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Based on the observational results summarised in Table 1 and in the previous sections, and on the theoretical models reviewed, for instance, by Brunetti (2004), Blasi et al. (2007) and Dolag et al. (2008), we have now a formation scenario for the different diffuse and extended radio sources in clusters. The current theories on the origin of the non-thermal component in galaxy clusters will be the starting point for new observational studies with the next generation radio telescopes (Sect. 5).

Table 1. Main observational properties of the different sources of diffuse radio emission in galaxy clusters. Note that linear polarisation levels of ~ 20-40% have been detected in filamentary structures of the radio halo in A 2255 by Govoni et al. (2005).

Position Size alpha Polarisation Example

Halo Centrally peaked gtapprox Mpc gtapprox 1 < few % Coma
Giant relic Peripheral ~ Mpc gtapprox 1 ~ 10-30% Abell 3667
Mini-halo Centrally peaked ltapprox 0.5 Mpc gtapprox 1.5 < few % Perseus
Phoenix Peripheral ~ 102 kpc gtapprox 1.5 ~ 10-30% Abell 85
AGN relic Close to the host galaxy few × 10 kpc gtapprox 1.5 ltapprox 20% Abell 133

Giant radio halos and relics are the most spectacular radio sources in clusters, and, as stated above, their synchrotron spectrum indicates the presence of cosmic rays that gyrate around magnetic field lines, frozen in the ICM. Therefore, relativistic particles cannot stream out from the gravitational field of the cluster, but they can still diffuse along magnetic field lines. It has been shown however (Völk et al. 1996, Berezinsky et al. 1997, Völk & Atoyan 1999) that typical relativistic electrons in radio halos and relics (with gamma ~ 1000 - 5000) have diffusion times which are longer than the Hubble time. They could therefore be simply diffused over cluster scales from one or more active radio galaxies (Jaffe 1977, Rephaeli 1977). However, the steep radio spectra of these sources indicate short lifetimes for the radiating particles (~ 108 yr), which lose energy not only via synchrotron emission, but also due to interactions with the Cosmic Microwave Background (CMB) photons (via Compton scattering emission) and with the ICM (via Coulomb interactions and Bremsstrahlung emission). The main radiative losses of electrons are due to Compton scattering of the CMB, and synchrotron emission; the former process dominates for B < 3 µG (the field equivalent of the CMB energy density). The radiative lifetime of a relativistic lepton with a Lorentz factor gamma < 108 is thus approximately given by (e.g. Longair 1981, Meisenheimer et al. 1989)

Equation 2 (2)

Since the expected diffusion velocity of the relativistic electrons is of the order of 100 km s-1 (Alfvén speed), cosmic rays do not have the time to propagate over the Mpc-scales of giant cluster radio sources. This excludes the hypothesis that relativistic electrons are produced at a localised point, and requires in situ acceleration mechanisms. Basically, two classes of models have been proposed:

The observational properties of radio halos and relics (see Sects. 2.1 and 2.3) are more in favour of primary models. The strongest point leading to this conclusion is the fact that diffuse and extended radio emission has been detected up to now only in merging clusters. A strong connection between cluster mergers and cosmic ray production is required in primary models, and is not expected in secondary models. In this respect, the fact that halos and relics are quite rare in clusters is again disfavouring the hadronic collision hypothesis, based on which we should expect electron acceleration to be possible in all galaxy clusters.

Since the shape of the synchrotron spectrum depends on the last acceleration phase of cosmic rays, detailed studies of the spectral index distribution in radio halos and relics provide important information on acceleration mechanisms acting in clusters. Primary models predict a radial steepening and a complex spatial distribution of the spectral index alpha, due to the existence of a) a maximum energy to which electrons can be accelerated (gamma < 105, Blasi 2004 and references therein), and b) different re-acceleration processes in different cluster regions. Secondary models assume that cosmic ray protons are accelerated during structure formation over cosmological epochs and accumulated in clusters. The collision of these protons with the thermal ICM would continuously inject electrons, resulting in a spectral index distribution unrelated to the intracluster magnetic field strength, and thus not dependent on the position in the cluster. The radial spectral steepening and/or the patchy structure of spectral index maps observed in several radio halos (Sect. 2.1) are clearly favouring primary models.

The hadronic collision hypothesis predicts power-law spectra flatter than primary models (alpha ltapprox 1.5) and magnetic field values higher than a few µG. Observational results are controversial concerning these points, due to the observed intermediate values of the radio spectral index (alpha ~ 1 - 1.5), and the widely differing estimates for mean intracluster magnetic field values (Sect. 4). Finally, emission of gamma-rays is expected in secondary models (Blasi et al. 2007), a challenging point to be tested observationally (Rephaeli et al. 2008 - Chapter 5, this volume). On the other hand, it has been suggested (Bykov et al. 2000, Miniati 2003) that the detection of gamma-ray emission from clusters may not necessarily reflect the hadronic origin of cosmic rays, since it could be related to the Compton scattering of CMB photon from shock-accelerated, intracluster electrons.

Given our current observational and theoretical knowledge, cosmic rays in giant radio relics (bottom panel of Fig. 5) are most likely originating from Fermi-I diffuse shock acceleration of ICM electrons (e.g. Hoeft & Brüggen 2007, Bykov et al. 2008 - Chapter 7, this volume). These radio sources would therefore trace the rim of shock fronts resulting from cluster mergers in the ICM, and they have been named "radio gischt 3" by Kempner et al. (2004). Firstly, this hypothesis is in agreement with the morphology and the position of most of the detected giant relics, which appear as elongated, sometimes symmetric, radio sources in the cluster periphery, where we expect to find arc-like shock fronts resulting from major cluster mergers (e.g. Schindler 2002). Secondly, the quite strong linear polarisation detected in giant relics would be in agreement with the model prediction of magnetic fields aligned with the shock front. Based on some observational results, however, a clear association between shocks and giant radio relics is not always straightforward. This is true in the case of the "exotic" giant relics mentioned in Sect. 2.3 (e.g. those with circular shapes, or located in intracluster filaments). Additionally, Feretti & Neumann (2006) did not detect a shock wave corresponding to the radio relic in the Coma cluster. They suggested that, similarly to radio halos (see below), the radio emission of this relic source is instead related to turbulence in the ICM. Currently, the main observational limitation to test the origin of giant radio relics comes from X-ray data. The sensitivity of X-ray instruments is not high enough to detect shock waves in the external regions of clusters, where the gas density and thus the X-ray surface brightness are very low, and where most of radio relics have been detected (see for instance the radio relic found in A 521 by Ferrari (2003); see also Ferrari et al. 2006a, Giacintucci et al. 2006).

The second class of relic sources pointed out in Sect. 2.3 (middle panel in Fig. 5), characterised by smaller sizes than giant relics (~ 102 kpc vs. 103 kpc), are most likely originating from adiabatic compression in cluster shocks of fossil radio plasma, released by an AGN whose central engine has ceased to inject fresh plasma ltapprox 1 Gyr ago (Enßlin & Gopal-Krishna 2001). The old non-thermal electrons, that would be undetectable at high (~ GHz) frequencies, are actually re-energised by the shock. This class of sources are therefore also called "Phoenix" (Kempner et al. 2004). The main physical difference with giant relics is related to the fact that, in the latter, shocks accelerate thermal ICM electrons to relativistic velocities through Fermi-I processes, while, in the case of Phoenix sources, the shock waves energise the relativistic plasma by adiabatic compression. The sound speed inside the fossil radio plasma is actually so high that shock waves cannot penetrate into the radio cocoons. These sources are rare because they require shocks and fossil plasma in the same region of the cluster. Moreover, adiabatic compression is efficient in re-accelerating electrons only if the time elapsed since the last AGN activity is not too long, i.e. less than about 0.1 Gyr in the cluster centre and less than about 1 Gyr in the periphery. All this also explains why we detect Phoenix sources in the external regions of clusters.

Among the sources called "radio relics" in the literature, only the smallest (several tens of kpc) are real "AGN relics" (top panel in Fig. 5). They are extinct or dying AGNs, in which the central nucleus has switched off, leaving the radio plasma to evolve passively (e.g. Murgia et al. 2005, Parma et al. 2007). Their spectrum becomes steeper and steeper, making the source more and more difficult to be detected at high frequencies, until it disappears completely (see Sect. 2.3). Due to the short radiative lifetime of their electrons (~ 107 - 108 yrs), these sources are usually located close to their host galaxy, which did not have time enough to move far away in the cluster potential.

In the case of giant radio halos, spectral index maps show no evidence of flattening at the location of shocks detected in X-rays (A 665: Feretti et al. 2004a, Markevitch & Vikhlinin 2001; A 2744: Orrù et al. 2007, Kempner & David 2004). This agrees with theoretical results showing that shocks in major mergers are too weak to produce relativistic particles uniformly over the whole central ~ 1 Mpc area of clusters (Gabici & Blasi 2003). Although it cannot be excluded that shock acceleration may be efficient in some particular regions of a halo (e.g. Markevitch et al. 2005), it has been suggested that cluster turbulence generated by cluster mergers may efficiently accelerate electrons in the cluster volume (e.g. Cassano & Brunetti 2005). The observed steepening of the spectral index with the distance from the cluster centre, and the few available spectral index maps showing flatter spectra in the regions influenced by merger processes (Sect. 2.1) support the scenario that ICM turbulence supplies the energy for the radiating electrons.

However, if the predictions of primary models better agree with the observational results, the merging event cannot be solely responsible for electron re-acceleration in giant radio halos and relics, because gtapprox 40% of clusters show evidence of a disturbed dynamical state (Jones & Forman 1999), while only ltapprox 10% possess radio halos and/or relics. As we have seen in Sect. 2.1, the power of observed radio halos Pnu seems to correlate with the mass M of their host cluster. The energy available to accelerate relativistic particles during cluster mergers is a fraction of the gravitational potential energy released during the merging event, that in turn scales as ~ M2. The Pnu - M relation could thus suggest that only the most massive mergers are energetic enough to efficiently accelerate cosmic rays (Buote 2001). A recent model by Cassano & Brunetti (2005) is in agreement with this conclusion, showing that only massive clusters can host giant radio halos. The probability to form these extended radio sources increases drastically for cluster masses M geq 2 × 1015 Modot since the energy density of the turbulence is an increasing function of the mass of the cluster. Based on the scenario of hierarchical structure formation, massive clusters result from a complex merging history, during which each cluster-cluster collision could have contributed to provide energy for cosmic ray acceleration.

Finally, as we have seen in Sect. 2.2, radio mini-halos have also been observed in clusters. They are located at the centre of cooling flow clusters and surround a powerful radio galaxy. Similarly to giant radio halos and relics, the electrons in radio mini-halos have short radiative lifetimes due to the high magnetic fields present in cooling cores (Taylor et al. 2002). The observed radio emission is thus not due to the radio lobes of the central AGN. Unlike the giant sources, mini-halos are typically found in clusters not disturbed by major mergers (Sect. 2.2). Again, two possible classes of models have been proposed. Relativistic electrons could have again an hadronic origin (Pfrommer & Enßlin 2004). Or they could be a relic population of (primary) relativistic electrons re-accelerated by MHD turbulence, with the necessary energy supplied by the cooling flow (Gitti et al. 2002). The re-acceleration model by Gitti et al. (2002) has been successfully applied to two cooling flow clusters (Gitti et al. 2002, 2004). The observed correlation between the mini-halo and cooling flow power has also given support to a primary origin of the relativistic electrons (Gitti et al. 2004, 2007a). However, there also seems to be some observational and theoretical evidence to support hadronic origin (Kempner et al. 2004 and references therein). Additionally, in two clusters (A 2142: Markevitch et al. 2000; RXJ 1347.4-1145: Gitti et al. 2007a, Gitti et al. 2007b), we got indications that cluster mergers and cooling flows may act simultaneously in powering mini-halo emission in the rare and peculiar clusters in which they coexist. Further theoretical and observational studies are indeed essential due to the low number of known radio mini-halos (Sect. 2.2).

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