How many radio relics await discovery?

Upcoming radio telescopes will allow to study the radio sky at low frequencies with unprecedent sensitivity and resolution. New surveys are expected to discover a large number of new radio sources. Here we investigate the abundance of radio relics, i.e. steep-spectrum diffuse radio emission coming from the periphery of galaxy clusters, which are believed to trace shock waves induced by cluster mergers. With the advent of comprehensive relic samples a framework is needed to analyze statistically the relic abundance. To this end, we introduce the probability to find a relic located in a galaxy cluster with given mass and redshift allowing us to relate the halo mass function of the Universe with the radio relic number counts. Up to date about 45 relics have been reported and we compile the resulting counts, N(>S_1.4). In principle, the parameters of the distribution could be determined using a sufficiently large relic sample. However, since the number of known relics is still small for that purpose we use the MareNostrum Universe simulation to determine the relic radio power scaling with cluster mass and redshift. Our model is able to reproduce the recently found tentative evidence for an increase in the fraction of clusters hosting relics, both with X-ray luminosity and redshift, using an X-ray flux limited cluster sample. Moreover, we find that a considerable fraction of faint relics (S_1.4<~10 mJy) reside in clusters with an X-ray flux below ~3e-12 erg/s/cm^2. Finally, we estimate the number of radio relics which await discovery by future low frequency surveys proposed for LOFAR and WSRT. We estimate that the WODAN survey proposed for WSRT may discover 900 relics and that the LOFAR-Tier 1-120 MHz survey may discover about 2500 relics. However, the actual number of newly discovered relics will crucially depend on the existence of sufficiently complete galaxy cluster catalogues.


INTRODUCTION
Some galaxy clusters show in their outskirts large-scale diffuse radio emission, which apparently does not originate from any individual galaxy. These objects are called 'radio relics'. Spectacular examples have been found for instance in A 3667 (Röttgering et al. 1997), A 3376 (Bagchi et al. 2006), and CIZA 2242(van Weeren et al. 2010. Diffuse sources are difficult to detect due to the low surface brightness and due to the steepness of the spectra. Moreover, they can be ⋆ E-mail: snuza@aip.de only classified as relics if galactic foreground and fossil radio galaxy emission can be excluded and the hosting galaxy cluster is identified. We give a list of currently known relics in Section 4.1. Radio relics show steep spectral slopes, which suggest that the origin of the radiation is synchrotron emission. Hence, radio relics indicate both the presence of relativistic electrons and magnetic fields. There are several approaches to estimate the strength of the magnetic field in the region of relics. Equipartition arguments have been applied leading to field strengths in the range ∼ 0.5 − 2 µG (Govoni & Feretti 2004). For the northwest relic in A 3667 upper limits of the c 0000 RAS hard X-ray flux in that region provide a lower bound for the magnetic field, namely 1.6 µG (Nakazawa et al. 2009). The Rotation Measure distribution of polarized emission from sources in the cluster volume or in the background allows to constrain the magnetic field strength and spectrum leading to values of ∼ 1-5 µG (Vogt & Enßlin 2003;Kuchar & Ensslin 2009;Bonafede et al. 2010).
Magnetic fields in galaxy clusters are either primordial (Grasso & Rubinstein 2001) or injected in the protocluster region by AGN and/or galactic winds (Völk & Atoyan 2000). Whatever the origin of the initial seed, some amplification mechanisms are required to account for their strength in clusters. Cosmological magnetohydrodynamics (MHD) simulations predict magnetic field strengths of the order of µG spread over the cluster volume (e.g. Dolag et al. 2005). These studies indicate that the amplification of the magnetic field resulting by pure adiabatic contraction is not sufficient to explain the observed magnetic field strengths. Merger events and accretion of material onto galaxy clusters are supposed to drive significant shear-flows and turbulence within the intra-cluster medium (ICM). This can in principle amplify magnetic fields up to at least µG levels (see Dolag et al. 2008, and references therein).
The morphology and temperature distribution of the Xray emission of the clusters which host radio relics indicate that relics only occur in systems with an ongoing or recent merger, e.g. A 754 . For A 3667 it has been shown that the relic is located where the bow shock of the moving sub-clump is expected (Vikhlinin et al. 2001). For some other clusters the density and the temperature jump of the shock front at the position of the relic have been identified (see e.g. Markevitch 2010, and references therein). This suggest the following scenario for the origin of the large-scale radio relics: cluster mergers lead to the formation of shock fronts which are responsible of electron acceleration causing the relic radio emission.
Two main mechanisms for the acceleration of electrons have been proposed to explain radio relics: (i) adiabatic compression of fossil radio plasma by the shock wave (Enßlin & Gopal-Krishna 2001;Enßlin & Brüggen 2002) or (ii) diffusive shock acceleration (DSA) by the Fermi-I process (Drury 1983;Blandford & Eichler 1987;Jones & Ellison 1991;Enßlin et al. 1998;Malkov & O'C Drury 2001). In the first scenario, radio relics should have toroidal and complex filamentary morphologies showing very steep, curved radio spectra due to inverse Compton (IC) and synchrotron losses. In the DSA scenario the electrons are accelerated by multiple crossings of the shock front (in a first order Fermi process) tracing shocks in the presence of ubiquitous magnetic fields. It is worth noting that other alternative scenarios have also been suggested (e.g. Keshet 2010).
The formation of radio relics in cosmological simulations has been studied e.g. by Hoeft et al. (2008), Battaglia et al. (2009) and Skillman et al. (2010). The latter studied structure formation shocks present in two cosmological boxes with a comoving volume of 64 3 h −3 Mpc 3 and 200 3 h −3 Mpc 3 , and use the non-thermal DSA radio model of Hoeft & Brüggen (2007) (hereafter HB07) to give an estimation of the radio relic luminosity function (RRLF) for z = 0 and 1 at 1.4 GHz. On the other hand, Cassano et al. (2010), using a Monte Carlo approach, studied the ocurrence of 'radio haloes' in merging galaxy clusters assuming that electrons are re-accelerated through MHD turbulence, posing interesting constraints for the upcoming Low Frequency Array (LOFAR) at 120 MHz. Here we would like to provide the appropiate scalings of the relic radiation as a function of cluster mass and redshift for a given frequency, as well as to give some plausible predictions for upcoming radio surveys within the context of the DSA scenario.
Currently, there are several new radio telescopes under construction, in particular, for the very low frequency regime. Moreover, several existing telescopes are getting significantly improved receivers or backends. For instance, in the Netherlands and neighboring countries LOFAR is almost completed. This instrument will survey the sky in the frequency range from 30 to 240 MHz. New receivers and electronics of the expanded Very Large Array (eVLA) will drastically improve the sensitivity of the Very Large Array (VLA). Furthermore, the Westerbork Synthesis Radio Telescope (WSRT) will be equipped with focal-plane array receivers which will be optimized for 1.4 GHz observations. This will significantly increase the field-of-view and will hence improve the survey speed tremendously rising the expected number of relic observations. In this work we develop a framework to relate the abundance of galaxy clusters in the Universe to the radio relic number counts. To this end we introduce the 'radio relic probability density', i.e. the probability of finding a radio relic with a given radio power located in a galaxy cluster of given mass and redshift. A large sample of observed relics would allow to fully determine the probability density function for a given halo mass function in the Universe. However, since only about 45 relics are presently known we therefore use the MareNostrum Universe cosmological simulation as a way of determining how the probability density scales with cluster mass and redshift. In order to normalize this function we compile a list of currently known radio relics.
From the resulting probability distribution we are also able to reproduce the recently found fractions of clusters with relics in the combined NORAS+REFLEX cluster sample presented by van Weeren et al. (2011c). Finally, we draw some conclusions on the amount of non-identified relics due to the fact that the hosting cluster is still not known. Moreover, we present some estimates on how many relics may be identified by upcoming radio surveys assuming plausible survey specifications. This paper is organized as follows. In Section 2 we derive the formalism to estimate the number of radio relics and introduce the 'radio relic probability function'. We also extend the usually assumed sharp transition for the flux detection limit to a 'discovery probability' which is more appropriate for relic samples. In Section 3 we present the cosmological simulation used in this work and briefly summarize the shock detection method and the non-thermal radio emission model adopted. In Section 4 we present the most up to date observed relic sample, discuss the normalization of our model counts and its comparison to observations and present our predictions for upcoming radio surveys. Finally, in Section 5 we close the paper with the summary.
In what follows we assume a flat ΛCDM cosmology with a matter density parameter ΩM = 0.27, an amplitude of mass fluctuations σ8 = 0.8 and a Hubble constant H0 = 100 h km s −1 Mpc −1 , with h = 0.7 (e.g. Komatsu et al. 2011).

HOW TO ESTIMATE THE NUMBER OF OBSERVABLE RELICS?
In this Section we present the formalism aim at estimating relic number counts in a given radio survey as well as the radio relic probability density.

Cumulative radio relic number counts
How many relics are seen in the sky above a given radio flux Sν at the observing frequency ν obs ? The flux of a source with luminosity per unit frequency Pν := dP dν (ν) located at redshift z is given by where ν is the rest frame frequency and d l (z) is the luminosity distance keeping in mind that the appropiate redshift correction for the frequency between the rest and observer frames needs to be considered. We introduce the luminosity function of 'radio relic clusters', i.e. the number of galaxy clusters per unit comoving volume and logarithmic relic radio power as a function of frequency and redshift where N is the number of clusters and Vc the comoving volume. The RRLF is obtained by the convolution of a halo mass function (e.g. Tinker et al. 2008) with the 'radio relic probability density' of finding a galaxy cluster of mass M , redshift z and relic radio power Pν. Therefore, the RRLF becomes where the relic probability density p(Pν, M, z) fulfills the following condition Hence, integrating Eq. (2) allows us to write the total abundance of relics per logarithmic flux bin as follows Note that we have used d log Pν = d log Sν since Pν depends linearly on Sν.
In observations low luminosity radio relics are hard to identify since the surface brightness may be too low to exhibit typical morphological features or spectral index variations. Moreover, a diffuse radio object is only identified as a relic when the galaxy cluster can be unambigously detected. Depending on the mass of the cluster this may also be challenging. As a consequence, we introduce a 'discovery probability', instead of a sharp flux-limit we use a smooth transition, which includes both the sensivity of the survey and the uncertanties present in the identification. We write this probability as where the effective sensitivity, S eff ν , basically gives the fluxlimit and w the width of the transition.
Finally, the cumulative radio relic function can be computed convolving Eq. (5) with the 'discovery probability' and multiplying by the sky fraction, fs, covered by the radio survey The radio flux-luminosity relation given by Eq.
(1) and the redshift integration of Eq. (5) are fully determined by the cosmological parameters. Since recent cosmological observations show that the resulting parameters are well constrained the procedure described above can be considered as a direct relation between the radio relic probability density and the observed number counts.

Radio relic probability density
If we were to build a perfect radio telescope that could detect even the faintest radio emission, which radio power distribution function linked to structure formation shocks should we expect? Merger shocks can persist in the cluster periphery basically forever, hence, every cluster should show some relic radio emission. On the other hand, very bright and very faint relics are most likely rare events. We therefore expect that there is a typical radio luminosity for a cluster with given mass and redshift although the related flux is evidently below current detection limits. As a consequence, we assume that the probability density to find a relic is given by a log-normal distribution: where σP is the standard deviation of the logarithmic radio power andPν is the mean radio power that scales with hosting cluster mass, observed frequency and redshift respectively. We parametrize this function as follows The radio power normalization is given by the 'reference radio power' P0 while the scaling with hosting cluster mass, redshift and observing frequency is governed by CM , Cz and Cν respectively. Formally, a different functional form could have been chosen for the radio relic probability density as long as the condition given in Eq. (4) is fulfilled. However, we will show in Section 3.3 that a log-normal function describes reasonably well the radio power distribution of our simulated relic samples.

SIMULATING RADIO RELICS
In order to simulate radio relics we need to use a galaxy cluster sample extracted from a cosmological simulation and apply an emission and magnetic model to the present shock waves. In this section we present our cosmological simulation, the method used to detect shocked gas within the simulated volume and our radio power emission model. Finally, we estimate the parameters of the relic probability density using our cosmological simulation.

The simulated galaxy cluster sample
Our simulated galaxy cluster sample was selected from the MareNostrum Universe cosmological simulation which is a non-radiative hydrodynamical run of a representative region of the Universe (Gottlöber & Yepes 2007). The simulation was run with the smoothed particle hydrodynamics (SPH) Gadget-2 code (Springel 2005). The adopted cosmology is in agreement with a flat ΛCDM scenario having a matter density parameter ΩM = 0.3, a baryon density parameter Ω b = 0.045, an intial matter power spectrum characterized by a scalar spectral index n = 1 and normalized to σ8 = 0.9, and a dimensionless Hubble parameter h = 0.7. The simulation started at z = 40 using a linear density field represented by 2 × 1024 3 gas and dark matter particles in a comoving box of 500 h −1 Mpc on a side. The resulting mass resolution for gas and dark matter particles is 8.3 × 10 9 h −1 M ⊙ and 1.5 × 10 9 h −1 M ⊙ respectively. Identification of bound structures is done using the parallel friends-of-friends (FoF) algorithm described in Klypin et al. (1999) with a linking length of 0.17 in units of the mean interparticle separation. In order to generate our galaxy cluster catalogs as a function of cosmic time we consider five different redshifts up to z = 1, namely z = 0, 0.25, 0.5, 0.75 and 1, and take the 500 most massive galaxy clusters present at each cosmic time. In this way, the range of cluster masses we are able to probe goes from ∼ 10 14 h −1 M ⊙ up to ∼ 2.5 × 10 15 h −1 M ⊙ , meaning that the baryonic component of the systems inside the virial radius is typically resolved with thousands of gas particles for the less massive clusters and with several tens of thousands for the most massive ones.

Shock finding and radio emission in the simulation
The cosmological SPH code Gadget clearly accounts for shock dissipation as shown by shock tube simulations (e.g. Springel et al. 2001). To this end, artificial viscosity has been introduced into SPH, which evaluates the local velocity field to estimate the dissipation (Monaghan 1992). However, this technique is not able to determine the Mach numbers, which are needed for combining SPH simulations with parametric models for radio emission of relics. Two methods have been introduced for locating shock fronts and estimating their strength: Pfrommer et al. (2006) uses the increase of entropy with time while Hoeft et al. (2008) evaluates spatial entropy gradients in single snapshots of the simulation. Here we apply a slightly modified version of the latter approach. Briefly, our scheme for locating shock fronts can be summarized as follows. For a given gas particle we evaluate its  Table 1. Best fit parameters for the radio relic probability density given by Eq. (8) using our set of MareNostrum clusters for magnetic field scaling models 'a' and 'b' (see text). The reference radio power, P 0 , is obtained in our models using available relic observations for normalization (see Section 4.2). As a comparison shown are the radio power scaling parameters obtained by Skillman et al. (2010) in the same redshift range (they assume model 'a' and an acceleration efficiency ξe = 0.005).
pressure gradient and define the shock normal of the particle as n ≡ −∇P/|∇P |. In case that the pressure gradient corresponds to a true shock front several conditions must be fulfilled. In particular, we demand that (i) the velocity field shows a negative divergence, (ii) the density increases from the upstream to the downstream region and (iii) that the latter is also valid for the entropy. Utilizing the Rankine-Hugoniot jump conditions for hydrodynamical shocks (see e.g. Landau & Lifshitz 1959) these requirements allow to determine the Mach number Mi. For a conservative estimate we compute the Mach numbers according to all three conditions and then take the minimum. We wish to avoid the overestimation of the Mach number since this could lead to spurious strong radio emission. We apply this shock detection scheme to all gas particles inside a cube of size 10 h −1 Mpc (comoving) centred in the centre of mass of the systems available in our FoF catalogs. We consider here merger shocks, i.e. shocks introduced by cluster mergers, which are found to have typical Mach numbers around ∼ 2.5 − 3 (Araya-Melo et al. submitted). We note that also fast galaxies in a rather cold ICM may generate shock fronts. In order to predict the radio power of the simulated shock fronts we need to know the magnetic field strength in the downstream area of the shock fronts. Following our previous work in Hoeft et al. (2008) and that of Skillman et al. (2010) we assume that the magnetic field is given by where ne is the local electron density, B0 is a magnetic field reference value and η is the slope of the density scaling. This dependence is motivated by the assumption that in average the magnetic field in the ICM is frozen in and that the gas motions distribute the magnetic field even to the outskirts of the cluster where luminous radio relics are generated. In fact, using Faraday rotation measures in the Coma cluster Bonafede et al. (2010) found evidence that magnetic fields are spread over the entire ICM. In this work we explore two magnetic models. In the first place we assume B0 = 0.1 µG and η = 2/3 (e.g. Hoeft et al. 2008) which typically leads to ∼ µG values at the ouskirts of galaxy clusters (model 'a'). We also adopt the scaling found by Bonafede et al. (2010) that produces higher magnetic field values ( µG) at these locations (model 'b'). Their best-fit model indicates a slightly lower exponent than before but a stronger field for an electron density of 10 −4 cm −3 , namely, η = 1/2 and B0 = 0.8 µG, respectively. It is worth mentioning that the upper limits for the IC emission in the hard X-ray band for the northwest relic in A 3667 indicate higher magnetic field strength ( 1.6 µG; Nakazawa et al. 2009) than obtained here for a typical electron density of ∼ 10 −4 cm −3 . Hence, this could lead to an overestimation of the hard Xray flux for a similar relic in the simulation. However, it is not currently known if the relic in A 3667 hosts an exceptionally strong magnetic field or if relics show in general field strengths of a few µG or more.
As mentioned in the introduction the emission scenario (ii) states that thermal (or mildly relativistic) electrons are accelerated at the shock front by DSA. Important evidence for this mechanism comes from the relic in galaxy cluster CIZA 2242 (van Weeren et al. 2010). In this case the observed gradient in the spectral index is consistent with electrons accelerated at the shock front, while synchrotron and IC losses cause the steeper spectral index in the downstream region. The relic in CIZA 2242 also shows that radio emission originates from a rather small volume in the downstream region of the shock with an extent less than 50 kpc. In HB07 we have worked out the relation between the radio emission and the properties of the shock front and downstream plasma within the context of the DSA model. Assuming that the relativistic electron population is advected with the downstream plasma and cooled down due to synchroton looses and IC scattering with CMB photons we are able to estimate the total radio emission. In particular, the radio power per unit frequency contributed by a SPH gas particle i can be written as follows In this formula, Ai represents the surface area given by the SPH particle, ne,i is the electron density, ξe is the electron acceleration efficiency, si is the shock compression factor, T d,i is the post-shock temperature, B d,i is the post-shock magnetic field, BCMB is the magnetic measure of the CMB energy density and Ψ(Mi) is a function that depends on the shock strength.
We would like to note that the efficiency, ξe, for electron acceleration, denotes the fraction of the energy dissipated at the shock front that is transferred to supra-thermal particles. The lower energy threshold for supra-thermal particles is computed from the condition that the power-law distribution of supra-thermal electrons must meet the thermal electron distribution at the lower energy threshold, see HB07 for more details. As a result of this approach the radio emission decreases drastically for Mach numbers lower than 3. For the computations that follow we simply adopt ξe = 0.005. We encourage the reader to see HB07 for details.

Estimation of the expected radio power scalings using the MareNostrum Universe
Our aim is to estimate the probability of finding relic radio emission coming from an arbitrary galaxy cluster having mass M , located at redshift z and observed at frequency ν obs . To this aim, we analyse the radio emission produced in the different clusters of our synthetic samples. As mentioned in Section 3.1 we take the 500 most massive clusters at each considered redshift (i.e., z = 0, 0.25, 0.5, 0.75, 1) and identify the shock fronts in each one of them. At all redshifts, we evaluate the radio power emitted from cluster relics as a function of hosting galaxy cluster mass and observing frequency. Note that when computing the radio power we consider all the luminosity caused by structure formation shocks within a distance of 3.6 h −1 Mpc (comoving) from the centre of mass without distinguishing between different relics. However, in each cluster, there are typically only one or two prominent relics which contribute most to the final radio emission. We considered five different frequencies in our analysis, namely ν obs = 0.12, 0.15, 0.21, 0.325 and 1.4 GHz. Fig. 1 shows the radio power distribution of relics at z = 0 and ν obs = 1.4 GHz in the case of magnetic model 'a', where the three panels show results for different hosting cluster mass. Best fit log-normal functions are also shown. We explore the parameter space of our relic cluster samples, given by the cluster mass, redshift and observing frequency, and repeat the fitting procedure of Fig. 1 (for simplicity we assume a constant value for σP ). In this way, we are able to find a set of parameters for the radio relic probability density capable to reproduce the mean radio power scalings of our synthetic radio relics (see Eq. (9)). The best-fitting scaling parameters for the two magnetic field models adopted are shown in Table 3.2. In general, the derived scalings show a good agreement. However, magnetic model 'b' displays lower radio power scalings with mass and redshift in comparison with model 'a', which is most noticeable for the redshift evolution. The reason for this can be understood in terms of the stronger magnetic field values achieved within the context of magnetic model 'b'. Since according to Eq. (11) the radio emission saturates for large magnetic field values the resulting radio power scaling is not so pronounced in this case.
As can be seen from Fig. 1 log-normal functions reproduce reasonably well the radio power distribution of the synthetic relics. However, since there are possibly more small shocks, a better resolution in the simulation may serve to alleviate the observed skewness in the radio power distribution at low cluster masses. Additionally, this could let us extend the cluster mass range studied to estimate the mean radio power scalings. In the following section we present the currently known observed relic sample to further normalize our theoretical expectations with observations.

Compilation of currently known relics
By definition, relics are diffuse radio emission in the periphery of galaxy clusters without any optical counterpart. Hence, relics are commonly searched for by correlating radio surveys with large catalogues of galaxy clusters. The Westerbork Northern Sky Survey (WENSS) has been carried out at 325 MHz covering the north sky for declinations higher than 28.5 • . The noise level of this survey is 3.6 mJy (Rengelink et al. 1997). The NRAO Very Large Array Sky Survey at 1.4 GHz (NVSS) covers the sky north of −40 • and has a noise level of 0.45 mJy (Condon et al. 1998). Systematic searches for diffuse radio emission in galaxy clusters have been undertaken, for instance, by inspecting a sample of 205 X-ray bright Abell-type clusters in the NVSS catalogue (Giovannini et al. 1999), by analyzing the WENSS data at the position of all Abell clusters (Kempner & Sarazin 2001), and by searching for steep spectrum sources in the VLA Low-frequency Sky Survey (VLSS; Cohen et al. 2007) catalogue (van Weeren et al. 2009c).
As described above in more detail, current models for the formation of relics are not able to predict the actual number of observable relics by themselves, because both, the number density of relativistic electrons and the strength of magnetic fields, are in general poorly constrained quantities. Therefore, we wish to normalize the radio relic number counts, N (> Sν ), using the number of known radio relics. To this end, we have compiled a list of all radio relics reported in the literature, as far as we are aware of, which can be seen in Table 2. We have included all types of radio relics, i.e. Mpc-scale single and double relics in the periphery of clusters as well as smaller relics inside the cluster volume. A few of the small relics might be attributed to the compression of fossil radio plasma (known as 'radio phoenix' class in the terminology of Kempner et al. 2004). However, we do not include phoenixes when computing the relic number counts.
For each cluster in Table 2 we give the flux of the diffuse emission which has been classified as 'relic' while the contribution of radio haloes has been excluded. In the cases where halo and relic radio emission are on top of each other due to projection effects we only estimate the flux density of the relic emission. In many clusters the diffuse relic emission is fragmented into multiple pieces, e.g. A 2255 (Pizzo et al. 2008), or shows some prominent patches and very extended emission as well (e.g. CIZA 2242; van Weeren et al. 2010). Instead of separating individual relics in a single galaxy cluster, we combine the flux, Sν, of all relics in the cluster which is consistent with defining the radio luminosity probability for diffuse radio emission in clusters instead of that for relics (in the same way as done in Section 3.2). For our analysis it is not useful to introduce relics as self-contained objects, since their identification depends inevitably on observational parameters such as sensitivity and resolution. Hence, we give in column (3) of Table 2 the entire radio relic flux present in each cluster. To normalize the relic number counts, N (Sν), we use the cluster flux at 1.4 GHz because most of the measurements available are done in the 21 cm band. We also estimate the radio power of the relics at 1.4 GHz (see column (6) of Table 2) assuming a spectral slope of -1.2 which is consistent with the parameter Cν given in Table 3.2. Interestingly, A 3667 displays an outstanding high flux. However, this object is not the most radio luminous relic as can be seen in Table 2. The ten most luminous relics have fluxes S1.4 100 mJy. It is worth noting that several of these luminous relics have been detected in recent years, namely 1RXS 06, CIZA 2242, and MACS 0717. On the other hand, the faintest relics known to date have a flux of ∼ 6 mJy.

Normalizing the radio relic number counts
Basically, we would like to normalize the predicted radio relic number counts by using the bright-end of the observed number count distribution. As noted above amongst the ten brightest relics there are however three relics which have been identified only recently. This could indicate that even the bright-end of the relics list does not contain all bright relic sources on the sky which may introduce an offset in the global counts. Hence, in order to estimate number counts we assume a fiducial flux of 100 mJy to centre the discovery probability. This means that at S eff 1.4 = 100 mJy half of the radio relic emission has been detected. Although this choice is arbitrary we take this value as a compromise between the lowest and brightest relics in the observed distribution. Furthermore, since the lowest flux of known radio relics is of a few mJy, we can set w = 0.8, which ensures that the discovery probability virtually vanishes below these values.
The non-detection of a relic could be due to several reasons. For instance, part of the sky might not be covered by deep radio surveys, galactic foreground radiation or bright sources in the cluster may obscure the diffuse emission, the surface brightness of the diffuse emission may be too low, or the related cluster could have not been identified yet. All these possibilities for the non-detection of existing diffuse radio patterns in a galaxy cluster are comprised in the complementary discovery probability 1 − φ.
Having already a model for the relic discovery prob-ability we are now able to normalize the number counts to the present observed sample (which we dub as 'NVSS' since many candidates have been found by means of that survey). In order to do so we take from Table 2 all confirmed relics above a declination of −40 • without including phoenixes. As can be seen in Fig. 2 the observed number of radio relics is well reproduced when using a normalization given by log P0 = 21.35 and 21.53 for magnetic models 'a' and 'b' respectively. However, a degeneration between the normalization parameter and the detection threshold exists: a higher value for the normalization would imply a threshold higher than 100 mJy if one is willing to reproduce observations. This would mean that the majority of relics with this higher flux has not been detected yet. We consider this possibility unlikely. On the other hand, we will show below that within the context of the magnetic models considered here a lower normalization can be ruled out as a result of the analysis of an X-ray flux limited cluster sample and their associated relics. The obtained low reference radio power (log P0 ∼ 21.4) is enough to reasonably describe the observed distributions (see Section 4.3). Therefore, for the set of parameters derived above for the relic radio power probability (CM , Cz, Cν, σP ) we are able to constrain the normalization very well. It is worth noting that to determine radio powers for the simulated clusters we had to assume an acceleration efficiency, ξe = 0.005 (with ξe as defined in HB07 model), and we had to assume average scalings for the magnetic field, see Eq. (10). For our MareNostrum clusters these assumptions lead to a reference radio power of log P0 = 22.23 and 24.13 in the case of magnetic models 'a' and 'b' respectively. Since, as mentiond above, to reproduce observations we require lower values for the normalization this implies that the acceleration efficiency must be ξe 0.001. In particular, for model 'b' the observed acceleration efficiency could be about ξe ∼ 10 −5 which is more in line with theoretical expectations of DSA in Type Ia supernova remnants (Edmon et al. 2011). Further increase of the magnetic field values, as suggested by Nakazawa et al. (2009), would reduce the required efficiency even more.
Most of the observed radio relics have a redshift lower than 0.3. In the sample there are only five relics with higher  Table 2).
redshift, and only one of them is located at z > 0.5. We wish to compare these numbers to the predictions according to the radio relic probability distribution. We simply split the result into the redshift intervals given above, as can be seen in Fig. 3. Apparently, our models predict more relics than observed for z > 0.3. This might indicate that relics in distant clusters are more difficult to detect (e.g. due to resolution effects), that clusters would need to be more Xray luminous to be found, or that our scaling parameter Cz derived from the simulations does not agree with the actual redshift evolution. In particular, model 'b' seems to better reproduce observations at all redshifts. However, it is important to realize that the number of both predicted and observed relics with z > 0.3 is very small, so we should be cautious with any interpretation. Much more extensive catalogues of relics are needed to draw a significant conclusion about the redshift evolution.
The most distant cluster which hosts a relic is MACS 0717. Fig. 3 indicates that in our model the highest redshift relics should have fluxes S1.4 within the range 10 − 50 mJy. Instead, the relic in MACS 0717 has a flux of about 140 mJy, indicating that this system is an outstanding radio relic. In fact, it is the most luminous relic known to date with P1.4 ∼ = 2 × 10 26 W Hz −1 . For instance, using scalings resulting from magnetic model 'a', the mean relic radio power of clusters having the mass and redshift of MACS 0717 isP1.4 ∼ = 4.1 × 10 24 W Hz −1 (see Eq. (9)). Hence, the luminosity of the relic is about 2σP higher than the mean relic luminosity, so it is a rare event but still reasonably likely considering all clusters in the Universe.
In general, we can study the deviation between the estimated radio power from observations and the expected mean radio power at a given redshift and cluster mass. We can quantify this deviation in terms of ∆P := log P1.4/P1.4 /σP , which measures the difference between the logarithmic radio power estimated from observations and the peak of the radio relic probability distribution in units of the parameter σP . In order to make a simple esti-mate we adopt the cluster X-ray luminosities given in column (8) of Table 2 to compute the cluster masses. In what follows we use a LX − M relation similar to that given by Pratt et al. (2009) that will be presented in Section 4.4. In column (7) of Table 2 we give ∆P for all relics in our observed sample adopting the scalings derived with the magnetic model 'a' to estimate the mean radio power. As expected most of the relics display a mean deviation of ∼ 2σP since we are observing the brightest (or close by) relics in the sky. Some of the relics show an unexpected high deviation from the mean radio power,P1.4, given by the hosting cluster mass and redshift. This may serve as an indication of the need to further investigate these systems in more detail to confirm their relic nature. However, the large deviations may also come from uncertanties in the cluster mass estimate. We do not pretend here to give a rigorous derivation of the hosting cluster masses but only assess the global tendency of the sample.

The X-ray -radio power relation
For giant radio haloes in massive galaxy clusters a close correlation between radio power and X-ray luminosity has been found (e.g. Venturi et al. 2007). In particular, Enßlin & Röttgering (2002) suggested that the radio power of observed haloes scales with cluster X-ray luminosity according to with parameters aν = 5.36 and bν = 1.69. In contrast, the radio power of relics show a large scatter for a given X-ray luminosity or temperature as can be seen in Fig. 4. This fact precisely reflects our starting point, namely, the recognition that the radio power of relics varies strongly for a given galaxy cluster mass. Therefore, this motivated us to introduce the radio power probability distribution. Formally, we Figure 5. Luminosity function of radio relics for models 'a' and 'b' (solid and dashed lines respectively) and radio haloes (dot-dashed line) at 1.4 GHz and z = 0. The radio halo luminosity function is an analytic approximation taken from Enßlin & Röttgering (2002) under the assumption that a constant fraction f rh = 1/3 of the clusters contain radio haloes.
can relate the mean radio power,P1.4, to the X-ray luminosity of clusters at z = 0 using the LX − M relation in Eq. (9) for the parameters given in Table 3.2. Comparing this result with Eq. (12) we get for aν and bν the values 1.3 × 10 −3 and 1.5, respectively. Interestingly, we find a similar exponent but a much lower proportionality constant. This seems to indicate that there are much less bright radio relics than haloes. However, we have to keep in mind that when computing the RRLF not only the mean radio power but the radio power distribution function needs to be taken into account, see Eq. (3). As a consequence we expect more radio relics than haloes as can be seen in Fig. 5.

Radio relics and an X-ray selected cluster sample
The radio power probability density introduced above allows us to predict the fraction of clusters in an X-ray selected sample which host a radio relic. As a first step, we introduce the differential distribution with respect to cluster X-ray flux In a similar way as done in Eq. (5) we can write Note that radio power is a function of radio flux and redshift, similarly, the X-ray luminosity is a function of X-ray flux and redshift. Using the LX − M relation (see below) to estimate the cluster mass, based on its X-ray luminosity, we can write p(Pν, M, z) = p(Pν(Sν, z), M (SX , z), z). We now introduce an X-ray flux threshold, S th X and assume that only clusters with a flux above the threshold are detected. This allows us Figure 6. Cumulative fraction of clusters which host diffuse relic emission. The cumulative number is depicted as a function of Xray flux, measured in the ROSAT 0.1 − 2.4 keV band. The curves give the cumulative number for a relic flux of 1, 10, and 100 mJy. In addition the completeness limit of the REFLEX cluster sample is given.
to determine the cumulative fraction of clusters with SX > S th X that host diffuse relic emission with a given flux Sν . Integrating the previous equation leads to where the normalization factor, NX (Sν), is determined by the condition FX (> 0, Sν) = 1. Fig. 6 shows the cumulative fraction of clusters for three different radio fluxes (for the sake of simplicity we assume only the radio power scaling parameters that result from model 'a' throughout this section; adopting those of model 'b' do not modify our main conclusions). About 80% of the clusters which host diffuse relics with 100 mJy have an X-ray flux larger than 3 × 10 −12 erg s −1 Hz −1 , which corresponds to the completeness limit of the REFLEX cluster sample ). Since candidate radio relics are commonly identified by cross-correlating radio and X-ray catalogues, the NORAS and REFLEX cluster catalogues are well suited to identify luminous relics. In contrast, for faint relics of about 10 mJy only ∼ 40% of the hosting clusters are expected to have fluxes above the REFLEX X-ray flux limit. This means that, if upcoming surveys will allow the detection of diffuse radio structures with fluxes of about 1 mJy, significantly deeper X-ray cluster catalogues will be needed to identify the majority of radio relics.
In a similar way we can determine the cumulative fraction of clusters which host diffuse relic emission as a function of radio flux. To this end we introduce the fraction of galaxy clusters with detectable relics at a given mass and redshift where φ is the discovery probability introduced earlier. Note that for w → 0 we can mimic a Heaviside-function, i.e. only relics above S eff ν are detected. We can now determine the cumulative fraction of clusters hosting radio relics per Xray luminosity bin, ∆ log LX , as follows (17) where n M,f := nM f φ (M, z) and the normalization factor, Nν (LX ), is given by the condition Fν (> 0, LX ) = 1. Fig. 7 shows the cumulative fraction for different cluster Xray luminosities. As expected, only a small fraction of clusters show radio relics for current detection limits of about 10 mJy. The fraction decreases strongly with the cluster Xray luminosity. For instance, ∼ 20% of clusters with an X-ray luminosity of about 3 × 10 45 erg s −1 are expected to host a relic with a flux of 10 mJy or brighter, while only ∼ 0.3% of clusters with 3 × 10 44 erg s −1 are expected to do so. Note that in Eq. (17) we have assumed that all clusters with given X-ray luminosity can be detected. To compare the result to X-ray selected cluster samples we need to introduce an X-ray flux limit. As a result, a large number of faint relics residing in distant clusters falls below the flux limit. In a recent work van Weeren et al. (2011c) selected 544 clusters from the NORAS (Böhringer et al. 2000) and the REFLEX ) cluster samples with an X-ray flux above 3 × 10 −12 erg s −1 cm −2 and located outside the galactic plane. Up to this flux the REFLEX sample is virtually complete. On the other hand the NORAS sample is almost 50% complete. Interestingly, these authors show that 16 out of the 544 clusters of the combined list contain at least one radio relic and found evidence for an increase of the fraction of clusters which host relics with cluster X-ray luminosity and redshift.
Eq. (17) allows us to determine the fraction of clusters with relics. The X-ray flux limit imposes an upper limit in the redshift integral, i.e. allowed redshifts must fulfill the condition z < z(S th X , LX ). First we wish to reproduce the dN cl /dLX and dN cl /dz distributions of the cluster sample selected in van Weeren et al. (2011c). To this end we rewrite Eq. (17) where N cl is the number of clusters and fs indicates the sky fraction covered by the selected cluster sample, we estimate it to be 35% 1 . The integration boundaries, z th (S th X , LX ) and L th X (S th X , z) are obtained from the flux limit in the survey, see Eq. (1). To perform the integration we need to relate cluster mass and X-ray luminosity. We assess the cluster luminosity and redshift distributions by choosing an appropriate LX (M ) relation. Recently, Pratt et al. (2009) investigated cluster scaling relations in detail. They provide best 1 The sky fraction is estimated by f sky × f int × (1 − f gal ), where f sky is the sky fraction covered by the survey which overlaps NVSS, f int is the completeness of the survey, and f gal is the fraction of clusters located at a galactic latitude lower than 20 • . For NORAS and REFLEX we assume 50%, 50% and 33%, and 34%, 90%, and 33%, for these quantities respectively. The sum of the two contributions leads to the estimated value. fit parameters for LX,500 − M500, where both quantities are measured within R500. However, our sample differs in several respects to theirs: (i) the halo mass function, nM , is given for M200, (ii) we do not extrapolate the luminosities to R500, (iii) our sample extends up to z ∼ 0.5 while Pratt et al. (2009) use only clusters with z < 0.2, and (iv) we use a slightly different value for ΩM. Adopting the scaling relation with E(z) = ΩM(1 + z) 3 + ΩΛ we are able to reproduce the observed distributions reasonably well (see Fig. 8).
To compute the abundance of clusters which actually hosts a detectable relic we have to use n M,f = nM f (M, z), instead of nM in Eqs. (18) and (19). Fig. 8 (red solid lines) shows the resulting X-ray luminosity and redshift distributions. Note we use here a lower effective sensitivity and smaller width in φ(Sν) because the clusters are known. As discussed above we may miss the discovery of a bright relic, since the cluster has not yet been identified. For instance, the double relic in PLCK G287.0 has been discovered only after the detection of the hosting cluster with Planck satellite even if the diffuse emission is clearly visible in NVSS. Since we consider here the relics in the NORAS+REFLEX sample all clusters are known by construction. We model the discovery probability in the following way: (i) we argue that the width is smaller than for the overall sample, we take w = 0.2, (ii) we adjust S eff 1.4 to reproduce the fraction of clusters with relics found in the sample, using S eff 1.4 = 27 mJy we obtain a fraction of 3%. Hence, the effective sensitivity adopted corresponds to 60 times the r.m.s. noise level in the NVSS survey. In this way we find that the fraction of relics in an X-ray flux limited cluster sample should indeed increase with X-ray luminosity and redshift as shown in Fig. 8. The redshift distribution is a crucial test for the mean radio power scaling parameter Cz.

Predictions for upcoming surveys
In Table 3 we have summarized specifications for planned surveys with LOFAR and the WSRT. We would like to give some plausible estimates for the expected number of relics to be discovered by upcoming radio surveys. However, we have to remember that there may be several uncertainties difficult to quantify. As we already mentioned the determination of p(Pν, M, z) is affected by the limitations of our simulation and by the adopted physical model used to relate the Mach number with the relic radio power. However, since we are able to reproduce the trends found for the NORAS+REFLEX sample we conclude that our approach has resulted in a reasonable set of parameters. In the present paper we use the radio relic probability density estimated from the MareNostrum simulation and leave for future work a more extensive modelling of the radio power emission.
To compute number counts for future surveys we also need to assess the discovery probability for each survey. Since it is beyond the scope of this work to model this in detail we simply adopt a conservative approach: we assume that the NVSS detection parameters hold similarly for the upcoming surveys, i.e. we take w = 0.8 (detection/nondetection transition of the instrument) and b := S eff ν /σν ∼ 200 (ratio of the effective sensitivity of the overall relic sample to the survey noise) as our fiducial parameters. The last condition means that for half of the clusters which host diffuse radio emission with a flux above 200 times the noise level of the survey a radio relic will be detected. At this point, an important remark needs to be done in relation to the sensitivity per beam achieved by a given radiotelescope. The next generation of radio surveys will presumably increase their beam resolution at least by a factor of a few.
In principle, for some of the relics, this could imply the requirement of higher b values than assumed here which may lead to an overestimation of the predicted number counts. For instance, in its final configuration the LOFAR telescope is expected to reach an angular resolution of ∼ 5 arcsec. However, since this instrument has many short baselines it is always possible to smooth images down to typical NVSS resolution values (i.e. ∼ 45 arcsec) without increasing too much the resulting r.m.s. sensitivity. Nevertheless, we have to keep in mind that the final predicted number counts will depend on the adopted radio power scalings and detection parameters. For instance, in the case of the LOFAR-Tier 1-120 MHz survey, if we let the effective detection threshold to vary in the range b = 150 − 300, keeping the remaining parameters fixed, our predictions will increase (decrease) by a factor of two (a half) with respect to the prediction corresponding to b ∼ 200. Similarly, one could assess the impact of varying some of the radio power scaling parameters keeping the rest unchanged. In particular, if we let the slope of the scaling with cluster mass to vary in the range CM = 1.5 − 3.5 we get brighter (fainter) relics located in clusters with masses below (above) 10 14.5 h −1 M ⊙ . The predictions in this case will also increase (decrease) in a similar amount as before.
For calculation purposes here we use the radio power scalings derived from the simulation assuming the fiducial detection paramaters presented above. Table 3 gives the total number of expected relics up to z = 0.3, as well as relics with 0.3 < z < 0.5, 0.5 < z < 1, and z > 1 for magnetic models 'a' and 'b' (in brackets). For z > 1 models with higher magnetic fields would generally produce less radio relics as a consequence of the resulting redshift evolution. Under these assumptions we expect that the LOFAR-Tier 1-120 MHz survey and WODAN large sky coverage survey should reveal several thousands of radio relics, due to the huge improvement in sensitivity that will presumably be achieved by these surveys. However, a given survey may provide candidates for radio relics, by cross-correlation with positions of known galaxy clusters. This means that deep follow up observations may be needed to confirm radio relics in the clusters. Interestingly, within the context of our simple scaling for the magnetic field, we also expect a significant number of relics with z > 0.5 and z > 1 to be detected. The actual number of relics at high redshift will in particular serve to constrain the redshift evolution of magnetic fields in clusters allowing us to further refine our model prescriptions.
As noted above, the number of relics detected in upcoming surveys will crucially depend on the cluster database used to correlate candidates displaying diffuse radio emission with known cluster positions. As a consequence, the final number of unambigously identified relics could be below our model expectations. To quantify this we estimate how many relics LOFAR should find in the Tier 1-120 MHz configuration for the NORAS cluster sample introduced before. Note that the LOFAR survey and the NORAS sample cover both the north sky. Based on the effective sensitivity found in the previous section for relics in this galaxy cluster sample we adopt here a flux threshold 60 times that of the r.m.s. noise level of the LOFAR-Tier 1-120 MHz survey (S eff 1.4 /σTier 1), which results in an effective sensitivity of 6 mJy (see dashed lines in Fig. 8). We find that LOFAR should discover relics in more than 50% of the clusters. In the case of luminous clusters the fraction can be as high as 90%.
Using Eq. (15) we can also estimate what is the required sensitivity in X-ray surveys to find at least a fraction of the relics that can be potentially discovered in a radio survey. Fig. 9 indicates that for relics with 20 mJy in the LOFAR-Tier 1-120 MHz survey, about 50% of the relics might be identified by cross-correlation if the X-ray surveys are complete up to at least 4 × 10 −13 erg s −1 cm −2 (in doing this calculation we have used magnetic model 'a' as done in the previous section). Hence, an all-sky X-ray catalogue with an X-ray flux limit one order of magnitude below that of the REFLEX sample is necessary to identify a considerable fraction of the relics.

SUMMARY
Radio relics are believed to trace merger shock fronts in galaxy clusters. The radio luminosity of shock fronts depends strongly on the Mach number of the shock, but also on the size of the front and on the magnetic field present in the downstream region. Even if in every cluster there are shock fronts related with past merger events, the actual radio luminosity caused by the shocks may vary strongly, from no detectable radio emission at all to the presence of luminous radio relics. To describe the large spread of radio luminosities more formally, we have introduced the radio power probability distribution, p(Pν, M, z), aim at assesing the likelihood of relics in galaxy clusters with a given mass and redshift in a given frequency. We use the MareNostrum Universe simulation to estimate the probability distribution. To this aim, we selected the 500 most massive clusters at 5 different redshifts up to z = 1 to detect shock fronts in assembling galaxy clusters. Then, we apply the scheme developed by Hoeft et al. (2008) for estimating their radio relic luminosity. Based on the distribution of radio relic luminosities of the simulated clusters we conclude that the radio power probability is well approximated by a log-normal distribution. Moreover, using our galaxy cluster samples, we are able to estimate how the radio relic distributions scale with cluster mass, redshift, and observing frequency.
Using the radio power probability distribution we wish to determine the relic number counts, N (> Sν ). Basically this is given by a convolution of the probability distribution and the dark matter halo mass function. However, radio relics are not straightforwardly identified in radio observations, therefore, even luminous relics are possibly present in radio catalogues, but not yet identified as relics. For instance, the relics in 1RXS 06+42 and in CIZA 2242 are bright systems present in the WENSS catalogue, but have only recently been reported as relics. We therefore introduce the discovery probability as a function of radio flux, φ(Sν). The number counts are obtained by a convolution of all three, the halo mass function, the radio power probability distribution, and the discovery probability.
It is important to remark that it is not possible to predict radio flux number counts of relics purely from cosmological simulations. In this regard a major source of uncertainty is the efficiency of the electron acceleration at the shock front. We therefore use the observed relic number counts to determine the reference normalization for the radio power probability distribution.
The resulting framework allows us to estimate the number of detectable relics in upcoming radio surveys. In the following we summarize our main conclusions: • To evaluate the MareNostrum Universe simulation we assumed an electron acceleration efficiency, ξe = 0.005 and two different magnetic field scalings with local electron   (1) Cluster name, (2) Classification (R: single relic; D: double relic; P: phoenix; C: diffuse radio emission detected [more observations are needed to confirm its nature]; F: probably misclassified as relic), (3) Radio flux (all relics in the cluster are considered), (4) Observed frequency, (5) Redshift, (6) Radio power (computed assuming a spectral index of −1.2), (7) Radio power deviation (model 'a'; see text), (8) Cluster X-ray luminosity, (9) Cluster X-ray temperature, (10) Checkmark if within NVSS relic sample (δ > −40 • ), (11) Checkmark if within NORAS+REFLEX cluster sample (in brackets if below flux limit S R = 3 × 10 −12 erg s −1 cm −2 ), (12) References (radio/X-ray). c 0000 RAS, MNRAS 000, 000-000  The colums correspond to: name of the radio survey, observing frequency, survey noise level, observed sky area (or declination limit), corresponding sky fraction and approximate number of expected relics using our set of fiducial parameters as a function of redshift for magnetic models 'a' and 'b' (in brackets). It is worth noting that these are only plausible estimates for upcoming radio surveys but are not meant to be definitive values (see Section 4.5).
density in the simulation. Normalizing the radio power probability distribution by the list of known NVSS relics resulted in lower values for the reference radio power which can be interpreted as evidence for a low electron acceleration efficiency. In particular, we find that ξe 0.001. According to the magnetic scaling proposed by Bonafede et al. (2010) in the case of the Coma cluster (model 'b') the acceleration efficiency could easily reach values of ξe ∼ 10 −5 . However, there are many uncertainties which may affect the P0 value, e.g. the actual discovery probability.
• After normalizing the radio relic number counts, N (> Sν ), we split the obtained number counts into redshift bins. As a result we expect more relics for z > 0.3 than being observed. This might indicate that the B(ne) relations assumed in the simulation show an additional dependence on redshift. However, magnetic model 'b' seems to agree better with observational results which would point toward µG magnetic fields in the location of radio relics. We consider this approach as a very promising diagnostics of the evolution of magnetic fields in galaxy clusters but larger relic samples are needed to draw any robust conclusion.
• The observed relic number counts are reasonably reproduced assuming an effective sensitivity of 100 mJy. Candidates for many relics may have been first identified in the NVSS survey, which has a r.m.s. noise level of 0.45 mJy. We adopt therefore that the effective sensitivity for finding relics is generally about 200 times the r.m.s. noise of a survey. We apply this to the specifications of the proposed LOFAR and APERTIF surveys. Under these assumptions we find that the LOFAR-Tier 1-120 MHz has the potential to find more than a thousand radio relics.
• More than 50% of the relics expected to be found with LOFAR-Tier 1-120 MHz survey should reside in clusters with z > 0.3 and there should be even more than 100 relics in clusters with z > 1. Hence, in principle this survey will allow to discover sufficient relics to analyze the evolution of magnetic fields in clusters in a statistical way.
• To confidently discover a relic, the clusters which host the diffuse radio emission need to be identified. Many of the relics which are detectable by the LOFAR-Tier 1-120 MHz survey may reside in faint clusters. More precisely, we predict that about 50% of the relics with 20 mJy will reside in clusters with an X-ray flux below 4 × 10 −13 erg s −1 cm −2 .
• Following van Weeren et al. (2011c) we study an X-ray flux galaxy cluster sample based on the NORAS+REFLEX catalogues. About 4% of the galaxy clusters in the sample host radio relics. This value is significantly lower than the one obtained for the overall sample since we consider only known clusters here. We found that we can reproduce the relic fraction assuming an effective sensitivity of 27 mJy at 1.4 GHz. As discussed in van Weeren et al. (2011c) we also find that fraction of clusters which host a relic increases with cluster X-ray luminosity and redshift.
• We expect that the LOFAR-Tier 1-120 MHz survey will find radio relics in around 50% of the NORAS+REFLEX clusters. Furthermore, for the most massive clusters this fraction can be as high as 90%.