Main expectations

Acceleration of CRe from MHD turbulence is a a rather inefficient mechanism (Brunetti & Jones 2014) that boosts the energies of electrons only up to ∼10 Gev, due to the competition with radiative (IC and synchrotron) losses. This implies that a high frequency cut off in the synchrotron spectra of radio halos, with νb being the

break frequency. Consequently, the possibility to detect a radio halo is related to the observing frequency, νobs and, in particular, the spectral steepening challenges

the detection of radio halos with νs ∼ few × νb < νobs, where we have defined νs

as the frequency at which the steepening becomes severe. A crucial prediction of the model proposed by Cassano & Brunetti (2005) and Cassano et al. (2006) is the existence of radio halos with different spectral properties. As a reference, if we assume a simplified homogeneous model (i.e. typical acceleration efficiency and magnetic field in the ICM), the break frequency of radio halos is expected to be proportional to the electron acceleration coefficient:

νb ∝ hBiγ2max∝

hBiχ2

(hBi2_{+ B}2

CM B)

2 (2.2)

where γmax is the maximum energy of the emitting electrons. χ, in turns, is related

to the energetics of the merger event, and thus to the mass of the clusters involved and to the kind of merger (major or minor merger). Therefore, high values of νb are

expected for massive clusters and in connection with major mergers. Consequently, radio halos with νb >GHz are expected to form only in the most massive clusters,

undergoing energetic merger events, that however are rare in the Universe (Fig. 2.2). At the same time, an unique prediction of this model is the existence of

Figure 2.1: Top: merger trees obtained from Monte Carlo simulations for clusters with present- day mass M0 = 2.5 × 1015M. Bottom left : electron spectra calculated at different redshift for

the same cluster. Bottom right : Evolution with redshift of the electron-acceleration coefficient resulting from the combination of the merger events that contribute to the ICM turbulence at a given redshift.

Figure 2.2: Cartoon of radio halo spectra expected in the framework of turbulent re-acceleration models. Yellow regions represent two possible observing frequency bands.

radio halos with νb <GHz, generated in less massive clusters and/or minor mergers.

These radio halos, when observed at ∼GHz frequencies should show extremely steep spectra (α < −1.5) and they are expected to constitute the class of USSRHs. Less energetic merger events, involving clusters of smaller masses, are more common in the Universe, therefore, given the existence of USSRHs, the number of radio halos is expected to increase at low observational frequencies (Fig. 2.2, see Sect. 2.1.1.1) .

2.1.1.1 Probability of forming radio halos

Since the break frequency depends on the electron acceleration coefficient, the above-mentioned condition for the detectability of radio halos becomes a condi- tion on the acceleration coefficient: χ ≥ χmin(νobs). Each merger tree of each

cluster in the model, has a corresponding χ(z) (Fig. 2.3) that reflects the formation history of the cluster. We can calculate the fraction of time that the cluster spends with χ ≥ χmin in a given mass and redshift range (∆M and ∆z in Fig.2.3). The

total probability of forming a radio halo is obtained from the combination of all the contributions of the merger trees relevant for that mass and redshift bins. This probability is then weighted with the Press & Schechter mass function to take into account the number density of clusters as a function of mass.

The basic prediction of the model is that the probability of forming radio halos increases with the mass of the host cluster (Fig.2.4, left panel, Cassano et al. 2006). Even though these calculations are complex and based on a number of unknown

Figure 2.3: Top: example of two merger trees (black and red lines) of the same cluster. Bottom: evolution of the electron acceleration coefficient, χ(z), associated with the two merger trees re- ported above. For a given ∆M (and ∆z) selected on the merger trees (bold face lines in the upper panel), we selected the fraction of time that each cluster spends with χ ≥ χmin (bold face lines in

Figure 2.4: Left panel : probability of forming radio halos as a function of the cluster mass in three redshift bins: z = 0 − 0.1 (black line), z = 0.2 − 0.3 (blue line) and z = 0.4 − 0.5 (green line). Right panel : Probability of forming radio halos as a funtion of redshift in two mass bins: M = 1 − 2 × 1015_M

 (cyan line) and M = 2 − 4 × 1015M (blue line).

parameters, this is a clear prediction of the re-acceleration model, because the mass, which defines the energetic of the merger events, sets the amount of energy available for particle acceleration. On the other hand, the probability of forming radio halos is expected to decrease with redshift due to the increasing IC losses (Fig. 2.4, right panel). Indeed, at high redshift, the electron energy losses via IC largely prevail the synchrotron energy losses (BCM B ' 3.28 × (1 + z)2 µG).

Another crucial expectation of turbulent re-acceleration models is that the frac- tion of clusters with radio halos should increase at low frequencies (Fig. 2.5, Cassano et al. 2010a, 2012). In particular, such increase is expected to be more significant for relatively low mass systems and at higher redshift (Fig. 2.5). This is due to the presence of a population of USSRHs, that are only visible at low radio frequency. In fact, USSRHs are expected to be generated in smaller systems and/or at relatively higher z with respect to radio halos emitting up to GHz frequencies. In low mass clusters, the merger driven turbulence is not sufficient to accelerate particles up to the energy necessary to emit synchrotron radiation at GHz frequencies, and at high redshift IC losses limit the maximum energy of the accelerated electrons and thus decrease the break frequency. This prediction is particularly relevant for radio telescopes such as the LOFAR (R¨ottgering et al. 2011; van Haarlem et al. 2013; R¨ottgering et al. 2013) and the MWA (Tingay et al. 2013; Wayth et al. 2015) that are opening a new observational window, at low frequencies.