ray electrons (CRe) and protons (hadrons or CRp) . In addition, clusters host other accelerators of CRs, ranging from ordinary galaxies (especially as a byproduct of star formation) to active galaxies (AGN) and, potentially, regions of magnetic reconnection . The long lifetimes of CRp (and/or nuclei) against energy losses in the ICM and their likely slow diffusive propagation through the disordered ICM magnetic field, together with the large size of galaxy clusters, make clusters efficient storehouses for the hadronic component of CRs produced within their volume or within the individual subunits that later merged to make each cluster . The consequent accumulation of CRs inside clusters occurs over cosmological times, with the potential implication that a non-negligible amount of the ICM energy could be in the form of relativistic, non-thermal particles. An important result of trapped CRp above a few hundred MeV kinetic energy is that they will necessarily produce secondary pions (and their decay products, including e and -rays) through inelastic collisions with thermal target-protons. Consequently, they can be traced and/or constrained by secondary-particle-generated radio and -ray emission.

Fig. 1: Multi-frequency view of the Coma cluster: the thermal ICM emitting in the X-rays (top left, adapted from ), the overlay between thermal SZ signal (colors) and X-rays (contours) (top right, adapted from ), the optical emission from the galaxies in the central region of the cluster (bottom left, from Sloan Digital Sky Survey, credits NASA/JPL-Caltech/ L.Jenkins (GSFC)), and the synchrotron (radio halo) radio emission (contours) overlaied on the thermal X-rays (colors) (bottom right, adapted from ).

The most direct way to pin-point CRp in galaxy clusters is through the detection of -ray emission generated by the decay of secondary particles. However to date, despite the advent of the orbiting Fermi-LAT and deep observations from ground-based Cherenkov arrays, no ICM has been firmly detected in the -rays. Cluster -ray upper limits, together with several constraints from complementary approaches based on radio observations suggest that the energy in the form of CRp is less than roughly a percent of the thermal energy of the ICM, at least if we consider the central Mpc–size region. This result contradicts several optimistic expectations derived in the last decades, based mostly on estimates for CRp production in structure formation shocks, and poses important constraints on the efficiency of CRp acceleration and transport in galaxy clusters.

On the other hand, the existence of CRe and magnetic fields in the ICM of many clusters is in fact demonstrated by radio observations. CRe are indeed very well traced to the ICM of clusters through their radio emission. Cluster-scale ( Mpc-scale) diffuse synchrotron emission is frequently found in merging galaxy clusters. It appears in the form of so-called giant radio halos, apparently unpolarized synchrotron emission associated with the cluster X-ray emitting regions, and giant radio relics, elongated and often highly polarized synchrotron sources typically seen in the clusters’ peripheral regions with linear extents sometimes exceeding Mpc. The locations, polarisation and morphological properties of radio relics suggest a connection with large scale shocks that cross the ICM during mergers and that may accelerate locally injected electrons or re-accelerate pre-existing energetic electrons to energies where they emit observable synchrotron emission. Electrons responsible for giant radio halo emissions, on the other hand, require virtually cluster-wide generation, as we outline below. There are good reasons to believe that radio halos trace gigantic turbulent regions in the ICM, where relativistic electrons can be re-accelerated through scattering with MHD turbulence and/or injected by way of inelastic collisions between trapped CRp and thermal protons . As we will discuss in this review, one of the most interesting consequences of the present theoretical scenario for CRe production in clusters is that cluster-scale radio emission should be more common than presently seen, especially at lower radio frequencies. Specifically, we may expect to find many more clusters through radio emission in the frequency range that will be explored in the next few years by the new generation of low frequency radio telescopes, such as LOFARaaahttp://www.lofar.org/, MWAbbbhttp://www.mwatelescope.org/ and LWAccchttp://www.phys.unm.edu/ lwa/. Another important consequence of current ICM CR models is that, despite a current dearth of detections, clusters should be sources of high energy photons at a level that could be detectable by the next generation of X-ray and -ray telescopes. Successful, firm detection of galaxy clusters in the hard X-rays and in -rays would lead to a fundamental leap forward in our understanding, as it will provide a unique way to measure the energy content of magnetic fields, as well as CRe and CRp in cosmic large scale structure.

Since it seems likely that radio halos and relics are signposts of the dissipation of gravitational energy into non-thermal components and emission, they can also be used as valuable (indirect) probes of the merging rate of clusters at different cosmic epochs . In this respect the upcoming radio surveys both at lower and higher radio frequencies, with LOFAR, MWA, and ASKAPdddhttp://www.atnf.csiro.au/projects/askap/, and on longer time-scales with the SKAeeewww.skatelescope.org, have the potential to provide important complementary data for cosmological studies.

Diffuse synchrotron radio emission on smaller scales, kpc, known as radio mini-halos, is also found at the centers of relatively relaxed clusters with cool cores . The existence of mini halos indicates that mechanisms other than major cluster-cluster mergers can power non-thermal emission in the ICM. As we will discuss in this review, also in this case, gravity is likely to provide the ultimate energy reservoir to power the non-thermal emission, potentially extracted, for example, from the sloshing of the gas in response to motions of dark matter cores in and near the cluster. However, AGNs that are usually found at the center of these sources and also frequently distributed over larger cluster volumes may be players. In addition, at the present time any relationships between mini and giant halos are still poorly defined.

Starting from this background, the goals of this review are to discuss the most relevant aspects of the origin and physics of CRs and non-thermal emission in galaxy clusters and to elaborate on the present theoretical framework. We will place emphasis on the most important current observational constraints, along with the observational prospects for the near future. In Sect.2 we will discuss the physics of CRs acceleration by different sources/mechanisms, whereas in Sect.3 we will discuss the relevant energy losses and the dynamics of CRs in the ICM. In Sect.4 we will discuss the most important observational properties of diffuse radio sources in galaxy clusters, their origin and the main prospects for the near future. Specifically giant radio halos, mini halos and relics are discussed in Sects. 4.2, 4.3, and 4.4, respectively, whereas in Sect. 4.1 we will briefly discuss current observational constraints on the magnetic field in the ICM. In Sect. 5 we will discuss current observational constraints on the high-energy emission from galaxy clusters, the most relevant theoretical aspects and prospects for the near future. Sect. 6 provides our Summary.

2 Cosmic ray sources and acceleration

Consensus has been reached in the past decade that shocks produced during the hierarchical formation of the large scale structure in the universe are likely sources of CRs in galaxy clusters, thus implying a direct connection between the generation of CRs and the formation and evolution of the hosting clusters. Similarly, there is consensus on the fact that turbulence can be induced in the ICM as a result of the same processes of clusters formation and that such turbulence affects the propagation of CRs, while also providing a potentially important mechanism for re-acceleration of CRp and CRe .

Several additional sources can supply (inject) relativistic particle populations (electrons, hadrons or both) into the ICM. For instance, particles can be accelerated in ordinary galaxies as an outcome of supernovae (SN) and then expelled into the ICM with a CRp luminosity as high as . Alternatively, high velocity outflows from AGNs may plausibly contribute up to in CRs over periods of years .

2.1 Galaxies, Starbursts and Active Nuclei

Individual normal galaxies are certainly sources of CRs as a consequent of current and past star formation. Massive clusters of galaxies contain more than a hundred galaxies where SN and pulsars accelerate CRs. The efficiency of CR acceleration at these sites is constrained from complementary observations of Galactic sources. However, the amount of CR energy available to the ICM depends also on the way these CRs are transported from their galactic sources into the ICM.

Voelk et al. 1996 pioneered the studies of the role of SN explosions, including starbursts, in cluster galaxies. The number of SNe experienced by a typical cluster since its formation epoch, , can be estimated from the metal enrichment of the ICM, assuming those metals are released by SNe. This gives a total energy budget in the form of CRp :

(1)

where is the mass of iron in the ICM ( is the iron abundance, the typical metallicity measured in galaxy clusters, and the baryon mass of the cluster), is the iron mass available to the ICM from a single SN explosion, erg is the SN kinetic energy and is the fraction of SN kinetic energy in the form of CRp; is constrained from observations of SN in the Galaxy. Note that the efficiency for acceleration of CRe in SNRs is apparently several orders of magnitude smaller than for CRp . Eq.2.1 implies a ratio between the CRp and thermal energy budget in galaxy clusters , assuming (appropriate for type II SN) and K. This is an optimistic estimate of the expected energy content of CRp in galaxy clusters from this source, because it does not account for adiabatic losses in the likely event that CRp from SN are transported into the ICM by SN-driven galactic winds.

On the other hand, clusters of galaxies contain AGNs, which, by way of their synchrotron-emitting jets and radio lobes, are known to carry CRe . The majority of cool-core clusters contain central, dominant galaxies that are radio loud. The radio lobes of these AGNs are seen frequently to coincide with X-ray dark volumes (“cavities”) that have turned out to be the best calorimeters of the total energy deposited by AGN outflows. The cavities, being filled with relativistic and some amount of very hot thermal plasma at substantially lower density than their surroundings, are poor thermal X-ray emitters. Such cavities have been seen in something like of the clusters observed by Chandra , despite the fact that they often exhibit low contrast. From an assumption of pressure balance between the cavity and the surrounding ICM the cavity energy contents have been estimated generally in the range erg . These approach % for an entire ICM in some cases. Dynamical estimates of cavity lifetimes are typically yr, roughly representing buoyancy timescales. These lead to AGN power deposition estimates within an order of magnitude of the X-ray cooling rate of the host cluster , at least while the AGN jets are active. The bubble forming duty cycles, estimated from the fraction of cool-core clusters that harbor clear bubbles, range as high as 70% . Simulations suggest that roughly of the power of the AGN outflow is immediately deposited irreversibly as ICM heat through shocks and entrainment. These may also drive ICM turbulence that would contribute to CR acceleration and to the dynamics of CRs (Sects. 2.2.2, 3.2). Given their large energy inputs, AGN outflows are widely invoked to account for heating needed to limit the effects of strong radiative cooling in cluster cores, e.g.. So, the total energy deposition into the ICM by AGNs is likely to be substantial.

However, the energy in CRe and CRp is harder to establish in radio lobes of cluster radio galaxies. In a few cases the absence of observed inverse Compton X-rays has been used to establish that most of the energy filling the radio lobes must be in some form other than radiating electrons , although CRe energy fractions as high as 10% are not ruled out. Meaningful CRp energy content estimates in the lobes do not exist at present, although recent detailed comparisons of internal-lobe and external pressure suggest that models in which CRp transported by the jet dominate lobe energetics are unlikely. Remarkably, various theoretical arguments have been made suggesting that much of the direct energy flux in AGN jets is carried by cold, non-radiating particles or electromagnetic fields . Even if much of the energy filling the cavities is carried by CRs, it is not yet clear how efficiently those CRs can be dispersed through diffusion and convective/turbulent mixing over the full cluster volume, and how much of the energy would remain in CRs, after accounting for adiabatic and other energy losses. Large scale magnetic fields, for instance, can help confine lobe contents . A connected problem that will be discussed in Sect.4.3 is the possible role of relativistic outflows from the central AGNs in the origin of radio mini halos in cool core clusters.

While most discussions of AGN energy deposition in clusters have focused on central, dominant galaxies, there are other populations of AGNs in clusters that could contribute to the CR population, either directly or indirectly. Low luminosity AGNs are quite commonly distributed throughout clusters. Stocke et al. have argued, in fact, that virtually all bright red sequence galaxies in rich clusters are likely to be low-luminosity blazars with relativistic jets that could collectively dominate AGN energy inputs to the ICM. In that case their CR outputs would be more easily distributed across the cluster by way of ICM turbulence and large scale “weather” (e.g., sloshing). In addition, tailed radio galaxies, quite common and widely distributed in both relaxed and merging clusters, show clear evidence of strong interactions with the ICM that includes entrainment and the generation of turbulence.

Fig. 2: Projection of matter density for a volume of size 187 Mpc/h simulated with ENZO AMR (velocity and density refinement technique) with peak spatial resolution 25 kpc/h. The two panels on the left are 8x8x2 Mpc/h zooms of the central region showing the projected density (top-left) and the kinetic energy-flux (bottom-left). The two panels on the right are 8x8x0.025 Mpc/h zooms of the same region showing the temperature distribution (top-right) and the overlay of shock map and turbulent velocity-vectors (obtained with a filtering of laminar motions on 300 kpc scale). Images are obtained at from simulations presented in .

2.2 Particle acceleration in the ICM

The current prevailing view is that the process of structure formation may contribute directly and indirectly most of the energetics of non-thermal components (CRs, magnetic fields and turbulence) in galaxy clusters . Mergers between two or more clusters are observed to heat clusters through shocks . Although more difficult to constrain observationally, the additional process of semi-continuous accretion of material onto clusters, especially from colder filaments, is expected to drive quasi-stationary, strong shocks and turbulent flows at Mpc distances from cluster centers that should impact on the ICM physics and acceleration of CRs over wide volumes.

Particle acceleration during mergers should occur at shock waves that are driven to cross the ICM. Particle acceleration is also expected to result from several mechanisms that may operate within turbulent regions also driven in the ICM during these mergers (e.g., turbulent acceleration and magnetic reconnection, etc). The intricate pattern of shocks and large-scale turbulent motions and their interplay is still difficult to establish observationally, but can be traced in some detail by cosmological simulations of galaxy cluster formation. Fig.2 provides a view of the complex dynamics of the ICM as seen in simulations. In particular, a complex pattern of strong and weak shocks is naturally driven in the ICM, largely by gravity variations reflecting dark matter dynamics (Sect. 2.2.1). This shock distribution, where most of the kinetic energy flux is dissipated within clusters, is morphologically correlated to some extent with the distribution of the turbulent motions in the ICM, that are, indeed, partly driven by those shocks (Sect. 2.2.2). In the following we will focus on the physics of shocks and particle acceleration at shocks (Sect. 2.2.1), and on the physics of turbulence and turbulent acceleration in galaxy clusters (Sect. 2.2.2).

Fig. 3: Distribution of the energy flux at shocks surfaces as a function of the shock-Mach number from numerical (cosmological) simulations (adapted from ). Units are in ergs Mpc. Shocks are divided into internal and external categories. External shocks are defined as shocks forming when never-shocked, low-density, gas accreted onto nonlinear structures, such as filaments etc. Internal shocks form within the regions bounded by external shocks.

2.2.1 Shocks in galaxy clusters as CRs accelerators

The total gravitational energy dissipated by baryonic matter in a merger of two clusters with roughly equal mass, , is erg. With the assumption that the gaseous components of the initial clusters are at the associated virial temperatures, it is easy to show that the merging components approach each other at slightly supersonic relative speed, therefore implying the formation of weak, , shock waves . Those shocks will typically strengthen moderately as they emerge into lower density and low temperature regions outside the cluster cores. Additional, remote, accretion shocks that result from the continuous accretion of matter at several Mpc-distances from clusters center (also called “external shocks” when they form due to the accretion of never-shocked gas), are typically much stronger; that is, they have higher Mach numbers, since they develop in cold, un-virialised external cluster regions. On the other hand, since gas densities are also quite low in those environments, the energy available for dissipation through such shocks is relatively smaller than through lower Mach number shocks that dissipate energy in higher density regions closer to cluster centers during mergers. Simple (analytical or semi-analytical) but accurate estimates of the amount of kinetic energy associated with accretion/external shocks are very challenging. A leap forward in understanding in this area, however, has been achieved in the last decade through extensive cosmological simulations that allow one to study the formation of shocks in clusters, from their outskirts to more internal regions with increasing detail.

Figure 3 illustrates these points. It shows the kinetic energy flux through shock surfaces, that is, , measured in clusters formed during cosmological simulations. Here, is the upstream gas density, while and are shock velocity and surface area, respectively. The distribution of energy fluxes shows that most of the gravitational energy is dissipated at relatively weak, “internal” (merger) shocks, with Mach number . A modest fraction of the kinetic energy-flux passes through related stronger internal-shocks developed during merger activity as they propagate outwards after merging cores have their closest approach. Figure 3 shows that only a few % of the energy flux is dissipated at strong, “external” shocks.

If even a small percentage of the merger and accretion shock-dissipated energy can be converted into non-thermal particles through a first order Fermi process, then the ICM could be populated with an energetically significant population of non-thermal, CR particles . This prospect has drawn considerable focus to potential consequences of shock generated CRs and the refinement of early estimates of CR production in ICMs.

2.2.1.1.   Shock acceleration of CRs

The acceleration of CRs at shocks is customarily described according to the diffusive shock acceleration (DSA) theory . In effect diffusing particles are temporarily trapped in a converging flow across the shock if their scattering lengths across the shock are finite but much greater than the shock thickness. Particles escape eventually by convection downstream. Until they do, they gain energy each time they are reflected upstream across the shock, with a rate determined by the velocity change they encounter across the shock discontinuity and a competition between convection and diffusion on both sides of the shock. The hardness (flatness) of the resulting spectrum reflects the balance between energy gain and escape rates. In other words, it depends on the energy gain in each shock crossing combined with the probability that particles remain trapped long enough to reach high energies. Mathematically this balance can be conveniently described through the diffusion-convection equation for a pitch angle averaged CRs distribution function in a compressible flowfff is the number of CRs per unit phase-space volume, that is :

(2)

where is the modulus of the particle’s momentum, is the velocity of the background medium (assuming , with the Alfvén velocity), while is the unit vector parallel to the local magnetic field, and is the particle spatial diffusion coefficient (see Sect. 2.2.2). The 2nd and 3rd terms account for convection and diffusion, respectively, while the right hand side takes account of the adiabatic energy gains (losses) suffered by particles in a converging (expanding) flow. As written, Eq.2.2.1 omits non adiabatic losses, such as from radiation, that can be important especially for CRe (CRs energy losses are discussed in Sect. 3.1), momentum diffusion and effects such as CR energy transfer to wave amplification/dissipation (turbulent-CRs coupling is discussed in Sect. 2.2.2).

Fig. 4: Time evolution of the spectrum of protons accelerated from a Maxwellian upstream distribution at shocks with Mach number 2 (left) and 30 (right) (adapted from ). The spectral slope predicted by (test-particle) DSA is represented as a red-dashed line.

Under these conditions, if all particles are injected at low energies and “see” the same velocity change across the shock, the steady state spectrum of test-particle CRs at a plane shock is a power law in momentum, , where the slope is

(3)

is the Mach number of the shock. For strong shocks, , this slope tends to . Thus, in the strong shock limit, the energy and pressure in the resulting CRs are broadly distributed towards the highest energies that are achieved. On the other hand, for weak shocks, with , this tends to . In this case the fractional velocity jump across the shock is small, so the energy in CRs accelerated from suprathermal values is concentrated in the lowest energy CRs. That is, the CRs gain relatively little energy before they escape downstream. Consequently, for the same number of CRs and the same kinetic energy flux through the shock, , the energy input to locally injected CRs through DSA is much greater in strong shocks than in weak shocks.

As a consequence of this theory, it is apparent that even a modest injection of particles at a strong shock can lead to a substantial fraction of the kinetic energy flux into strong, initially purely hydrodynamical shocks going into CRs. Those, in turn backreact on and modify the structure of the shocks themselves. Under these conditions the process of particle acceleration is described using nonlinear theory . The main outcome of that development is the formation of a compressive precursor to the shock, leading to an increase in the total shock compression, upstream turbulence and magnetic field amplification, followed by an actual weakening of the fluid shock transition (the so-called ‘sub-shock”). In a highly CR-modified shock a large part of the DSA process at high CR energies actually takes place in the precursor when the spatial diffusion coefficient, is an increasing function of particles momentum. Then, the subshock is responsible mostly for the acceleration process at low energies and injection of seed DSA particles.

The importance and detailed outcomes of nonlinear evolution in strong shocks depend on the size of the CR population at the shock, the hardness of the CR spectrum being accelerated at the shock, the efficiency and distribution of turbulent magnetic field amplification upstream of the shock and the geometry of the shock . These physical details are important, since they regulate how much energy is extracted from the flow into the shock and, accordingly how much pressure will develop from these CRs and amplified magnetic field within the shock transition. On the other hand, unless they include much larger total CR populations or interact with a pre-existing CR population with a hard spectrum, weak shocks are minimally affected by nonlinear effects, because of the steeper CR spectra generated in these shocks. Fig.4 shows the time evolution of CRp spectra accelerated at simulated weak and stronger shocks. In the case of weak shocks the spectrum agrees with the prediction of test particle DSA theory, while the spectrum becomes concave and flatter than test particle DSA for stronger shocks, due to the non-linear back-reaction of CRp. The quality of comparisons between real and theoretical strong, DSA-modified shocks is still an open question (see the discussion at the end of this section).

The acceleration time-scale at the shock (i.e. the time necessary for CRs of energy to double that energy) depends on the time interval between shock crossings for the CR, , is the particle velocity, and on the ratio of the CR velocity to the fluid velocity change across the shock, so . Thus, it primarily depends on the spatial diffusion coefficient of particles, and inversely on the shock velocity, , and the compression through the shock. In the simplified case that the spatial diffusion coefficient does not change across the shock, the mean acceleration time to a given momentum in an unmodified shock can be written as :

(4)

that approaches for strong shocks.

Fig. 5: The life-time of CRp in the ICM as a function of energy due to photo-pion and photo-pair production (solid curves) are compared to a reference life-time of clusters Gyrs, to the maximum diffusion time-scale of CRp (assuming Bohm-diffusion on 3 Mpc scales and using an optimistic value G) and to the minimum acceleration time-scale of CRp by shocks (assuming an optimistic configuration with Bohm-diffusion with G, and a shock velocity km/s). The allowed region, marking the energies of CRp that can be obtained via shock acceleration, is highlighted in blue.

In order to derive a maximum energy of the accelerated CRs in a given time interval we assume a Bohm diffusion coefficient, , (see also Sect 2.2.2) where, , is the particle Larmour radius. This is optimistically small, thus giving us an optimistic upper energy bound, since it assumes a mean free path equal to the CR gyroradius. The spatial diffusion coefficient for relativistic particles then becomes in practical terms,

(5)

resulting in an acceleration time scale to GeV energies of the order of 1 yr, if we assume a typical shock velocity in galaxy clusters, km/s, and G. Then, from eqs.4-5, GeV, implying for realistic available acceleration times ( years) that the power law distribution of the accelerated particles should extend up to very high energies, where energy losses or diffusion from the acceleration region quenches the acceleration process (spatial diffusion in the ICM is discussed in Sect. 3.2).

The energy losses for CRe, due especially to synchrotron and inverse Compton processes (see Sect. 3), are much more significant than those for CRp. For CRe the maximum energy accelerated at shocks in galaxy clusters can be of the order of several tens of TeV . Once these CRe are advected downstream of the shock, radiative cooling due to inverse Compton scattering (ICS) with the CMB (and if the magnetic fields are strong enough, synchrotron, Sect. 3) will steadily reduce the maximum electron energy, so that it scales asymptotically as , where is the propagation distance downstream from the shock reached over a time, . This causes the “volume integrated” electron spectrum to steepen by one in the power law index, above energies reflecting this loss over the lifetime of the accelerating shock . Consequently, the “spatially unresolved” synchrotron spectrum “from the downstream region” will show a steepened spectral slope, .

By contrast, CRp in these shocks are not subjected to significant energy losses until they reach extremely high energies where they suffer inelastic collisions with CMB photons. CRp with energies above a few hundred PeV will produce when they collide with CMB photons, limiting their life-times to a period below a few Gyr (Figure 5). Most important, at energies above about eV, such collisions produce pions, and each such collision extracts a significant fraction of the CRp energy (this is the same physics that determines the so-called Greisen-Zatsepin-Kuzmin (GZK) cutoff in the ultra-high energy CR spectrum). For example, for energies above eV the CRp life-time drops rapidly below yr (Fig. 5). In Figure 5 we compare these relevant time-scales with the shock-acceleration time-scale (eqs. 4-5) and with the life-time of shocks and clusters themselves. It follows that CRp acceleration at cluster shocks can reach at most maximum energies of a few eV. We note that this is an optimistic estimate, as we are implicitly assuming that such high-energy CRp are still effectively confined in galaxy clustersggge.g., confinement of CRp with energies eV for a Hubble time would require conditions comparable to Bohm diffusion at distances of several 100 kpc from the shock (CRs diffusion/confinement is discussed in Sect. 3.2).

2.2.1.2.   Shock acceleration in the ICM & open questions

Particle acceleration efficiency at strong shocks is becoming constrained by studies of SN-driven shocks in our Galaxy . Those shocks transfer % or more of the energy flux though them into CRp. It is important to keep in mind that the shocks mostly responsible for acceleration of observable Galactic CRs are very strong, with Mach numbers upwards of , and that they are found in low beta-plasma, , environments. By contrast, and as discussed above, the ICM is a high- environment and most of the kinetic energy flux penetrating galaxy cluster shocks is associated with much weaker shocks where, the acceleration efficiency is probably much less, although still poorly understood . In this respect galaxy clusters are special environments, as they are unique laboratories for constraining the physics of particle acceleration at (Mpc-scale) weak shocks. It remains an open issue whether these weak shocks can accelerate CRs in the ICM at meaningful levels.

A critical, unresolved ingredient in shock acceleration theory is the minimum momentum of the seed particles that can be accelerated by DSA; i.e., the minimum momentum that leads to diffusive particle transport across the shock. This, along with the detailed processes that control this minimum momentum are crucial in determining the efficiency with which thermal particles are injected into the population of CRe and CRp. Particles must have momenta at least several times the characteristic postshock thermal ion momenta in order to be able to successfully recross into the preshock space. Quasi-thermalized particles are inherently less likely to recross in the upstream direction in weak shocks than strong shocks, because of the weaker dissipation in weak shocks; i.e., the ratio of the postshock thermal speed to the postshock convection speed is relatively smaller. Injection is expected to depend sensitively on the charge/mass ratio of the injected species, since that determines the rigidity () of particles at a given energy. For this reason, nonrelativistic electrons appear to be very difficult to inject from the thermal population (because ), so are likely to be far fewer than injected protons. Typically some kind of upstream, pre-injection process, often involving protons reflected by the shock that generate upstream waves that can resonate with nonrelativistic electrons, is invoked to enable electron injection at shocks . The processes “selecting” the particles that can recross define so-called thermal-leakage injection. It is important to realize that they are poorly understood, especially in the relatively weak shocks with large expected in cluster media. Existing collisionless shock simulations, hybrid and PIC simulations, have focused on strong shocks or low beta-plasma . The orientation of the magnetic field with respect to the shock normal is also important, since it strongly influences the physics of the shock structure .

A connected, unresolved ingredient in nonlinear CR shock theory is the level of amplification of the magnetic field and its distribution within the shock due to CR-driven instabilities. The evolution of the magnetic field through the full shock structure is important, since the magnetic field self–regulates the diffusion process of supra–thermal particles on both sides of the shock and also affects the injection process . There are several proposed models to amplify magnetic fields significantly within the CR-induced shock precursor ; none of them applies until some degree of shock modification already takes place. This means they only apply in strong shocks, so probably not in cluster merger shocks. Some magnetic field generation and/or amplification downstream of curved or intersecting shocks may result when the electron density and pressure gradients are not parallel (the so-called Biermann Battery effect ), due to the Weibel filamentation instability or when the downstream total plasma and pressure gradients are not parallel (so-called baroclinic effects that amplify vorticity ). Amplification of turbulence and magnetic fields only downstream, however, have minimal impact on DSA, which depends essentially on those properties on both sides of the shock.

As a final remark we mention that there is still discussion on the spectrum of CRs resulting from shock acceleration. Although calculations in the last decade agreed on the conclusion that the spectra of CRs accelerated at strong shocks are concave (flat) as a result of the dynamical back-reaction of CRs (Figure 4), current data for SNRs seem to favour steeper spectra and suggest a partial revision of the theory . Specifically, the absence of CR spectral concavity in modified shocks requires a relative reduction in acceleration efficiency or increased escape probability for the highest energy CRs.

2.2.2 Turbulence in the ICM and CR reacceleration

Galaxy clusters contain many potential sources of turbulence. These include cluster galaxy motions , the interplay between ICM and the outflowing relativistic plasma in jets and lobes of AGNs , and buoyancy instabilities such as the magnetothermal instability (MTI) in the cluster outskirts . However, the most important potential source of turbulent motions on large scales is the process that leads directly to the formation of galaxy clusters . Mergers between clusters deeply stir and rearrange the cluster structure. In this case turbulence is expected from core sloshings, shearing instabilities, and especially from the complex patterns of interacting shocks that form during mergers and structure formation more generally (Figure 2). Such a complex ensemble of mechanisms should drive in the ICM both compressive and incompressive turbulence, as also supported by the analysis of numerical simulations ; in Figure 6 we report a sketch of the turbulent properties of the ICM.

Large scale turbulent motions that are driven during cluster-cluster mergers and dark matter sub-halo motions are expected on scales comparable to cluster cores scales, kpc, and might have typical velocities around km/s, e.g. . These motions are sub–sonic, typically with , but they are strongly super-Alfvénic, with , e.g. . This implies a situation in which magnetic field lines in the ICM are continuously advected/stretched/tangled on scales larger than the Alfvén scale; that is, the scale where the velocity of turbulent eddies equals the Alfvén speed, ( is the slope of the turbulent velocity power-spectrum, ). Below this scale, turbulent, “Reynolds stresses” are insufficient to bend field lines and turbulence becomes MHD (Fig. 6). Under these conditions the effective particle mean-free-path in the ICM should be rather than the value of the classical Coulomb ion-ion mean free path, kpc (Fig. 6)hhhthe effective mfp should be the smaller of the two scales, however under typical ICM conditions , see.

However the ICM is a “weakly collisional” plasma and can be very different from collisional counterparts, because it is subject to various plasma instabilities, e.g. . In many cases, as a result of plasma instabilities the (relatively weak) magnetic field is perturbed on very small scales. That provides the potential to strongly reduce the effective thermal particle (and CRs) mean free path and also the effective viscosity of the fluid, below the classical Braginskii viscosity determined by thermal ion-ion Coulomb collisions.

All these considerations about the velocities of large-scale motions and the effective particle mean-free-path in the ICM allow us to conclude that the effective Reynolds number in the inner ICM is ; that is, much larger than it would be if it were determined by the classical ion–ion mean free path (). Theoretically this suggests that a cascade of turbulence and a turbulent inertial range could be established from large to smaller scales (Fig. 6). In addition plasma/kinetic instabilities in the ICM generate waves at small, resonant, scales (Fig. 6). Among the many types of waves that can be excited in the ICM we mention the slab/Alfvén modes that may be excited for example via streaming instability and gyro-kinetic instability, and the whistlers that may be excited for example via heat-flux driven instabilities. Also coherent wave phenomena in MHD turbulence can generate non-linear electrostatic waves; for example lower hybrid electrostatic waves in the ICM might be excited through the non-linear modulation of density in large amplitude Alfvén wave-packets. Both large-scale motions (and their cascading at smaller scales) and the component of self-excited turbulent waves at small scales have a strong role in governing the micro-physics of the ICM through the scattering of particles and the perturbation of the magnetic field.

Current X-ray observations do not allow one to derive stringent constraints on the turbulent motions in dynamically active (i.e. merging or non cool core) clusters (see however the pioneering attempt by ) (constraints on cool core clusters are discussed in Sect. 4.3). This will hopefully change in the next years thanks to the ASTRO-H satelliteiiihttp://astro-h.isas.jaxa.jp/en/ that will allow measurement of ICM turbulence through the Doppler broadening and shifting of metal lines induced by turbulent motions and the effect of turbulence on resonant lines properties.

Fig. 6: A schematic view of turbulence in the ICM. The transition from hydro- to MHD turbulence is marked (see text). The expected spectral features of both solenoidal and compressive turbulence generated at large scales are illustrated : solenoidal turbulence develops an Alfvénic cascade at small (micro-) scales whereas the compressible part (fast modes in the MHD regime) is presumebly dissipated via TTD resonance with electrons and protons in the ICM (or eventually via TTD resonance with CRs in case of reduced effective mean free path, see text). A schematic illustration of relevant examples of “self-excited” modes, excited via CRs- or turbulent-induced instabilities, is also shown together with the relevant scales: slab modes, lower hybrid electrostatic waves, and whistler waves (see text). A schematic illustration of the scales of the Alfvén scale, , classical mean-free-path due to Coulomb ion-ion collisions, , and reduced particles mean-free-path is also given.

2.2.2.1   Turbulent Acceleration

Turbulence in the ICM can potentially trigger several mechanisms of particle acceleration. The non-linear interplay between particles and turbulent waves/modes is a stochastic process that drains energy from plasma turbulence to particles . In addition, reconnection of magnetic fields may be faster in turbulent regions than in stationary media or laminar flows, potentially providing an additional source of particle acceleration in the ICM . In this Section we will focus on stochastic re-acceleration of CRs due to resonant interaction with turbulence, in particular with low-frequency waves, because this turbulent-acceleration mechanism has the best developed theory and is the most commonly adopted in the current literature. Acceleration of CRs directly from the thermal pool to relativistic energies by MHD turbulence in the ICM is very inefficient and faces serious problems due to associated energy arguments . Consequently, turbulent acceleration in the ICM is rather a matter of re-acceleration of pre-existing (seed) CRs rather than ab initio acceleration of CRs, e.g.

The (re)acceleration of CRs by turbulence is customarily described according to the quasi-linear-theory (QLT), where the effect of linear waves on particles is studied by calculating first-order corrections to the particle orbit in the uniform/background magnetic field , and then ensemble-averaging over the statistical properties of the turbulent modes . In QLT one works in the coordinate system in which the space coordinates are measured in the Lab system and the particle momentum coordinates are measured in the rest frame of the background plasma that supports the turbulence and in which turbulence is homogeneous. Then the gyrophase-averaged particle density distribution, , is the cosine of the particle pitch angle, evolves in response to electromagnetic turbulence according to the Fokker-Planck equation :

(6)

where , and are the fundamental transport coefficients describing the stochastic turbulence–particle interactions. These are determined by the electromagnetic fluctuations in the turbulent field.

Much attention has been devoted to the interaction with the low-frequency Alfvén and magnetosonic MHD wavesjjjFollowing here we define low frequency waves as those having frequency , where is the Larmor frequency of nonrelativistic ions. For these waves the relevant Fokker-Planck coefficients are given by :

(7)

where is the wave frequency, the wave-number, and the wave-number components perpendicular and parallel to the background field, , ( is the non-relativistic gyrofrequency) and where we define as the argument for the Bessel functions, . The relevant electromagnetic fluctuations are :

(8)

and

(9)

where and indicate right-hand and left-hand wave polarizations.

Under conditions of negligible damping and the integral in eq.(7) is

(10)

where selects the resonant conditions between particles and waves; namely, (gyroresonance that is important for Alfvén waves) and (Transit Time Damping, TTD, or wave surfing that is the most important for magnetosonic waves) . In the MHD approximation the polarisation and dispersion properties of the waves are relatively simple. For Alfvén waves, , and , while for (fast) magnetosonic waves, , and (see for details); in high beta-plasma such as the ICM.

In the case of low-frequency MHD waves with phase velocities, , much less than the speed of light the magnetic-field component is much larger than the electric-field component, . Then the particle distribution function, , adjusts very rapidly to quasi-equilibrium via pitch-angle scattering, approaching a quasi-isotropic distribution. In this case the Fokker-Planck equation (eq. 6) simplifies to a diffusion-convection equation :

(11)

where the momentum diffusion coefficient parallel to the magnetic field, , is:

(12)

and the spatial diffusion coefficient, , is :

(13)

and where we added two terms in eq. (11), and , that account for energy losses (see Sect. 3.1) and injection of CRs, respectively. To avoid confusion, we mention that the basic physics behind the two diffusion convection equations (2) and (11) is similar. The principal differences are that eq. (11), which targets CR interactions with local turbulence, ignores large-scale spatial variations in the background velocity, , while eq.(2) ignores the momentum diffusion coefficient, , because in strongly compressed flows at shocks it is sub-dominant (note that eq.2 also omits energy losses and injection of CRs).

From eqs. (7)-(9) and (12) and (13) it is clear that the momentum and spatial diffusion coefficients depend on the electric field and magnetic field fluctuations, respectively. Simple, approximate forms for these coefficients can be written in some circumstances that are useful in several astrophysical situations, including galaxy clusters.

For instance, if one assumes isotropic pitch angle scattering by resonant (linearly polarized and undamped) Alfvén waves with , the spatial diffusion coefficient from eq. (13) can be written for relativistic CRs as :

(14)

where represents the net amplitude of (resonant) magnetic field fluctuations defined in eq. (8) and where (in Sect.3.2 we will give a equivalent formula in terms of the Alfvén wave spectrum, eq. 25). We note that for the result is equivalent to the classical Bohm diffusion formula, , that has been used in Sect. 2.2.1.

Similarly, if we assume momentum diffusion due to Transit-Time-Damping, TTD, () interactions with isotropic magnetosonic waves, we can write approximately for relativistic CRs :

(15)

where () depends on details of the turbulence, is the scale on which magnetosonic waves are dissipated, and now represents magnetic field fluctuations associated with those waves through eq.(8) (, e.g. , where is the velocity of large-scale turbulent eddies).

Fig. 7: The coupled evolution with time of the spectra of CRp (upper-left; is in cgs units), CRe (right) and Alfvén waves (bottom-left); Alfvén waves are continuously injected assuing an external source (adapted from ). Panels highlight the non-linear interplay between the acceleration of CRs and the evolution of waves that, indeed, are increasingly damped with time as they transfer an increasing amount of energy to CRs. Saturation of CRe acceleration at later times is due to the combination of radiative losses and the damping of the waves that limits acceleration efficiency.

2.2.2.2.   Application to the ICM & open questions

To account for the turbulence–particle interaction properly, one must know both the scaling of turbulence down to resonant interaction lengths (Eqs. 89), the changes with time of the turbulence spectrum on resonance scales due to the most relevant damping processes, and the interactions of turbulence with various waves produced by CRs. This is extremely challenging. However, in the last decade several modeling efforts attempted to study turbulent acceleration in astrophysical environments, including galaxy clusters, by using physically motivated turbulent scalings and the relevant collisionless damping of the turbulence.

Several calculations suggest that there is room for turbulence in galaxy clusters to play an important role in the acceleration of CRs. This outcome depends on the fraction of the turbulent energy that goes into (re)acceleration of CRs in the ICM.

Presumably the many types of waves generated/excited in the ICM, both at large and very small scales (Fig. 6), should jointly contribute to the scattering process and (re)acceleration of CRs. Much attention has been devoted to CR acceleration due to compressible (fast mode) turbulence that is driven at large scales in the ICM from cluster mergers and that cascades to smaller scales. Under this hypothesis, and as expected for typical conditions in galaxy clusters (i.e., high temperature, high beta plasma and with likely forcing and dissipation scales), it is the compressible fast modes that are the most important in the acceleration of CRs in ICMs. Most of the energy of these modes is converted into the heating of the thermal plasma via the TTD resonance. However current calculations show that TTD can drain as much as a few to 10 percent of the total turbulent–energy flux into the CR component of the ICM plasma. In this case CRe in the ICM can be re-accelerated up to energies of several GeV, provided that the energy budget of fast modes on scales of tens of kpc is larger than about 3-5 percent of the local thermal energy budgetkkkWe note that if compressive turbulence is generated at larger (e.g. a few 100 kpc) scales, this condition in terms of energy budget, essentially implies that a large fraction of the energy of such turbulence is transported to smaller scales. It implies, for example, that weak shocks generated by compressive turbulence and viscosity do not dissipate most of the turbulent energy budget into heat.. We expect that this conclusion is important for understanding the origin of radio halos (Sect. 4).

Moreover there are several circumstances under which the fraction of the turbulent energy that is transferred into CRs may be much larger than a few %, thus making the acceleration process also more efficient than in the previous case. For example, if we consider the scenario discussed above, where compressive (fast mode) turbulence is generated at large scales, a large fraction of the energy of the fast modes can be converted into the acceleration of CRs if the CR particle collision frequencies in the ICM are much larger than those due to the classical process of ion-ion Coulomb collisions. Potentially this may occur if the interactions between particles are mediated by magnetic perturbations generated by plasma instabilities. Instabilities may be driven by compressive turbulence and CRs in the ICM . Another case where a large fraction of the turbulent energy is potentially dissipated into the (re)acceleration of CRs in galaxy clusters is that of incompressible turbulence, where acceleration is driven by Alfvén modes via gyro-resonance (). In this case the important caveat is that Alfvén modes develop an anisotropic cascade toward smaller scales, below the Alfvén scale, that quenches the efficiency of the gyro-resonance, , scattering (acceleration) process . Consequently models of Alfvénic acceleration assume that waves are generated in the ICM at small (quasi-resonant) scales (Fig. 6), although it remains still rather unclear whether such small-scale waves can be efficiently generated in the ICM.

In all these cases, where a fairly large fraction of the turbulent energy is drained into CRs, the efficiency of acceleration is essentially fixed by the damping of turbulence by the CRs themselves. Figure 7 shows an example of CRp and CRe acceleration under these conditions. As time proceeds during the acceleration, the CRs gain energy and extract an increasing energy budget from the turbulent cascade. Consequent modifications are induced in the spectrum of the turbulence, causing a decrement in the acceleration efficiency. This differs from the cases where a smaller fraction of turbulent energy is dissipated into CRs acceleration. In these cases indeed the turbulent properties and the efficiency of turbulent acceleration depend only weakly on the CRs properties and energy budget.

2.3 Generation of secondary particles

If we assume that CRp remain in galaxy clusters for a time-period, , the grammage they encounter during their propagation is g cm. On cosmic time scales that would often be comparable in the ICM with the nuclear grammage required for an inelastic collision, g cm. This implies that the generation of secondary particles due to inelastic collisions between CRp and thermal protons in the ICM is an important source of CRe.

The decay chain for the injection of secondary particles is :

A threshold reaction requires CR protons with kinetic energy just larger than MeV to produce . The injection rate of pions is given more generally by :

(16)

where is the CRp spectrum, is the spectrum of pions from the individual collisions of CRp (of energy ) and thermal protons and is the pp cross section for , and .

Neutral pions decay into –rays with spectrum :

(17)

where . Neutral pions produced near threshold will decay into a pair of s with average energy MeV. This provides a rough measure of the low energy end of the expected -ray spectrum. Charged pion decays produce muons, which then produce secondary electrons and positrons (as well as neutrinos) as they decay. The injection rate of relativistic electrons/positrons then becomes:

(18)

where is the spectrum of electrons and positrons from the decay of a muon of energy produced in the decay of a pion with energy , and is the muon spectrum generated by the decay of a pion of energy .

Secondary electrons continuously generated in the ICM are subject to energy losses (Sect. 3.1). If these secondaries are not accelerated by other mechanisms, their spectrum approaches a stationary distribution because of the competition between injection and energy losses :

(19)

where accounts for CRe energy losses (see Sect. 3.1). Assuming a power law distribution of CRp, , the spectrum of secondary electrons at high energies, , is , with , where approximately accounts for the log–scaling of the p-p cross–section at high energies . The radio synchrotron emission from these electrons ( actually) would have a spectral slope, .

3 The life-cycle of CRs in galaxy clusters

In this Section we discuss the energy-evolution and dynamics of relativistic CRp and CRe that are released in the cluster volume from the accelerators described in the previous Section. In this Section we also discuss the constraints on CR acceleration efficiency and propagation that come from the current -ray and radio observations.

3.1 Energy Losses

Cosmic rays, especially electrons for energies above GeV, are subject to energy losses that limit their life-time in the ICM and the maximum energy at which they can be accelerated by acceleration mechanisms.

3.1.1 Electrons

The energy losses of ultra-relativistic electrons in the ICM are essentially dominated by ionization and Coulomb losses at low energies

(20)

where is the number density of the thermal plasma protons, and and by synchrotron and inverse Compton losses at higher energies,

(21)

where is the magnetic field strength in units of , and we assumed isotropic magnetic fields and distributions of CRe momenta. The factor in the square brackets can alternatively be expressed as , where G is the equivalent magnetic field strength for energy losses due to ICS with CMB photons.

The life-time of CRe, , from Eqs. 2021, is :

(22)

This depends on the number density of the thermal medium, which can be estimated from X-ray observations, on the IC-equivalent magnetic field (i.e., redshift of the cluster), and on the magnetic field strength, which is important only in the case and eventually can be constrained from Faraday rotation measures (Sect. 4.1).

3.1.2 Protons

For relativistic CRp, the main channel of energy losses in the ICM is provided by inelastic p-p collisions (Sect. 2.3). This sets a CRp life–time

(23)

is the inclusive p–p cross-section .

For trans-relativistic and mildly relativistic CRp, energy losses are dominated by ionization and Coulomb scattering. CRp more energetic than the thermal electrons have

(24)

where is the RMS velocity of the thermal electrons, while is the corresponding thermal proton velocity.

Fig. 8: Life-time of CRp (red) and CRe (blue, lower curves) in the ICM at redshift , compared with the CR diffusion time on Mpc scales (magenta, upper curves) (adapted from ). The most relevant channels of CR energy losses at different energies are highlighted in the panel. Adopted physical parameters are : cm, (solid) and G (dashed). Diffusion is calculated assuming a Kolmogorov spectrum of magnetic fluctuations with kpc and .

3.1.3 General energy loss considerations

Figure 8 shows the (total) time scales for losses of CRe and CRp. CRp with energy 1 GeV – 1 TeV are long-living particles with life-times in the cores of galaxy clusters several Gyrs. At higher energy the CRp time-scale gradually drops below 1 Gyr, while at very high energy, in the regime of ultra high energy CRp, the life-time is limited by inelastic collisions with CMB photons, as discussed in Section 2.2.1 (Fig. 5).

On the other hand, CRe are short-lived particles at the energies where they radiate observable emissions, due to the unavoidable radiation energy-losses (mainly ICS and synchrotron). The maximum life-time of CRe, about 1 Gyr, is at energies MeV, where radiative losses are roughly equivalent to Coulomb losses. On the other hand, CRe with energy several GeV that emit synchrotron radiation in the radio band (GHz), have shorter life-times, 0.1 Gyrs. The life-times of CRe at high energies do not vary much from cluster cores to periphery, because for weak magnetic fields they are determined by the unavoidable losses from ICS off CMB photons. On the other hand, CRe ICS lifetimes will scale strongly and inversely with cluster redshift according to (eqs. 21-22).

3.2 Dynamics of CRs in the ICM

The propagation of CRs injected in the ICM is mainly determined by diffusion and convection.

The time necessary for CRs to diffuse over distances is . This implies that the spatial diffusion coefficient necessary for diffusion of CRs over Mpc-scales within a few Gyrs is extremely large, cms. For CRs with GeV energy this is several orders of magnitude larger than that in our Galaxy. This simple consideration suggests that galaxy clusters are efficient containers of CRs.

More specifically, according to QLT the diffusion coefficient for gyro-resonant scattering of particles with Alfvénic perturbations of the magnetic field is (see also Sect. 2.2.2):

(25)

where is the power spectrum of turbulent field-perturbations on a scale (that interacts resonantly with particles with momentum ), such that and ; is the minimum wavenumber (maximum scale) of turbulencelllWe note that if turbulent field-perturbations are on scales , i.e. , and eq. (25) is the coefficient of parallel Bohm diffusion, . In Figure 8 we show a comparison between the life-time of CRs and their diffusion-time on Mpc scales assuming a Kolmogorov spectrum of the Alfvénic fluctuations, , with a maximum scale kpc and . The diffusion time of CRs in this case is substantially larger than a Hubble time, implying that CRp can be accumulated in the volume of galaxy clusters and that their energy budget increases with time, as first realized by . Figure 8 suggests that even a fairly small level of magnetic field fluctuations in the ICM, , should be sufficient to confine most of the CRs in the gigantic volume of galaxy clusters for a time-period comparable to the age of clusters themselves.

Fig. 9: Left panel: spatial distribution of tracers particles in a galaxy clusters at that have been uniformly generated in the cluster region in the simulated box at 30, 6 and 0.1. The bottom-right panel shows the gas density at (adapted from ). Right panel: volume-filling distribution of the turbulent spatial-transport coefficient, , that is estimated from the analysis of numerical simulations of galaxy clusters. Curves refer to cases where turbulent motions are driven by cluster mergers, AGN-jets and sloshing (from ).

Formally eq.25, which has been originally adopted to support CRs confinement in clusters, assumes an isotropic distribution of Alfvénic fluctuations at resonant scales. This is not true in the case of the strongmmmWe note, from the MHD turbulence literature, that the notion “strong,” refers to the strength of the non-linear interaction between turbulent waves, not to the amplitude of the turbulent fluctuations. incompressible turbulence, because its cascading process is anisotropic with respect to the mean field . Consequently, particle scattering is strongly reduced . However, in general this does not imply that CRs scattering is inefficient because, even limiting to the particular case of strong incompressible turbulencennnIf a fraction of the turbulence is compressible, scattering is efficient and dominated by fast modes., we can think that the non-resonant mirror interactions with the slow-mode perturbations provide a lower limit to the rate of scattering that is still orders of magnitude more efficient than that due to the gyroresonant scatter calculated according to QLT We also note that Alfvén waves can be excited directly at resonant scales, for example, due to the streaming instability that is driven by CRs streaming along the field lines, eg. (Fig. 6). In this case the streaming speed (the effective “drift speed”) of CRs gets limited to the Alfvén speed. If this process is efficient in the ICM, CRs drift at the Alfvén speed and the time-scale necessary for CRs to cover Mpc distances is Hubble time using a reference value, cm/s. In the presence of background turbulence, the streaming instability can be partially suppressed . This is because turbulence suppresses the waves responsible for self-confinement of CRs, since they cascade to smaller scales before they have the opportunity to scatter CRs. Such a background turbulence, however, also limits the “free flight” of CRs, through the scattering with the fluctuations of the magnetic field induced by turbulence itself. In the most “favourable” case, where we necglect small scale fluctuations, CRs can “fly” along the magnetic field lines over maximum distances that are of the order of the smallest scale on which the magnetic field is effectively advected by turbulent motions. That is the MHD, Alfvén scale, , where is the turbulent-injection scale and is the Alfvénic Mach number of the turbulence on that scale, eg. (Sect. 2.2.2, Fig. 6). The resulting maximum diffusion coefficient is cm s, still implying a diffusion time over Mpc-scales of several Gyrs. In conclusion, we believe that the wealth of waves that can be naturally generated in the ICM, on both small and large scales, supports the paradigm of confinement of CRs in galaxy clusters; namely, that most of the energy budget of CRp is accumulated in the cluster volume over the cluster life-time.

Assuming, in the light of this discussion, that parallel diffusion of CRs of some energy is strongly suppressed by fluctuations on relatively small scales, the cluster-scale dynamics of these CRs is controlled by advection via gas flows accompanied by a process of turbulent-transport (Figure 9, left). This process is analogous to the transport of passive scalars by a turbulent flow, and it induces the CR particles to exibit a random walk behaviour, eg. , within the bulk flow. This regime is known as Richardson diffusion in hydro-turbulence . In this regime the transport is super-diffusive, , on scales smaller than the injection scale of turbulent eddies and diffusive, , on larger scales. Because it is controlled by the fluid motions, the CR transport-coefficient is energy-independent and can be estimated as , where and are the velocity and scales of the largest turbulent eddies. According to numerical simulations of galaxy clusters the largest values, cms, are derived in the case of merging systems (Figure 9, right) as a result of large scale motions and mixing generated during these events in the ICM. This has the potential implication that turbulence might transport CRs on a scale that could be of the order of cluster-cores, thus, potentially inducing a spatial distribution of CRs that is broader than that of the thermal plasma. Similarly, we note that the same mechanism could also spread metals through the ICM leading to the formation of the flat spatial distributions of metals that are observed in non-cool core clusters .

To avoid confusion from the complexity of the above discussion, we clarify that the relative importance of particle scattering-based diffusion parallel to the local mean magnetic field and bulk fluid turbulence-based transport of CRs in the ICM depend on particle energies and the fluid turbulence properties, with the dynamics of very high energy CRs being dominated by scattering-based diffusion.

3.3 Limits on CRp

The expected confinement and accumulation of CRp generated during cluster formation (Sect. 3.2) motivate the quest for CRp in galaxy clusters. Most of the thermal energy in the ICM is generated at shocks, as previously noted. Estimates of CRp acceleration efficiency suggest as much as 10% of the kinetic energy flux at cosmological shocks may be converted into CRs. Then, one might claim that the resulting energy budget of CRs should be a substantial fraction of the ICM thermal energy. If true, the presence of the CRp could influence many aspects of ICM dynamics, including, for examples, contribution to ICM pressure support and a partial quenching of radiative cooling in core regions.

The most direct approach to constraining the energy content of CRp in ICMs consists in the searches for -ray emission from the decay of the neutral pions due to CRp-p collisions in the ICM. Early space-based -ray upper limits from EGRET observations provided limits in several nearby galaxy clusters . Subsequently, more stringent limits have been derived from deep, pointed observations at energies 100 GeV with ground-based Cherenkov telescopes . These limits, unfortunately, depend on the unknown spectral shape of the CRp-energy distribution and the spatial distribution of CRp in the clusters. The most stringent limits are obtained assuming () and a linear scaling between CRp and thermal energy densities (Fig. 10); under these assumptions a particularly deep limit is derived for the Perseus cluster. Constraints are significantly less stringent for steeper spectra and for flatter spatial distributions of the CRp component in the cluster. The recent advent of the orbiting Fermi-LAT observatory has greatly improved the detection prospects thanks to its unprecedented sensitivity at MeV/GeV energies. However, after almost 5 years of operations, no firm detection of any ICM has been obtained. Only upper limits to the -ray emission have been obtained for both individual nearby clusters and from the stacking of samples of clusters. Under the assumption that the spatial distribution of CRp roughly follows that of the thermal ICM, in the most stringent cases (including stacking procedures) these limits constrain about 1% (Fig. 10), with only a weak dependence on . Similar limits are derived by assuming a spatial distribution of CRp that is slightly broader than that of the ICM, such as that expected from simulations by. Limits on the CRp total energy become gradually less stringent if one assumes a flatter spatial distribution compared to the ICM, since then more CRp reside in regions where the number density of thermal-targets protons is lower and where they consequently produce fewer and -rays.

Fig. 10: A collection of representative upper limits to the ratio of the energy in CRp and thermal ICM as derived from -ray and radio observations. Radio-based upper limits depend on the magnetic field strength in a Mpc volume. Limits from observations with Cherenkov telescopes include the case of Coma , A85 and Perseus . In and a spatially constant ratio of CRp and ICM energy densities are assumed. In the case of Coma we report the limits obtained by these authors by adopting a spatial distribution and spectrum of CRp from numerical simulations. In the case of Perseus the reported region (green) encompasses limits obtained using and a spatially constant ratio of CRp and ICM energy densities and limits obtained by adopting a spatial distribution and spectrum of CRp from numerical simulations. Limits from EGRET and Fermi-LAT are taken from and , respectively. In limits are obtained assuming a spatially constant ratio of CRp and ICM energy densities, whereas we report the limits obtained by by assuming the spatial distribution of CRp from simulations. Upper limits derived from radio observations are taken from . The sensitivity level to CRp from future SKA 1 observations is also reported for clusters at .

Radio observations of galaxy clusters also provide limits on , since these observations allow one to constrain the generation rate of secondary CRe in the ICM . Only a fraction of clusters show diffuse, cluster scale synchrotron emission at the sensitivity level of present observations (Sect. 4). Most do not. Radio upper limits to the cluster-scale emission limit the combination of the energy densities of the magnetic field and secondary CRe, and consequently the energy budget in the form of primary CRp as a function of the magnetic field strength. Faraday rotation measurements provide an indication that the central, Mpc-volume, regions of galaxy clusters are magnetized at G level. This information allows one to break the degeneracy between CRp and magnetic field energy densities, resulting in limits few (Fig. 10).

Current upper limits violate optimistic expectations for the CRp energy content and –ray emission from galaxy clusters derived in the last decade . Consequently the available limits now suggest that the efficiency of CRp acceleration that has been previously assumed for the most important mechanisms operating in galaxy clusters was too optimistic, or that eventually CRp diffusion and turbulent-transport (Sect. 3.2) play an important role. In this respect we notice that current observational constraints refer mainly to the innermost (Mpc) regions of clusters where both the number density of thermal protons (targets for production) and the magnetic field are largest. In fact no tight constraints are available for the clusters outskirts where the CRp contribution might be relatively larger.

As a final remark on this point we note that future radio telescopes, including the phase 1 of the SKA, will have a chance to obtain constraints one order of magnitude deeper. These constraints will be better than current constraints from -ray observations even if we assume that the magnetic fields in the ICM are significantly weaker than those estimated from Faraday rotation measurements (Fig. 10). We note however that current limits on the hard X-ray and -ray emission from galaxy clusters exclude the possibility that the magnetic fields in the ICM are smaller than 0.1-0.2 G. Weaker fields than these in clusters hosting giant radio halos would, as we explain in Sect. 5, necessarily require inverse Compton hard X-rays above current upper limits from the same CRe population responsible for the radio synchrotron emission.

4 Diffuse synchrotron radio sources in galaxy clusters: radio halos and relics

Steep spectrum (, with ), diffuse radio emission extended on cluster scales is observed in a number of galaxy clusters. The emission is clearly associated with the ICM and not individual sources, implying the existence of relativistic electrons and magnetic fields mixed with the ICM. Without entering in the details of the morphological zoology that is observed, see e.g., in this Section we focus on the two main classes of diffuse radio sources in galaxy clusters: radio halos and giant radio relics.

Halos and relics have different properties and presumably also a different origin. Radio halos are classified in giant (Figure 11) and mini radio halos (Figure 15), peaking in intensity near the centre of galaxy clusters and having good spatial coincidence with the distribution of the hot X–ray emitting gas. Radio relics (Figure 16) are typically elongated and located at the cluster periphery. Some clusters include both. The two classes of radio sources differ also in their polarization properties. Halos are generally unpolarised, while relics are strongly polarised. Synchrotron polarisation in the relics is a signature of significant anisotropy in the magnetic field on large scales, probably due either to compression (e.g., shocks) or possibly to shear (e.g., tangential discontinuities). The absence of observed polarization in radio halos and their morphological connection with the thermal X-ray emission suggest that the relativistic plasma that generates that synchrotron radiation occupies a large fraction of the volume filled by the hot X-ray emitting ICM.

Faraday rotation measurements and limits to ICS X-rays (and -rays) indicate that the ICM is magnetised at G levels (Sects. 4.1, 5), in which case the relativistic CRe emitting in the radio band have energies of a few GeV (Lorentz factor ) that have life-times 0.1 Gyr in the ICM (Sect. 3, Figure 8). This short life-time, combined with the excessively long time that is needed by these CRe to diffuse across a sizable fraction of the Mpc-scale of the observed emissions, requires that the emitting particles in halos and relics are continuously accelerated or generated in situ in the emitting regions ; this is known as the diffusion problem.

The hierarchical sequence of mergers and accretion of matter that leads to the formation of clusters and filaments dissipates enormous quantities of gravitational energy in the ICM through processes on microphysical scales. Even if a small fraction of this energy is converted into CRs acceleration we may expect non-thermal emission from galaxy clusters and from the Cosmic Web more generally. Cluster-scale radio sources may probe exactly this process and consequently they are extremely important crossroads of cosmology, astrophysics and plasma physics. However, the very low radio surface brightness of the Mpc-scale radio sources in galaxy clustersooo Jy arcsec at 1 GHz, combined with their steep radio spectra (average values reported for are in the range 1.2–1.4), make their detection difficult. Presumably this implies that we are currently detecting only the tip of the iceberg of the non-thermal radio emission from the Cosmic Web , which also implies that current observational classification is possibly subject to a revision in the next years (for observational hints of diffuse emission on possibly very large scales, such as radio bridges and complex emissions, see also and ref. therein).

Theoretically several mechanisms that are directly or indirectly connected with the formation of galaxy clusters may contribute to the origin of the observed radio emission. In this respect it is commonly accepted that radio halos and relics are due to different mechanisms and consequently probe different pieces of the complex physics in these environments. Specifically giant radio halos probably trace turbulent regions in the ICM where particles are trapped and accelerated by some mechanism, while, on the other hand, radio relics are associated with cosmological shock waves where particles can be accelerated. As discussed in Sect 2.2 shocks and turbulent motions in the ICM are tightly connected, and this ultimately may provide a possible physical link between relics and halos .

4.1 A brief note on B estimates in the ICM

In the following Sections we shall use cluster-scale radio sources as probes of the physics of CRs in the ICM. The origin and distributions of magnetic fields in galaxy clusters are not the main focus of our review. However, for clarity we briefly comment on the current ideas for the origin of magnetic fields in these systems. In particular, they rely on the possibility that seed magnetic fields, of cosmological origin or injected in the volume of galaxy clusters by galaxies and AGNs, are mixed and amplified by compression and stretching through accretion and turbulent-motions induced by shocks and clusters dynamics.

However, given the importance of ICM magnetic field strength estimates from Faraday rotation in constraining CR populations, it is appropriate here to outline briefly how they are currently obtained. They depend, of course, on the circular birefringence at radio frequencies of a magnetized plasma. Propagating through such a medium, the plane of linear polarization (generally in synchrotron emission) rotates through an angle , where , with the magnetic field component along the line of sight in G, is the electron number density in units of cm, and the differential path in kpc. Examination of the wavelength variation in polarization seen through the ICM is used to measure distributions of RM across the observed sources. The RM distribution carries essential information about the ICM magnetic field strength and structure when combined with X-ray data to establish the electron density distribution. As already noted (Sects. 2-3), the ICM magnetic field is expected to be turbulent. In that case will fluctuate (around zero if the turbulence is isotropic), reducing the mean rotation measure, , towards zero if the mean aligned field vanishes, but adding a nonzero dispersion, , where the mean number density, the rms strength of the line-of-sight magnetic field, the path length, and , with the 3D correlation length of the magnetic field . The units are the same as before. In practice the magnetic field estimates depend on estimations for (or sometimes ) and . Accuracies in these are controlled by limited statistics in the RM distribution (since available coverage of the ICM is generally rather sparse), by the indirect connection between the 2D RM coherence scale and the 3D magnetic field correlation scale, (but see ), and often by uncertainties in the total ICM path to a polarized source, if it is embedded. Typically, RM samples are taken from several moderately extended background and embedded synchrotron sources distributed across an area comparable to or larger than the cluster core. By adopting simple models for the systematic radial scaling between the magnetic field and the ICM density, along with simple parameterization of the magnetic field turbulence spectrum several such analyses have been conducted, with typical resulting central ICM magnetic field values G