USING RADIO HALOS AND MINIHALOS TO MEASURE THE DISTRIBUTIONS OF MAGNETIC FIELDS AND COSMIC RAYS IN GALAXY CLUSTERS

The spectral slope near the center of most radio halos is α ≃ −1 (see Section 5.2). A synchrotron spectrum of this type is emitted by rapidly cooling CREs, injected with approximately constant energy per decade in particle energy E_e. The synchrotron emissivity corresponding to a logarithmic CRE energy density injection rate Q ≡ du_e/(dt dln E_e) = constant is given by

$$\begin{eqnarray} &&\nu j_\nu \equiv \nu \frac{du_{\gamma }}{dt\,d\nu } = \frac{Q/2}{1+(B_{\rm cmb}/B)^{2}}, \end{eqnarray} \tag{ 9 }$$

where the denominator accounts for Compton losses, B_cmb ≡ (8πu_cmb)^1/2 ≃ 3.2(1 + z)²μG is the amplitude of the magnetic field which has the same energy density as the CMB, and u_j is the energy density of component j (γ for radio photons, e for CREs, etc.). In the B ≫ B_cmb limit, the synchrotron emissivity (Equation (7)) extracted from the observations yields a direct estimate of the CRE injection rate:

$$\begin{eqnarray} &&Q \simeq \frac{Q}{1+(B_{\rm cmb}/B)^{2}} = 10^{-31.1\pm 0.2}\, n_{-2}^2 \mbox{ erg}\mbox{ s}^{-1}\mbox{ cm}^{-3}.\qquad \end{eqnarray} \tag{ 10 }$$

As CREs do not have time to diffuse before they lose their energy, this result applies locally. It holds in both GHs and MHs.

The remarkably small scatter found in the radio–X-ray correlations among different GHs has been interpreted as implying a robust radio mechanism that keeps the synchrotron emissivity narrowly distributed around a universal function of the plasma density and perhaps also temperature. Equation (10) and the narrow scatter in its normalization suggest that the universal quantity in radio halos—both GHs and MHs—is Q/n².

Our results strongly emphasize the robustness of the radio emission mechanism. First, we find a tight radio–X-ray correlation not only in GHs but also in MHs. As the latter have physical properties that are considerably different from GHs (smaller size, higher ambient density, nearby AGN, CF association), the radio emission mechanism must be sufficiently robust to reproduce the same levels of Q/n² over a wide range of physical conditions. Second, we confirm that a tight radio–X-ray correlation exists not only in the total cluster luminosity but also in the local surface brightness, and that the ratio η between radio and X-ray brightness is fairly uniform within each cluster. This is particularly striking in MHs, where the coincident density and temperature profiles are steep.

K09 have pointed out that in the strong magnetization regime, defined as B > B_cmb, radio emission from GHs is independent of the precise value of B, and so the tight correlation only constrains the CRE injection Q. In contrast, in the weak magnetization regime B < B_cmb, the product QB² should be universal, requiring a physical mechanism that carefully balances CRE injection and magnetic fields both with each other and with the ambient gas. Furthermore, in a strongly magnetized halo model, the transition from a constant to a rapidly declining (∝B²) behavior as B drops below B_cmb naturally explains the GH bimodality observed (K09).

The similar values of η we find in GHs and in MHs strengthen this argument considerably because it is difficult to come up with a double feedback mechanism (CRE–magnetic fields–ambient plasma) that operates identically in the different environments of GHs and MHs, without fine tuning. The bimodality argument does not apply to MHs, however, as their distribution has not been shown to be bimodal and may in fact be continuous if high magnetization is ubiquitous in CC centers (see discussion in Section 6.1).

Two types of models have been proposed for CRE injection: (1) secondary production by hadronic collisions involving CRIs (Dennison 1980) and (2) in situ turbulent acceleration or reacceleration of primary CREs (Enßlin et al. 1999). These models typically assume fixed ratios between the energy densities of the primary particles (either CRIs or CREs), magnetic fields, and thermal plasma u_th ∼ nT. Other possibilities involve a primary CRI distribution that has energy density u_i ∝ n (such a scaling is less likely for the magnetic fields or the rapidly cooling CREs) or a magnetic field frozen into the plasma B ∝ n^2/3. The synchrotron emissivity in each of the nine model variants corresponding to these three primary distributions folded with different magnetization levels and scalings is shown in Table 2.

Table 2. Radio Emissivity in Model Variants with Different Distributions of CREs and Magnetic Fields

Primary Scaling →	CREs	CRIs	CRIs
↓ Magnetic Scaling	ue ∝ nT	ui ∝ nT	ui ∝ n
$B_{\rm cmb}>B\propto \sqrt{nT}$	jν ∝ n2T2	jν ∝ n3T2	jν ∝ n3T
Bcmb > B ∝ n2/3	jν ∝ n7/3T	jν ∝ n10/3T	jν ∝ n10/3
B > Bcmb	jν ∝ nT	jν ∝ n2T	jν ∝ n2

Download table as:  ASCIITypeset image

Among these model variants, only the two secondary CRE models in which the magnetic field is strong (models highlighted as boldface in the table) are consistent with the scaling Equation (7)—j_ν ∝ n²T^κ where κ = 0.2 ±  0.5—and with Equation (10). Both models are consistent with our data. A slightly better fit to the GH profiles is obtained with u_i ∝ n (i.e., κ = 0; see Section 3), but more data are needed in order to establish the thermal dependence of the CRI distribution with sufficient statistical significance.

An independent argument in favor of these two models is the environment of MHs. MHs are found in relaxed clusters such as A2029, where only CFs reveal deviations from hydrostatic equilibrium. Such CFs reflect subsonic bulk shear flows and strong magnetization, but were not associated with particle acceleration. The CF–MH connection therefore supports both secondary CREs and strong magnetization in MHs. The similarity in the values of η we find in GHs and in MHs implies, transitively, that the same holds for GHs.

Additional evidence supporting the presence of strong magnetic fields and the absence of primary CREs is found by examining the morphological and spectral properties of halos, as discussed in Section 5. Note, for example, that the brightness correlation shown in Figure 2 is strongest in the centers of halos, but diminishes at larger radii, where turbulent activity associated with mergers is expected and where merger shocks and relics are found. In primary CRE models, one would not expect the correlation to preferentially tighten away from the turbulent regions.

The preceding discussion is strictly valid only as long as the timescale for substantial changes in the magnetic field is longer than the cooling time of the CREs, t_cool. For CREs that emit synchrotron radiation received with characteristic frequency ν,

$$\begin{eqnarray} &&t_{\rm cool} \simeq 0.13 \left[\frac{4\left(\frac{B\sqrt{3}}{B_{\rm cmb}}\right)^{-3/2}}{1+\left(\frac{B}{B_{\rm cmb}}\right)^{-2}}\right] \nu _{1.4}^{-\frac{1}{2}}(1+z)^{-\frac{7}{2}} \mbox{ Gyr}, \end{eqnarray} \tag{ 11 }$$

where the term in square brackets peaks at unity when $B=B_{\rm cmb}/\sqrt{3}$, and scales as B^−3/2 for B ≫ B_cmb. In GHs, t_cool is much shorter than the ≳1Gyr timescale characteristic of the halo lifetime (K09).

However, significant local variations in the magnetic field can take place on a shorter timescale, of the order of the sound crossing time of the turbulent eddies t_s ∼ l/c_s. This is shorter than t_cool for a sound velocity c_s ∼ 10³kms⁻¹ and eddy length scales l ≲ 100kpc. (Note that substantial magnetic power is measured on coherence scales ∼10kpc; see Section 6.1.) The radio emission should therefore be averaged over t_cool and over the beam. Nevertheless, as long as many eddies contribute to the emission and the variations in magnetic energy density remain of order unity, this correction would be small. Note that fast changes in magnetic configuration over a light crossing time, if present, may be observed as temporal radio variations by next generation telescopes such as the Square Kilometre Array (SKA). Indeed, the milliarcsecond resolution attainable by SKA (Schilizzi et al. 2007) corresponds to a light crossing time of less than a year, for nearby halos at redshift z ≲ 0.02.

In MHs, the magnetic fields are probably associated with sloshing activity in the core (see Section 6.1). The characteristic timescale for the decay of core sloshing is ≳1Gyr (e.g., Ascasibar & Markevitch 2006), much longer than t_cool. The timescale for the buildup of sloshing depends on its trigger mechanism; in a merger-induced scenario this is again ≳1Gyr. However, local variations in the magnetic field could occur over the radial sound crossing time, t_s ∼ r/c_s, or on the crossing time of the magnetic structures associated, for example, with CFs. These timescales could be shorter than t_cool in the central ∼50kpc (note that in these regions B usually significantly exceeds B_cmb; see Section 5.1). As in GHs, averaging over t_cool and over the beam could introduce small variations in radio brightness, in particular at the edges of MHs.

In both GHs and MHs, Q/n² is weakly sensitive to variations in CRE injection, through changes in the fractional energy of the CRIs u_i/u_th. Thus, Q/n² should be replaced by its value averaged over t_cool and over the beam. This correction should be very small, except near a cosmic-ray source.

In our model, both GHs and MHs are regions in which strong magnetic fields with B ≳ B_cmb indirectly illuminate the cluster's CRI population in radio waves. This explains the spatial coincidence between MH edges and CFs, which are present in more than half of all CCs. Bulk shear flow is believed to magnetize the plasma across and beneath the CFs; there is however no evidence for shear above the CFs (Keshet et al. 2010). Therefore, observations of sharp MH termination coincident with CFs reflect the transition from strong to weak magnetic fields. This is probably the reason for the rapid, nearly exponential η(r) cutoff in Perseus above r ≃ 0.05 r₅₀₀ ≃ 50kpc, shown in Figure 2. Note that CFs are not spherical; the radial decay may result from one or several CFs extending over various radii, seen projected and radially binned.

In GHs and in MHs away from CFs, it is natural to assume that the magnetic field decays with r, possibly as some fixed fraction of equipartition B² ∝ nT. At some distance r_b, B drops below B_cmb, leading to a ∼(B/B_cmb)² suppression in the radio emissivity. This can produce a radial break in the projected radio profile, with η ≃ constant at r ≪ r_b and ${\eta }\propto B^2 \propto r^{{{\kappa }_B}}$ at r ≫ r_b with some κ_B < 0. A power-law pressure profile $nT\propto r^{{\kappa }_P}$ would imply κ_B = κ_P; for an isothermal distribution κ_B = −2.

Such a decline in η is found in A2029 above r_b ≲ 120kpc, and possibly also in the MHs in A1835 and RXJ1347, as seen in Figure 2. The figure also shows a simple model for A2029, where we assume that B² = knT and that η asymptotes to its average GH value at small radii, η(r → 0) → η₀ = 10^−4.4, so the proportionality constant k is the only free parameter. This fit (dot-dashed curve) corresponds to a central magnetic field B₀ ≃ 2.1μG. However, due to the limited range of data in this cluster, there is a degeneracy between B₀ and η₀—higher magnetic fields are possible if the central value of η in A2029 is lower than the GH average, and vice versa. The exponential fit of M09 (dashed red curve) suggests higher magnetic fields and lower η₀; a fit with B₀ = 6.3μG is shown (dotted) to illustrate this. The corresponding low value of η₀ in this cluster could arise from its low star formation rate (SFR), as discussed in Section 6.2.

Assuming that B² ∝ nT for r < r_b, we can place a lower limit on the central magnetic field amplitude B₀ in each observed halo, given a pressure model. If no break in η is identified out to the halo radius R_ν, then

$$\begin{eqnarray} &&B_0> B_{0,\rm min} \equiv \bigg[ \frac{(n T)_{r=0}}{(n T)_{r=R_\nu }} \big(f_{{\rm ic}}^{-1}-1\big) \bigg]^{1/2} B_{\rm cmb}(z), \end{eqnarray} \tag{ 12 }$$

where we allowed a fraction f_ic ≃ 0.5 of inverse Compton losses at R_ν. The B_0,min values of the halos in our sample are presented in Table 1, assuming that R_ν = min{r_b, 3r_e} and adopting individual isothermal β-models for each cluster, as detailed in the table. In the MHs, the central magnetic field could be much higher than B_0,min because (1) the significant growth in nT toward the center (due to the density cusp) is not captured by the β-model and (2) the magnetic field immediately beneath the CFs could be very strong and near equipartition (Keshet et al. 2010). However, in MHs with only a declining η profile observed (e.g., A2029), adopting the upper limit on r_b may overestimate B₀.

Instead or in addition to a radial break in brightness, a halo in which the magnetic field is marginal, B ≃ B_cmb, could become clumpy or filamentary, appearing bright only in B ≳ B_cmb islands. This is probably the reason for the unusual clumpy or filamentary radio morphology observed in RXC J2003.5 − 2323 (Giacintucci et al. 2009), A2255, and A2319 (M09). Indeed, a decline in η is directly seen in A2319 around r = 150kpc (suggesting low magnetization; see Figure 2), and A2255 is the only strongly (20%–40%) polarized GH known to date (Govoni et al. 2005); the absence of strong beam depolarization suggests relatively weak magnetic fields (see, for example, Murgia et al. 2004). Additional evidence for the low magnetization in these three halos is their relatively steep spectrum, as discussed in Section 5.2.

Such marginally magnetized halos are particularly interesting because the magnetic field can be determined in multiple locations, and because the radio morphology directly gauges the magnetization or magnetic decay process. Fast putative variations in the magnetic field may be easier to detect in such regions, as they could involve temporal changes in the small scale radio morphology. For example, in A2319, SKA could resolve such changes over a light crossing time of ∼2 yr.

While radio emission rapidly declines outside the B > B_cmb region, an opposite signature is expected in inverse Compton emission from the same CREs, as they scatter CMB photons to high energies. This emission may be clumpy or filamentary in B ∼ B_cmb regions. However, as pointed out by Kushnir & Waxman (2010), such radiation cannot explain the hard X-ray excess detected in a number of clusters (for review, see Rephaeli et al. 2008), because the inverse Compton signal from secondary CREs has νI_ν comparable to that in the radio (see Equation (4)), ∼3 orders of magnitude lower than needed to account for the detected hard X-ray excess.

Spectral steepening with increasing r, increasing ν, or decreasing T has been reported in several radio halos (Feretti & Giovannini 2008; Ferrari et al. 2008; Giovannini et al. 2009, and references therein). Such trends naturally arise in our model due to the energy dependence of the inelastic cross section for collisions of CRPs with the intracluster gas at the relevant E_p ∼ 20GeV CRP energies.

Secondary CREs are mainly produced by charged pion production p + p → π^± + X, where X is any combination of particles, followed by mesonic decays $\pi ^\pm \rightarrow \mu ^\pm +\nu _\mu (\bar{\nu }_\mu)$ and leptonic decays $\mu ^\pm \rightarrow e^\pm +\nu _e(\bar{\nu }_e) \, + \, \bar{\nu }_\mu (\nu _\mu)$. Other processes, in particular He–p collisions, have a similar cross section per nucleon and should not modify our results by more than 10%. On average, the energies of the CRE, the π^±, and the CRP are related by E_e ≃ f_e|πE_π ≃ f_e|πf_π|pE_p, where f_e|π ≃ 1/4 and f_π|p ≃ 1/5 (Ginzburg & Syrovatsky 1961). The inclusive cross section for π^± production varies significantly as a function of E_p around E_p ≃ 10GeV, dropping from >40mb above 30GeV to zero at the threshold energy 0.3GeV (Blattnig et al. 2000).

We compute the radio spectrum based on two different methods. Model A uses the spectral fits for e⁺ and e⁻ production in inelastic p–p scattering according to Kamae et al. (2006, valid for 0.5GeV < E_p < 512TeV), with corrected parameters and cutoffs (T. Kamae & H. Lee 2010, private communication). Model B assumes that E_e = (E_p/20) and utilizes the inclusive cross sections for charged pion production according to Blattnig et al. (2000), which agree with experimental data for 0.3 < (E_p/GeV) ≲ 300 (Norbury 2009). Model A should be more accurate because f_e|πf_π|p is not independent of E_e and the CRP spectrum.

The results, depicted in Figure 3 for emitted frequency ν_e = (1 + z)ν = 1.4GHz and various power-law CRP spectra n_p(E_p) ∝ E^s_p, show that the radio spectrum steepens with increasing cosmic-ray electron/positron energy

$$\begin{eqnarray} &&E_e\simeq 4.2\nu _{e,1.4}^{1/2}(B/5\mbox{ $\mu $G})^{-1/2}\mbox{ GeV}. \end{eqnarray} \tag{ 13 }$$

Zoom In Zoom Out Reset image size

Model A features a spectral break at E_e ≃ 2GeV, attributed mainly to electron production through diffractive processes, in which one or both protons transitions to an excited state (discrete resonances and continuum). Note that even if this channel is blocked, a similar spectral break is found at E_e ∼ 1GeV. A more dramatic steepening than in either model is found if we assume that E_e = (E_π/4) and use the pion spectra produced in p–p collisions according to Blattnig et al. (2000); however, these formulae have not been tested beyond E_p = 50GeV.

Due to the limited energy range over which the energy index s_e ≡ dlog(n_e)/dlog(E_e) of the injected CREs can be computed, the radio spectrum in Figure 3 is calculated in the approximation where each CRE emits a single photon, such that α ≃ s_e/2 (e.g., Keshet et al. 2003). A more accurate computation, convolving the CRE distribution with the synchrotron emission function, would somewhat smear the spectral features.

The radio steepening with increasing E_e implies steepening with increasing ν or r, or decreasing T, as observed. Both models agree that spectral steepening by |Δα_ν|>0.2 indicates B > 20(ν_e/1.4GHz)μG magnetic fields in the region associated with a flatter radio spectrum. Measuring α across the E_e ≃ 2GeV spectral break can be used to unambiguously fix the CRP spectrum s. Similarly, measuring α at the spatial η break where B ≃ B_cmb can be used to determine s, as illustrated in the inset of Figure 3. Conversely, if a distinct spectral break exists as predicted by model A, it would directly gauge the magnetic field, once the CRP spectrum has been determined. Multi-frequency radio data could thus allow a sensitive mapping of the magnetic field, both above and below B_cmb, throughout the halo, using, for example, using deprojected maps at several radio frequencies.

Spectral measurements of radio halos should be interpreted with caution. Contamination by CRE sources such as shocks, relics, central AGNs, and radio galaxies is common. Spatial averaging often blends together different sources, depending on flux sensitivity and angular resolution. Therefore, with present observations, the spectrum can be reliably associated with a halo only when measured locally in uncontaminated regions; in our model these are regions of uniform η. Also note that radio emission above 1 GHz is increasingly suppressed by the Sunyaev–Zel'dovich effect (Enßlin 2002).

For example, η(r) is nearly constant in A665 within the range r < r_max ≃ 150'' examined in Figure 2. Spectral steepening with increasing r was identified in this cluster (Feretti et al. 2004b) by comparing radio maps at 0.3 and 1.4GHz frequencies. The spectral index steepens from α ≃ −1.0 in the halo's center to α ≃ −1.3 toward the south and toward the east, within the flat η region. Similar steepening was found toward the north and west beyond r_max, but the spectrum first flattens to α ≃ −0.9.

Similarly, the regular GH in A2744 exhibits a linear radio–X-ray correlation in brightness out to r ≃ 1Mpc (Govoni et al. 2001a). Along the main northwest elongation, the 0.3–1.4GHz spectral slope α(r < 200kpc) ≃ −1.0, slightly flattens to −0.9 around 500kpc, and steepens again to −1.5 as r → 1Mpc. An azimuthal average, however, includes a northeast relic tail and under-threshold regions, leading to an unrealistically uniform α(r < Mpc) ≃ −1 (Orrú et al. 2007). A similar behavior is found in A2219 (Orrú et al. 2007).

In these examples, the spectral index steepens with increasing r from α ≃ −1.0 to −(1.2–1.5) in the less perturbed/contaminated direction, sometimes after a mild flattening to α ≃ −0.9. Such steepening was also reported in A2163 (Feretti et al. 2004b) and in Coma (Giovannini et al. 1993).

Comparable steepening was found as a function of frequency in the GH in A754. Its spectrum measured between 74 and 330MHz, α^0.3_0.07 ∼ −1.1, steepens to α^1.4_0.3 ∼ −1.5 (Bacchi et al. 2003). More substantial steepening has been reported in other clusters, such as A2319 (Feretti et al. 1997) and A3562 (Giacintucci et al. 2005). However, contamination by extended radio galaxies was reported in these halos.

Significant steepening with increasing r or ν as in the examples above is more consistent with the E_e ∼ 2GeV spectral break of model A than with model B. In model A, strong magnetic fields with B ≳ 10(ν_e/700MHz)μG are present in regions where α_ν is flat. Moreover, an uncontaminated spectral break measured at some emitted frequency ν_b would imply that the local projected magnetic field is B ≃ 10(ν_b/700MHz)μG. Observations at several frequencies can thus be used to map the local projected magnetic field. As a preliminary example, interpreting the radio spectrum of A754 as a spectral break somewhere between ν_e = 200 and 900MHz near the center of the halo would imply 3 < B₀/μG < 14.

Figure 3 indicates that radio steepening from α = −1.0 to −1.3 corresponds to a CRP spectral index s ≃ −2.7. Steepening beyond −1.5, if uncontaminated, would imply a steep CRP spectrum with s < −3. The radio spectrum at high, E_e ≫ 10GeV CRE energies approaches α → s/2. However, inward of the spectral break at E_e < 2GeV, α ≃ −1 depends only weakly on s. This could explain the universally observed α ≃ −1 in the centers of halos.

Figure 4 shows the average ν ≲ 1.4GHz spectral indices 〈α_1.4〉 reported for GHs in our sample, as a function of their minimal central magnetic fields B_0,min calculated in Section 5.1. The results are also summarized in Table 1. A positive correlation is seen in the sense that halos with stronger central magnetization tend to have a flatter radio spectrum, as expected. Note that this correlation is stronger than, and not simply due to, a correlation we find between 〈α〉 and the halo size.

Zoom In Zoom Out Reset image size

For comparison, we also plot (dashed lines in Figure 4) the average radio spectrum 〈α_1.4〉 as a function of B₀, computed for a typical isothermal β-model with β = 0.7, assuming B² ∝ nT, spectral model A, and radio emission extending to R_ν ≃ 3r_c. Although this result cannot be quantitatively compared to the GH data (which are only lower limits on B₀), the trends qualitatively agree. This provides an independent indication that the CRP spectrum is steep, with s ≲ −2.7.

The radio spectrum is relatively steep in the five GHs and MHs for which we presented evidence for low magnetization in Section 5.1, in agreement with the 〈α〉–B₀ correlation. The halos that are not shown in Figure 4, include A2029 (〈α_0.2〉 ≃ −1.35; Slee & Siegman 1983), A2255 (〈α^1.4_0.3〉 = −2.06 ± 0.57; Kempner & Sarazin 2001, where we infer 〈α^1.4_0.6〉 ∼ −1.5 by comparing M09 with Harris et al. 1980), Perseus (〈α^1.5_0.3〉 ∼ −1.2; Gitti et al. 2004), and RXC J2003.5 − 2323 (〈α^1.4_0.2〉 ∼ −1.32 ± 0.06; Giacintucci et al. 2009).

Finally, a cautionary comment is in place regarding the extrapolation of the spectrum to low frequencies that correspond to E_e ≲ 1GeV CRE energies. Although E_e = 1GeV CREs are produced on average by E_p ∼ 20GeV CRPs (for s = −2.5), the contribution of low energy CRPs with E_p ≲ 3GeV is not negligible. We expect a CRP power-law energy spectrum n_p(E_p) ∝ E^s_p to be modified around the proton rest mass. Consequently, our s = constant approximation becomes increasingly unlikely at low E_e ≲ fewGeV CRE energies.

In the model derived above, the primary CRI population is long-lived and has very similar properties in different clusters and in different parts of a cluster. Hence, the defining property of halos is sufficiently high magnetization. Measuring strong, B > B_cmb magnetic fields without radio detection at the level given by Equation (4) would rule out the model, unless the CRI energy fraction is exceptionally low (e.g., due to a low level of star formation; see Section 6.2). The model can be tested by checking if the different magnetic field estimates it produces in a given halo are self consistent and agree with independent measurements.

In the model, GH clusters are associated with strong, B ≳ B_cmb ∼ 3μG magnetic fields, whereas weaker, B < B_cmb fields are present in clusters devoid of a halo. Magnetic fields of the order of B_cmb, extending over Mpc scales, are consistent with Faraday rotation measures (RM) in non-cool clusters, considering the uncertainties involved. Such RM studies suggest magnetic fields ranging between a few μG to 10μG, extending out to r ∼ 500kpc, in a sample of non-CCs, on 10–20kpc coherence scales (Clarke 2004). Weaker magnetic fields, typically ∼1μG to a few μG, were found using other methods, such as simulating the polarization of extended sources (Murgia et al. 2004) or using a Bayesian maximum likelihood analysis of the Faraday RM (Vogt & Enßlin 2005).

RM studies could potentially test the association between strong magnetization and the presence of halos, as implied by the model, although this is complicated by several factors. First, RM analyses involve substantial systematic and statistical errors associated with projection effects, the magnetic power spectrum, the range of coherence scales, and non-Gaussianity. Second, cluster fields are often reconstructed using peripheral RM sources, assuming some magnetic field scaling B ∝ n^μ that extends both inside and outside the halo; however, this assumption is uncertain and in some cases inconsistent with our model, e.g., in asymmetric or clumpy halos. Finally, the RM gauges the magnetic field amplitude integrated along the line of sight, whereas the radio emission scales differently with the magnetic field, e.g., as B² in B ≲ B_cmb regions.

It is interesting to compare the magnetic fields estimated in different clusters. Care must be taken to compare the same measures estimated under the same assumptions. For example, assuming a Gaussian, Kolmogorov magnetic power spectrum with μ = 1/2, we find 〈B(r < 3r_c)〉 = 1.8μG and B(3r_c) = 1.3μG in Coma (Bonafede et al. 2010), which harbors a GH, while weaker fields, 〈B(r < 3r_c)〉 = 1.2μG and B(3r_c) = 0.8μG are found in A2382 (Guidetti et al. 2008), which has no halo. Similarly, 〈B(r < 3r_c)〉 = 1.3μG and B(3r_c) = 0.8μG are found in A119 (Murgia et al. 2004), which does not harbor a GH, but this result is obtained for μ = 0.9, so comparison to the other two clusters may be misleading. Note that it is not clear, considering the inherent uncertainties, if the differences between these B estimates are significant.

The association between GHs and merger events is fairly well established (see Feretti & Giovannini 2008, and references therein). With present magnetic field estimates, it is quite plausible that a major merger event could amplify the intracluster magnetic field by a factor of a few, sufficient to exceed B_cmb and produce a GH. The model thus resolves the notorious discrepancy between magnetic field estimates based on Faraday RM and on previous GH analyses, which unnecessarily assumed equipartition between magnetic fields and CREs.

The very strong magnetic fields we infer in the centers of MHs are consistent with the RM observed in CCs, which typically imply B ≃ 10–40μG in the core, on coherence scales of a few up to 10kpc (Carilli & Taylor 2002). It is interesting to point out that the cooling flow power is correlated both with the RM (Taylor et al. 2002) and with the MH radio power (Gitti et al. 2004), while no strong correlation between MH and AGN power was identified (e.g., Govoni et al. 2009).

The association between MH edges and CFs strongly suggests that sloshing motions play a major role in magnetizing the core. Note that ∼10kpc scales are indeed characteristic of the shear magnetic amplification anticipated beneath CFs (Keshet et al. 2010). Sloshing could suppress cooling in the core, for example, by mixing the cold gas with a heat inflow (Markevitch & Vikhlinin 2007). In such a scenario, a stronger cooling flow may correspond to a larger magnetized region, leading to correlations between the cooling flow power and both RM and MH power, as observed, but not necessarily to an MH–AGN correlation.

The distribution of MHs (without the AGN component) among CCs should reflect the statistics of core magnetization, and therefore of sloshing. We expect a correlation between the presence of MHs and of CFs in a cluster, and a correlation between the MH size or power and the shear flow strength, manifest, for example, in the number and size of CFs and in the (measurable, see Keshet et al. 2010) shear across them. Assuming that some level of magnetization by core sloshing is always present in CCs, as suggested by the ubiquity of observed CFs, the steep magnetic profile associated with the density cusp would imply that every CC has some MH, even if small. The MH distribution would then be continuous and not bimodal as in GHs.

Shear amplification of magnetic fields parallel to the CF plane is expected mostly below the CF and in a thin boundary layer around it, in which the field can reach high, near equipartition levels (Keshet et al. 2010). The morphological association between MH edges and CFs, discovered by Mazzotta & Giacintucci (2008), is therefore expected to be ubiquitous. It is best seen where CFs are observed edge on; elsewhere it may be observed as morphological correlations between radio maps and spatial X-ray gradients, which trace projected CFs.

The magnetic field (the polarization) is expected to be parallel (perpendicular) to the CF, i.e., approximately tangential (radial). Polarization would be preferentially observable where beam deprojection is minimal, i.e., at large radii where the magnetic field weakens. Interestingly, nearly radial, 10%–20% polarization was detected in the MH in A2390, growing stronger with increasing radius (we refer to the spherical, r ∼ 150kpc component around the cD galaxy in this irregular MH; see Bacchi et al. 2003). We predict that near CF edges, where the magnetic field is particularly strong, the radio spectrum would be relatively flat (see Section 5.2).

The spectral steepening of the radio signal with increasing CRE energy E_e ∝ (ν/B)^1/2 provides a novel method for measuring the primary CRP spectrum. The radio steepening observed in some halos, roughly from α ≃ −1.0 to α ≃ −1.3, corresponds to a CRP spectral index s ≲ −2.7 at E_p ∼ 20GeV energies. The CRP energy fraction then becomes (cf. Equation (10))

$$\begin{eqnarray} \xi _{p}(>E_p) & \equiv & \frac{u_{p}(>E_p)}{u_{{\rm th}}} \simeq 10^{-3.6\pm 0.2}\left(\frac{E_p}{10\mbox{ GeV}}\right)^{-0.7}.\qquad \end{eqnarray} \tag{ 14 }$$

Note that with the uncertain and possibly contaminated radio spectra presently available, a steeper CRP spectrum with s ≃ −3 is possible. The model would be challenged if the uncontaminated spectrum of a substantial halo population turns out to be much steeper than α = −1.5, unless the corresponding steep CRP spectrum can be explained.

The CRP distribution in Equation (14) resembles (but has an energy fraction a few 100 times smaller than) the CRP distribution found in the solar vicinity above 1GeV nucleon⁻¹. This distribution could originate from sources that inject roughly equal energy per decade of CRP energy (s₀ ≃ −2.2), such as SNe, if energy-dependent diffusion is significant in the inner halo regions. For example, a simple estimate of CRI scattering off magnetic irregularities with a Kolmogorov power spectrum yields a diffusion coefficient D ≃ 10³⁰(E_p/GeV)^1/3(B/μG)^−1/3cm²s⁻¹ (Völk et al. 1996). This implies CRI diffusion over ∼0.5Mpc during a Hubble time and a steepening by Δs = −1/2. More substantial steepening is possible if the diffusion function has a stronger energy dependence. For example, the diffusion function is often assumed to scale as D ∝ E^1/2_p, which could lead to a Δs = −3/4 steepening in the CRI spectrum.

The CRI output of SNe can be crudely estimated (Völk et al. 1996) if we assume that a fraction f_II of the cluster's Z = 0.3 Z_0.3 solar metallicity is seeded by Type II SNe, which on average produce 0.1 M_☉ M_Fe,0.1 of iron and deposit a fraction ξ = 0.3ξ_0.3 of the 10⁵¹ E₅₁erg explosion energy in E_p > 10GeV CRIs:

$$\begin{eqnarray} &&\xi _{p}^{\rm {SN}} \simeq 0.03\, f_{{\rm \scriptsize {II}}}\, Z_{0.3}\, E_{51}\, M_{{\rm Fe},0.1}^{-1} \xi _{0.3}. \end{eqnarray} \tag{ 15 }$$

This can reproduce Equation (14) if over the cluster's lifetime, the CRIs diffuse to distances a few times larger than the radius R_ν of the radio halo. Note that if CRI diffusion is entirely absent, the CRIs accelerated in SNe would be confined to the cluster and adiabatic losses could only lower their energy density to the level of Equation (14). However, they would then retain their flat s ≃ −2.2 spectrum.

An SNe origin of CRIs can be tested by examining the correlations between η and (intensive) tracers of SNe activity among different halos. One possible tracer is the local metallicity measured at r = 0.1 r₅₀₀, tabulated in Table 1. We chose to use Z(0.1 r₅₀₀) because (1) it was measured for all the M09 halos with XMM-Newton profiles in Snowden et al. (2008), (2) the spatially averaged Z is not meaningful when temperature gradients are large, and (3) 0.1 r₅₀₀ lies well within the Z(r > 0.02 r₅₀₀) ∝ r^−0.3 decline typically found in both cool and non-CCs (Sanderson et al. 2009). While our sample is statistically small, Perseus, which shows a significantly higher η than in all the other halos in our sample, also shows a slightly higher Z(0.1 r₅₀₀) (Snowden et al. 2008). However, due to the large uncertainty in abundance measurements, the elevated metallicity in Perseus is not significant (<1σ) with respect to some halos. Moreover, at smaller radii r ≲ 0.025 r₅₀₀, the metallicity in A2029 appears to be higher than in all other halos and is significantly (>3σ) higher than in Perseus (Snowden et al. 2008).

Although better metallicity statistics may identify a more significant correlation between η and Z, metallicity is probably not the most useful tracer of the SNe contribution to CRIs in the halo. Metallicity provides a cumulative measure of SNe activity, tracing the metals released from all past SNe in the cluster. In a model where a significant fraction of the CRIs have already diffused away from the cluster's center, metallicity would not linearly trace the population of CRIs residing within the halo, especially in the more compact MHs. It is more appropriate to use an intensive tracer of recent SNe activity, such as the SFR normalized by the cluster's gas mass M_g or the fraction of star-forming galaxies. A correlation between an SNe measure and a CRI tracer, such as η or the deviation from the luminosity correlation $\nu P_\nu /\bar{L}_{X[0.1,2.4]}^{1.7}$, may be more useful than the metallicity in establishing or ruling out an SNe origin of halo CRIs.

As seen in Figure 2, the halos with the highest η in our sample are the MH in Perseus and the GH in A665, while the halos with the lowest η are the MH in A2029 and the GH in A773. Interestingly, the literature shows evidence for exceptionally high specific star formation in both Perseus and A665 and for a low specific SFR in A2029. (We found no relevant data for A773.)

In Perseus, which has the smallest M_g (by at least a factor of 4, see Fukazawa et al. 2004) and one of the most powerful cooling flows within our sample (e.g., White 2000; Allen et al. 2002), there is optical-to-UV evidence for a relatively high SFR (e.g., Bregman et al. 2006; Rafferty et al. 2008). In particular, the cD galaxy NGC1275 in Perseus has a high SFR of ∼30 M_☉yr⁻¹ (Dixon et al. 1996)—the highest in our sample when normalized by M_g. Note that the central galaxy in A1835 has a higher SFR of ∼100 M_☉yr⁻¹—the highest SFR known in such objects (Peterson & Fabian 2006). However, M_g is ∼10 times larger in A1835 with respect to Perseus (Fukazawa et al. 2004). Note that regions containing a high density of cosmic rays are directly observed in Perseus in the form of X-ray cavities, reaching distances r > 100kpc (M. Markevitch 2010, private communication). A high specific SFR is also inferred in A665, which was found to be the cluster with the highest dispersion in color–magnitude relation (a known tracer of star formation) in a sample of 57 X-ray bright clusters (López-Cruz et al. 2004). In contrast, A2029 has a low specific SFR, as it was shown to have an SFR ∼70 times lower than in A1835 (Hicks & Mushotzky 2005) while its gas mass is only ∼1.8 times smaller (Fukazawa et al. 2004).

While these trends support an SNe origin of CRIs, more work is needed in order to quantify their significance and compile a comparative statistical analysis. Note that the combination of a strong correlation of η with the specific SFR and a poor correlation with metallicity, if established, would directly imply that CRI diffusion is significant. Indeed, it is difficult to explain the steep s ≲ −2.7 spectrum without invoking CRI diffusion.

The above estimates of diffusive steepening assume that most of the CRIs produced by the sources presently dominating the halo have already escaped beyond it. This is consistent with the typical SFR peak at z ∼ 1, and with the above estimates of the halo CRI abundance and the total CRI output of SNe (cf. Equations (14) and (15)). However, such substantial diffusion would introduce some scatter in the radio–X-ray correlation, depending on the CRI production history of each cluster. Quantitative estimates of the SNe history of GH clusters, needed to compute this scatter, are beyond the scope of this work.

We have shown that the radio–X-ray correlation in GH luminosity (Equation (1)) can be generalized (Equation (2)) to hold for both GHs and MHs (Figure 1), by correcting for the halo size. A universal, linear relation between the radio and X-ray surface brightness, η = 10^{−4.4 ± 0.2}, was presented (Equation (4) and Figure 2). This, combined with the radial η and T profiles, implies a universal radio emissivity νj_ν = 10^{−31.4 ± 0.2}n²₋₂(T/T₀)^{0.2 ± 0.5}ergs⁻¹cm⁻³ (Equation (7)) near the center of halos. We argued that these results and their applicability to GHs and MHs alike strongly support one model for all halos, involving secondary CREs (injected according to Equation (10)) and strong magnetic fields with B ≳ B_cmb, while disfavoring other models (Table 2).

This model makes useful predictions without requiring additional assumptions or fine tuning. Radio emission rapidly fades in regions where B drops below B_cmb, producing a distinct radial break (e.g., in A2029 and in Perseus; Figure 2) or a clumpy/filamentary radio morphology (e.g., in RXC J2003.5 − 2323, A2255, and A2319) that can be used to map B ≃ B_cmb contours. Marginally magnetized regions with B ≲ B_cmb ∝ (1 + z)² are characterized by relatively high polarization and a steeper radio spectrum; their morphology traces the magnetic evolution and can potentially reveal a temporal signal. We expect a higher incidence rate of such transition regions at higher redshift, while no halos should exist at very high redshift.

Another direct consequence of the model is radio spectral steepening with increasing E²_e ∝ ν/B (Equation (13)), i.e., with increasing r or ν or decreasing T, as indeed is observed. Such steepening, and in particular a B ≃ 10(ν_e/700MHz)μG spectral break (Figure 3), gauges the magnetic field, roughly producing an additional B contour for each radio map frequency. The spectral break could be used to accurately map B throughout the halo, using future radio telescopes such as the Murchison Widefield Array,³ the Low Frequency Array,⁴ and SKA.

A pressure model can be used to extrapolate B throughout the cluster. This indicates central magnetic fields B₀ (Equation (12)) that exceed 10μG in most halos (see Table 1 for lower limits B_0,min). A correlation between the average radio spectral index 〈α〉 and B₀, implied by the model, was identified in GH data (Figure 4).

In our model, any source of strong (B ≳ B_cmb), persistent magnetic fields in the intracluster medium would have similar properties to radio halos, as long as it does not significantly inject additional cosmic rays. This may include some extended radio galaxies, which were recently found to exhibit properties similar to halos (Rudnick & Lemmerman 2009). Conversely, the universal value of η we predict for any highly magnetized, uncontaminated region in the intracluster medium provides a powerful test of the model.

The spectral steepening of the radio signal, the universality of α ≃ −1 in the centers of halos, and the correlation between 〈α〉 and B_0,min (Figure 4), indicate a steep CRI spectrum, s ≲ −2.7, and thus favor significant CRI diffusion. In a diffusion model, the most plausible source of the CRIs (Equation (14)) is SNe (e.g., Equation (15)). We show (in Section 6.2) preliminary evidence for a correlation between η and the SFR normalized by the gas mass M_g, supporting an SNe CRI origin. None of these properties is expected in an alternative model (K09), in which the secondary CREs arise from ∼1GeV CRPs, which are accelerated in the cluster's virial shock and advected inward with the flow, thus being compressed to ∼10GeV energies. Note that the data slightly favor a j_ν ∝ n²T⁰ scaling within each cluster (see Section 4), which is natural if CRIs originate in SNe, rather than the j_ν ∝ n²T¹ behavior anticipated if they are accelerated in the virial shock. Also note that adiabatic compression of CRIs produced at the virial shock and advected with the gas would lead to a radially increasing η ∝ n^−1/3 profile (due the soft equation of state of relativistic particles), which is not observed (see Figure 2).

We stress that although our model and the model of K09 disagree regarding the origin of CRIs, the CRI distribution, the spectral properties of halos, and the role of diffusion, we reach the same conclusions regarding the radio mechanism: emission from secondary CREs in strong magnetic fields. This conclusion is based on (1) the tight radio–X-ray correlation in total GH luminosity and the GH bimodality (K09), (2) the tight radio–X-ray correlation in both coincident luminosity and surface brightness, in both GHs and MHs, despite their different physical properties, (3) the strong magnetic fields inferred from Faraday RMs in MHs and possibly (see Section 6.1) also in GHs, (4) the tightening of the brightness correlation at small radii, away from merger shocks, radio relics, and their associated turbulence, (5) the j_ν ∝ n²T^κ scaling of the radio emissivity within each halo, where κ ≲ 1, (6) the coincidence between MH edges and CFs manifest as a sharp radial cutoff in η (e.g., in Perseus), (7) a power-law radial break where η(r ≪ r_b) ≃ constant and η(r ≫ r_b) ∝ B², possibly seen in the MHs in A2029, A1835, and RXJ1347, and in the GH in A2319, (8) the clumpy/filamentary morphology of some halos, where independent evidence for low magnetization is present, and (9) the spectral steepening and the correlation between 〈α〉 and B₀ (this suggests strong magnetization provided that the CRI spectrum is steep s ≲ −2.7).

Each aspect of our model can be tested in the near future. The association between the presence of halos and strong B ≳ B_cmb magnetic fields can be directly tested by comparing the magnetization levels independently estimated in halo and in non-halo clusters, as illustrated in Section 6.1. The secondary origin of the halos can be tested if the CRIs are detected directly through their π⁰ production; for example, such a detection of a CRI component substantially stronger than in Equation (14) would rule out our model. The SNe origin of the CRIs can be tested by carefully examining the correlation between a CRI measure such as η and an intensive SNe tracer such as the specific SFR in a sample of halo clusters.