Preamble

Vol. 43, No. 2

June 2025

Progress in Astronomy Vol. 43, No. 2 June 2025 doi: 10.3969/j.issn.1000-8349.2025.02.03

Diffuse Radio Emission in Galaxy Clusters

WU Wenyu¹,², XIE Fuguo¹
(1. Astrophysics Division, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China)

Abstract

Galaxy clusters, as the largest self-gravitationally bound systems in the Universe, serve not only as laboratories for studying galaxy formation and evolution but also as precise tracers of cosmic large-scale structure. This paper primarily introduces the observational characteristics and physical origins of diffuse radio emission from galaxy clusters, and its significance in studying cluster evolution. Cluster-scale diffuse radio emission originates from synchrotron radiation produced by non-thermal electrons in magnetic fields, including radio halos, radio mini-halos, and radio relics. Radio halos predominantly appear in merging clusters, and their origin is thought to be explained by merger-induced turbulent reacceleration models. Radio mini-halos, currently few in number, mainly exist in relaxed cool-core clusters, and their origin can also be explained by turbulent reacceleration models, except that the turbulence is induced by gas sloshing in the cluster core. Radio relics are mostly distributed in cluster peripheries, generally showing high polarization, and their origin is related to shocks generated during cluster mergers. Finally, we briefly prospect galaxy cluster research in the era of LOFAR 2.0 and SKA.

Keywords: galaxy cluster; radio halo; radio mini-halo; radio relic; synchrotron emission
Classification Number: P157.8
Document Code: A

1 Introduction

Galaxy clusters consist of hundreds to thousands of member galaxies and represent the largest self-gravitating bound systems that can maintain virial equilibrium in the Universe. The total mass of a galaxy cluster can reach $10^{15} M_\odot$, with spatial scales of several Mpc [1]. Clusters are crucial sites for galaxy formation and evolution, though member galaxies contribute less than 5% of the total cluster mass. The main components also include dark matter halos (approximately 80% of total mass) and the intracluster medium (ICM) pervading the vast space between galaxies (accounting for 15%–20% of total mass) [2]. From the perspective of cosmic large-scale structure, galaxy clusters located at nodes of cosmic filaments, together with non-bound galaxies forming cosmic filaments, constitute the cosmic web, making clusters fundamental building blocks of large-scale structure.

The ICM is a high-temperature ($T \approx 10^7 \sim 10^8$ K), low-density (number density $n \sim 10^{-3}$ cm$^{-3}$) plasma confined by the cluster's gravitational potential. These hot gases produce thermal bremsstrahlung radiation primarily in the X-ray band. Physically, during cluster formation or mergers, ICM gas accreting into central regions can convert gravitational potential energy into internal gas energy. The first galaxy cluster X-ray sources were observed by the Uhuru X-ray satellite in the 1970s, with well-known sources including the Perseus and Coma clusters [3]. Half a century of observations shows that galaxy clusters are typically bright extended X-ray sources with luminosities $L_X$ ranging from $10^{36} \sim 10^{38}$ J·s$^{-1}$ [1]. X-ray luminosity shows clear correlations with both cluster total mass and ICM gas temperature or the parameter $Y_X$ (product of ICM gas mass and temperature) [4–6]. Notably, the observed slopes of these correlations are steeper than theoretical predictions from self-similar gravitational collapse models, suggesting ICM properties may be influenced by additional or more complex physical processes such as star formation and feedback, active galactic nucleus (AGN) feedback, and cosmic ray heating [1, 5].

Galaxy clusters also exhibit rich observational information at radio wavelengths. On one hand, numerous thermal electrons in the ICM cause inverse Compton scattering of cosmic microwave background (CMB) photons, creating spectral distortions known as the SZ effect [7, 8]. Since the SZ effect depends on the total pressure of hot gas along the line of sight rather than cluster redshift, numerous clusters—particularly high-redshift ones—have been discovered at millimeter/submillimeter wavelengths. On the other hand, large-scale extended radio emission with smooth spectra has been detected in some clusters' ICM, indicating a new component: non-thermal cosmic ray particles. This diffuse radio emission originates from synchrotron radiation produced by high-energy relativistic electrons (Lorentz factor $\gamma > 10^3$) moving in cluster-scale magnetic fields of order $10^{-10}$ T. The prevailing view holds that non-thermal electrons in the ICM are primarily accelerated (or re-accelerated) by turbulence or low-Mach-number shocks resulting from cluster dynamical processes (mainly mergers) [9]. These diffuse radio sources exhibit diverse morphologies, classified into three types: radio halos, radio mini-halos, and radio relics [9]. Radio halos appear in merging cluster centers (not projection effects [10]) as large-scale ($\gtrsim 1$ Mpc) diffuse structures with regular morphology, low polarization, and no optical counterparts. Radio mini-halos are smaller-scale diffuse structures found in relaxed cool-core cluster centers (often around central radio galaxies), differing from halos not only in size (hundreds of kpc) but also in host cluster type, with regular morphology and low polarization. Radio relics are large-scale ($\gtrsim 1$ Mpc) diffuse structures in cluster peripheries with irregular, often elongated shapes, strong polarization, and no optical counterparts. All three sources feature low surface brightness and steep spectra.

Galaxy clusters can be divided into relaxed and merging systems based on dynamical state. Cluster mergers represent the most violent astronomical events since the Big Bang, releasing up to $\sim 10^{57}$ J of energy over billion-year timescales. This energy heats ICM gas, accelerates cosmic rays, and amplifies magnetic fields through turbulence or low-Mach-number shocks [11]. Thus, merger processes simultaneously affect both thermal and non-thermal components, manifesting across different wavebands from X-ray to radio. Indirect evidence includes the clear positive correlation between radio halo power and X-ray luminosity [9, 12]. Many radio relics also show direct spatial correspondence with shocks identified in X-rays, providing observational constraints on relic formation mechanisms [13]. Clearly, multi-wavelength observations are essential for comprehensively revealing cluster states, structures, and evolution. Based on X-ray characteristics, clusters can also be classified as cool-core and non-cool-core systems. The key difference is that cool-core clusters exhibit prominent peaks in central X-ray surface brightness. Since the bremsstrahlung emissivity scales as $j \propto n^2 T^{1/2}$ and the cooling timescale as $t_{\rm cl} \propto n^{-1}T^{1/2}$, this indicates centrally concentrated, high-density gas with short cooling times and temperatures significantly below the cluster's characteristic temperature. Some cool-core clusters also show arc-shaped or spiral cold fronts [11], representing contact discontinuities between gas clouds and surrounding hotter, more tenuous gas. Spatial correlations between radio mini-halos and X-ray cold fronts have been identified in several cool-core clusters (RX J1720.1+2638, MS 1455.0+2232, PSZ1 G139.61+24.20, A1068, A3444, etc.), with mini-halo emission confined within cold front boundaries [14–17]. This discovery has prompted discussion about whether mini-halos and X-ray cold fronts share a common origin [18].

Section 2 introduces large-scale magnetic fields in galaxy clusters, which play a crucial role in synchrotron radiation, and describes measurement methods. Section 3 discusses non-thermal electrons producing diffuse synchrotron radiation, focusing on particle acceleration mechanisms in the ICM environment. Section 4 examines observational results and origin models for radio halos, mini-halos, and relics. Section 5 provides a brief summary and outlook.

2 Large-Scale Magnetic Fields in Galaxy Clusters

Large-scale magnetic fields permeate the entire ICM and play important roles in accelerating high-energy particles. The origin of these fields remains uncertain. One possibility is that weak seed fields are significantly amplified through magnetohydrodynamic dynamo effects and adiabatic compression of magnetic field lines during hierarchical structure formation and cluster mergers [25]. Seed fields could be primordial—magnetic field fluctuations from the early Universe [26]—or diffuse into the ICM from early stars and protogalaxies [27].

Precise magnetic field measurements in galaxy clusters are extremely challenging. Equipartition estimation provides one method, assuming cosmic ray energy density $\epsilon_{\rm CR}$ equals magnetic field energy density $\epsilon_B$ in the radio-emitting region [28]. This method requires knowledge of the proton-to-electron number density ratio, which cannot be constrained observationally and depends on cosmic ray production mechanisms, introducing substantial uncertainties. This approach yields magnetic field strengths of $0.01 \sim 1$ nT [9].

Beyond radio synchrotron emission, other methods provide crucial clues about ICM cosmic ray electrons. Inverse Compton scattering of CMB photons by cosmic ray electrons produces hard X-ray emission with power-law spectra superimposed on the ICM thermal bremsstrahlung continuum. Comparing inverse Compton X-ray flux with radio synchrotron flux enables magnetic field strength calculations [29, 30]. This method assumes both radio and hard X-ray emissions originate from the same population of high-energy electrons. Since CMB photon density is known, this approach works for regions with both X-ray and radio emission. However, observations typically only yield upper limits on inverse Compton X-ray flux, currently allowing only lower limits on ICM magnetic fields.

A more reliable and direct method analyzes Faraday rotation of background radio galaxies, known as Faraday rotation synthesis. When polarized radiation from background sources passes through the ICM, its polarization angle $\chi$ undergoes Faraday rotation proportional to wavelength squared: $\Delta\chi = {\rm RM} \cdot \lambda^2$, where the rotation measure ${\rm RM} = \int n_{e,{\rm th}} B_\parallel dl$ ($B_\parallel$ being the line-of-sight magnetic field component). RM can be determined by comparing polarization angle variations across multiple radio bands [31]. In cluster studies, if no foreground media exist between observer and background source, RM depends solely on ICM properties [31]. Combined with X-ray observations providing thermal electron density distributions and modeling of magnetic field structures, RM measurements constrain cluster magnetic field properties.

Faraday rotation studies have greatly expanded understanding of cluster magnetic fields. With limited numbers of background polarized sources, current results are mainly statistical. It is now established that magnetic field strength generally decreases with distance from cluster centers, though fluctuations occur at certain scales. Statistically, central magnetic field strengths are $\sim 1$ nT in cool-core clusters and $\sim 0.1$ nT in merging clusters [9]. Limited background sources hinder understanding of large-scale magnetic field structures (coherence scales, power spectra, etc.).

Recent cosmological magnetohydrodynamics (MHD) simulations have advanced cluster magnetic field studies [32–34]. Weak primordial seed fields ($\lesssim 10^{-14}$ T) are amplified through small-scale dynamo effects and compression during structure formation. Simulations show amplified field strength distributions correlate closely with matter density, with field strengths in cluster halos exceeding theoretical values from simple magnetic freezing (flux conservation) by several orders of magnitude. Simulated magnetic field strengths decrease with cluster-centric radius, consistent with observations, though field power spectra are more complex than simple power-law forms commonly assumed in Faraday rotation synthesis [35]. Recent studies have applied power spectra from cosmological MHD simulations to interpret Faraday rotation measurements of ICM magnetic fields [36].

Future high-sensitivity low-frequency radio facilities like upgraded LOFAR and the Square Kilometer Array (SKA) will provide more (background) polarized sources, enabling detailed statistical studies and even deep measurements of individual cluster magnetic fields.

3 Non-Thermal Electrons and Particle Acceleration Mechanisms

3.1 Radiative Power and Characteristic Lifetime

Synchrotron radiation, produced by relativistic electrons spiraling in magnetic fields, is the common emission mechanism for three types of astrophysical radio sources: supernova remnants, large-scale jets, and ICM diffuse sources. The radiative power of a single electron depends on its energy (Lorentz factor) and magnetic field strength; higher magnetic fields require lower electron energies to produce radiation at a given frequency. Astronomical observations correspond to collective effects of numerous electrons. Given an isotropic, uniform non-thermal electron distribution $n_e(\gamma)$ and uniform magnetic field $B$, the synchrotron emissivity $J(\nu)$ can be calculated [19]:

$$
J(\nu) = \int_{\theta_{\min}}^{\theta_{\max}} \int_{\gamma_{\min}}^{\gamma_{\max}} n_e(\gamma) F(\nu/\nu_c) \sin^2\theta \, d\theta \, d\gamma,
$$

where $c$ is the speed of light, $e$ is electron charge, $m_e$ is electron rest mass, $\theta$ is the pitch angle between electrons and magnetic field $B$, $\nu_c = (3/2)\gamma^2 \nu_L \sin\theta$ ($\nu_L = eB/(2\pi m_e c)$) is the electron critical frequency, and $F$ is the synchrotron kernel function $F(x) = x \int_x^\infty K_{5/3}(y) dy$ ($K_{5/3}$ is the modified Bessel function of order 5/3). For cluster diffuse emission, considering a power-law electron energy distribution ($n_e(\gamma) \propto \gamma^{-\delta}$, where $\delta$ is the spectral index) and optically thin radio sources, the observed spectrum follows a power law $F_\nu \propto \nu^{-\alpha}$, with spectral index $\alpha$ related to particle spectral index $\delta$ by $\alpha = (\delta - 1)/2$ [19, 20]. Cluster diffuse radio spectra are typically steep, with $\alpha \gtrsim 1$, implying relatively few high-energy electrons ($\delta \gtrsim 3$).

In the ICM, non-thermal electrons lose energy through multiple mechanisms, primarily inverse Compton scattering of CMB photons. Through collisions with electrons, CMB photons gain energy becoming hard X-ray or gamma-ray photons while electrons lose energy. Both synchrotron and inverse Compton energy loss rates scale with the square of the Lorentz factor [21]; higher-energy electrons lose energy faster, causing the initial power-law spectrum to steepen progressively from high to low energies over time. Considering only these radiative loss mechanisms without acceleration, the characteristic lifetime (radiative cooling timescale) $t_{\rm age}$ of non-thermal electrons is [9]:

$$
t_{\rm age} \approx 3.2 \times 10^{10} \frac{B^{-3/2}}{\nu^{1/2}} \frac{1}{(1+z)^4} \left[ \left( \frac{B}{B_{\rm CMB}} \right)^2 + 1 \right]^{-1} \text{ yr},
$$

where $t_{\rm age}$ is in years, magnetic field strength $B$ is in units of $10^{-10}$ T, $B_{\rm CMB} \approx 3.25(1+z)^2$ is the equivalent magnetic field strength of the CMB, $z$ is source redshift, and observing frequency $\nu$ is in MHz. Higher frequencies (corresponding to higher electron energies $\gamma$) yield shorter lifetimes, meaning higher-frequency radiation diminishes more rapidly, causing observed diffuse radio spectra to often steepen at high frequencies. For galaxy clusters, typical $t_{\rm age} \lesssim 10^8$ yr. Based on Bohm diffusion, high-energy non-thermal electrons ($\gamma > 10^3$) have typical diffusion scales of $\sim 10$ pc [22], far smaller than diffuse source sizes. Even assuming ordered magnetic fields enabling more efficient diffusion than the Bohm case, the upper limit on diffusion scales remains far below 1 Mpc. Theoretical results indicate that non-thermal electrons producing diffuse radio emission cannot originate from small-scale regions (e.g., radio galaxies or AGN) but must be generated or (re)accelerated in situ, constraining possible particle acceleration mechanisms.

3.2 Particle Acceleration Mechanisms

Producing relativistic non-thermal electrons across $100$ kpc $\sim 1$ Mpc scales requires acceleration mechanisms connected to cluster-wide dynamical processes. This section briefly introduces several possible ICM particle acceleration mechanisms.

(1) First-order Fermi acceleration, also called diffusive shock acceleration (DSA), describes particle acceleration at magnetohydrodynamic shocks. Scattering centers (turbulence, plasma waves, or non-uniform magnetic/density structures) exist both upstream and downstream, enabling charged particles to cross the shock front repeatedly. Each crossing results in head-on collisions with the shock, gaining energy proportional to crossing次数. Multiple crossings of an ideal parallel shock by thermal equilibrium particles produce power-law non-thermal particle distributions, with spectral index $\delta$ related to shock Mach number $\mathcal{M}_s$ by $\delta = 2(\mathcal{M}_s^2 + 1)/(\mathcal{M}_s^2 - 1)$ [23].

(2) Second-order Fermi acceleration, or Fermi stochastic acceleration, describes random acceleration of particles scattering in magnetized turbulent fluids. Head-on collisions gain energy while trailing collisions lose energy. Random motions slightly favor head-on collisions, resulting in net acceleration, though the energy conversion efficiency is low due to stochasticity.

(3) Secondary models. First-order Fermi acceleration, AGN activity, and galactic outflows can produce high-energy non-thermal protons in the ICM. Since non-thermal protons have radiative lifetimes far exceeding those of electrons (approaching Hubble timescales) and most cannot escape clusters, substantial non-thermal proton populations can accumulate over a cluster's lifetime [24]. In hadronic models, high-energy relativistic non-thermal protons colliding with thermal ions produce secondary high-energy non-thermal electrons. Overall, in secondary models, non-thermal electrons are generated as secondary particles (decay products).

4 Diffuse Radio Sources: Observations and Origin Models

4.1 Radio Halos

4.1.1 Primary Observational Properties

Radio halos are diffuse extended sources with regular, smooth morphologies and brightness distributions roughly matching the ICM distribution. Larger, more dynamically disturbed clusters have higher probabilities of hosting giant radio halos [37]. Nearly 100 radio halos have been confirmed [FIGURE:1] shows a typical cluster radio halo; panel (a) displays the observed structure before subtracting discrete foreground sources, while panel (b) shows the halo emission after subtraction. Radio halos typically span $1 \sim 2$ Mpc with 1.4 GHz powers of $10^{23} \sim 10^{26}$ W·Hz$^{-1}$.

Spectral properties provide crucial clues to halo origins. Halo spectra are steep, with typical spectral indices $\alpha \sim 1.1 \sim 1.7$. Giovannini et al. [39] found a correlation between halo spectral index and ICM temperature: hotter clusters tend to host halos with flatter spectra. Halos with $\alpha > 1.5$ are called ultra-steep spectrum radio halos (USSRHs). Such steep spectra suggest high-frequency cutoffs; measuring near the cutoff yields artificially steep indices. Only the brightest halos are expected to have cutoff frequencies above $\sim 1$ GHz, making low-frequency, high-sensitivity observations essential for discovering more halos and revealing detailed spectral features.

Radio halo power correlates with host cluster X-ray luminosity and mass. In 2013, Cassano et al. [40] analyzed 25 radio halos (including 6 USSRHs), finding $P_{1.4} \propto L_{X,500}^{2.11 \pm 0.20}$ and $P_{1.4} \propto M_{500}^{3.70 \pm 0.56}$ at 1.4 GHz, where $L_{X,500}$ and $M_{500}$ are X-ray luminosity and total mass within $R_{500}$. USSRHs significantly affect these correlations, steepening them when included. After eight more years of observations, Duchesne et al. [41] compiled 86 halos in 2021, deriving $P_{1.4} \propto M_{500}^{3.21 \pm 0.39}$ and $P_{0.15} \propto M_{500}^{3.15 \pm 0.41}$ at 1.4 GHz and 150 MHz, respectively, without distinguishing normal halos from USSRHs. Future larger, more complete samples will help reveal whether normal halos and USSRHs differ in these correlations.

Beyond these correlations, Cassano et al. [40] studied a sample of 54 clusters at similar redshifts (including the 25 halo hosts) and discovered a bimodality: X-ray bright clusters ($L_{500} > 5 \times 10^{37}$ J·s$^{-1}$) separate into two branches—one with radio halos following the $P_{1.4}-L_{500}$ relation, and one without halos where diffuse radio upper limits fall far below the relation (so-called "radio-quiet" clusters). This bimodality persists even after subtracting central X-ray emission from cool cores and correlates with dynamical state: merging clusters preferentially host halos, while radio-quiet clusters are statistically more relaxed [37]. In 2021, Cuciti et al. [42, 43] selected 75 massive clusters ($M_{500} \gtrsim 6 \times 10^{14} M_\odot$) from the Planck SZ catalog at redshifts $0.08 \sim 0.33$, discovering 28 halos ($\sim 37\%$) and 5 candidates through radio observations. Combining X-ray data providing dynamical information (centroid shift $w$, power ratio $P_3/P_0$, concentration parameter $c$ [37]), they found over 90% of halos occur in merging clusters. The $P_{1.4}-M_{500}$ relation shows large scatter significantly contributed by dynamical state: the deviation from the relation (above or below) correlates with disturbance degree (more disturbed or relaxed). Clusters without detected halos are typically relaxed, with upper limits well below the correlation. This confirms the bimodality and its connection to dynamical state.

4.1.2 Origins

Two historical models for radio halo origins are the secondary model and turbulent reacceleration model. The secondary model was outlined in Section 3.2, but abundant observational evidence now challenges it, most directly from gamma-ray upper limits for the Coma cluster [44]. The mainstream turbulent reacceleration model posits that major mergers generate powerful turbulence that reaccelerates seed electrons via second-order Fermi acceleration (Section 3.2). The strong correlation between halo luminosity and cluster dynamical state supports this model, suggesting the mechanism disrupting cool cores and disturbing clusters is the same as that producing large-scale synchrotron emission.

Under turbulent reacceleration, assuming isotropic seed electron distributions, non-thermal electron spectral evolution can be described by the Fokker-Planck equation [45]:

$$
\frac{\partial n_e(\gamma, t)}{\partial t} = \frac{\partial}{\partial \gamma} \left[ D_{\gamma\gamma}(\gamma, t) \frac{\partial n_e(\gamma, t)}{\partial \gamma} \right] - \frac{\partial}{\partial \gamma} \left[ \left( \frac{d\gamma}{dt} \right) n_e(\gamma, t) \right] + Q_e(\gamma, t) - \frac{n_e(\gamma, t)}{t_{\rm esc}(\gamma, t)},
$$

where $D_{\gamma\gamma}$ describes the diffusion (reacceleration) coefficient from turbulence-electron interactions, and $Q_e$ and $t_{\rm esc}$ terms describe electron injection and escape. The first and second $D_{\gamma\gamma}$ terms represent systematic (advective) and random (diffusive) energy gains from turbulence, while $|d\gamma/dt|$ includes energy loss rates from synchrotron and free-free emission, inverse Compton scattering of CMB photons, Coulomb collisions with ICM thermal electrons, and adiabatic expansion. $D_{\gamma\gamma}$ depends on specific turbulence-electron interaction mechanisms. For turbulent magnetized plasmas, various MHD waves exist (Alfvén, fast/slow magnetosonic). Electron cyclotron resonance with Alfvén waves provides one acceleration mechanism, while transit-time damping (TTD)—where particle transit time equals wave period, causing resonant wave damping and particle energy gain—is another mechanism arising from fast magnetosonic waves [45, 46].

An unresolved question in turbulent reacceleration models concerns seed electron origins. Possible sources include galactic outflows and AGN activity, electrons accelerated in previous mergers or accretion shocks, or secondary electrons from proton-proton interactions. Their relative contributions remain unclear. Vazza et al. [47] used passive tracer particles in MHD simulations to study evolution of non-thermal electrons injected by radio galaxies into the ICM, finding these electrons can effectively fill the ICM over hundred-million-year timescales, providing a stable fossil electron reservoir.

4.2 Radio Mini-Halos

4.2.1 Primary Observational Properties

Radio mini-halos typically span $\sim 100 \sim 500$ kpc, primarily existing in relaxed cool-core clusters with emission often surrounding the brightest cluster galaxy (BCG). Their 1.4 GHz powers range from $10^{23} \sim 10^{25}$ W·Hz$^{-1}$. Only 25 mini-halos have been confirmed [FIGURE:2] shows a typical mini-halo. The limited number partly reflects observational challenges in distinguishing mini-halo emission from BCG radio emission, requiring high spatial resolution and dynamic range in radio observations plus detailed X-ray cavity studies.

Compared to radio halos, mini-halos generally have higher synchrotron emissivity. Though often non-spherical, both halo and mini-halo radial surface brightness profiles can be fitted by $I(r) = I_0 e^{-r/r_e}$. Murgia et al. [49] showed mini-halos have smaller scale radii $r_e$ than halos, higher central surface brightness $I_0$, and a larger $I_0$ value range. Mini-halo spectral properties resemble those of halos, being steep, with some studies observing high-frequency steepening [9].

Limited by small sample sizes, statistical studies of mini-halo properties remain preliminary. Unlike halos, no clear correlation exists between mini-halo radio power and cluster mass [15]. However, studies find positive correlations between mini-halo power and cluster X-ray luminosity, with slopes similar to halo correlations [48, 50, 51]. In 2020, Richard-Laferrière et al. [52] analyzed 28 mini-halos and 5 candidates, exploring connections with central AGN feedback. After decomposing BCG radio spectral energy distributions into flat-spectrum nuclear components (related to ongoing AGN accretion) and steep-spectrum components (from past AGN jet activity), they found mini-halo power correlates strongly with both, especially the steep-spectrum component, with a steeper correlation slope. X-ray cavities, thought to be produced by AGN jets, show cavity power representing work plus internal energy divided by buoyancy timescale. Mini-halo power correlates strongly with central X-ray cavity power, suggesting connections between BCG AGN feedback and mini-halos.

4.2.2 Origins

Despite their smaller size, mini-halos still require in situ acceleration due to short synchrotron electron lifetimes. Thus, mini-halo emission does not directly originate from central AGN, unlike radio lobes overlapping X-ray cavities. The mainstream view explains mini-halos through turbulent reacceleration similar to halos, but involving much smaller-scale turbulence induced by gas sloshing in relaxed cluster cores. This sloshing may be triggered by minor mergers with large impact parameters that disturb the core without disrupting the cluster's relaxed state. Seed electrons likely originate from the central AGN and are distributed over larger scales by gas sloshing. The observed spatial correlation between mini-halos and X-ray cold fronts supports this model. Numerical simulations provide supporting evidence: ZuHone et al. [18] used FLASH 3 to simulate a major merger between a $10^{15} M_\odot$ cluster and a $2 \times 10^{14} M_\odot$ subcluster (initial separation 3 Mpc, impact parameter 500 kpc). The merger induced gas sloshing in the main cluster core, generating significant turbulence and magnetic field amplification within sloshing cold fronts, while negligible turbulence occurred outside. Using passive tracer particles, they tested evolution of different seed electron distributions under gas sloshing, finding electrons could diffuse throughout the cold-front-confined core and be reaccelerated by MHD turbulence to observed mini-halo levels.

Despite differences, radio halos and mini-halos may be physically connected, possibly transitioning between states or coexisting. For example, during major mergers with large impact parameters, early phases may only trigger core gas sloshing producing mini-halos; as mergers progress, fully developed turbulence may replace mini-halos with halos. Alternatively, during strong minor mergers, both mini-halos within cold fronts and weaker peripheral halo-like emission may coexist. Special cases like the cool-core cluster CL1821+643 hosting a radio halo [53] and the non-major-merger cluster Abell 2142 showing both central mini-halo and peripheral weak halo components with different surface brightness and spectral indices [54] demonstrate close connections between these sources, facilitated by the complexity of merger processes.

4.3 Radio Relics

4.3.1 Primary Observational Properties

Radio relics predominantly appear in cluster peripheries with elongated shapes and largest linear sizes (LLS) of $0.5 \sim 2$ Mpc. Most show asymmetric transverse brightness profiles: sharp edges on the side farther from the cluster center, gradually fading toward the center. A subclass called double relics features two elongated, outwardly curved relics on opposite sides of the cluster center, often aligned with the merger axis. High signal-to-noise, high-resolution observations reveal ubiquitous filamentary substructures within relics. Radio relics have been observed in $\sim 60$ clusters [FIGURE:3] shows a cluster with radio relics.

Relic spectra are steep, typically with $\alpha \sim 1.0 \sim 1.5$. Spatial studies of spectral index reveal clear gradients across relic widths, with flattest indices on the side farthest from the cluster center, steepening toward the center. Figure 3b) demonstrates this clearly for the northern relic in the Sausage cluster.

Radio relics are among the most highly polarized extragalactic sources, with very elongated relics showing the highest polarization. At frequencies $\gtrsim 1$ GHz, polarization exceeds $\sim 20\%$, while lower frequencies show stronger depolarization. For large relics, intrinsic polarization angle distributions are remarkably ordered, with magnetic field vectors aligned along relic surfaces (Figure 3c).

Statistical studies of relic properties aid understanding of their origins. Feretti et al. [57] found correlations between relic power and cluster X-ray luminosity: $P_{1.4} \propto L_X^{1.2}$. Based on 17 double relics, Duchesne et al. [58] derived $P_{1.4} \propto M_{500}^{2.55 \pm 0.55}$ for total double relic power versus cluster mass. de Gasperin et al. [59] found correlations between individual relic LLS and projected distance from cluster center (X-ray brightness peak) for 15 double relics, with larger relics located farther out. Studying double relics reduces projection effects on distance measurements.

4.3.2 Origins

Based on relic shapes, positions, spectra, and polarization properties, their origin is attributed to diffusive shock acceleration (DSA) of electrons by merger-induced shocks in cluster peripheries [60]. In ideal head-on binary cluster mergers, initial approach and compression create equatorial shocks perpendicular to the merger axis. After cluster cores pass through each other, two merger shocks propagate outward along the axis ahead of the cores. Eventually, mutual gravity decelerates and merges the clusters while shocks continue outward propagation. Merger shocks accelerate electrons in situ, producing relics that trace shock morphologies. Equatorial shocks propagate into low-density regions surrounding merging clusters, unlikely to produce observable diffuse radio emission; reality is more complex.

This formation theory predicts shocks should coexist spatially with relics, identifiable in X-ray observations through sharp surface brightness discontinuities and significant temperature/density jumps. Over 20 X-ray shocks coincident with relic positions have been detected [9]. Furthermore, Rankine-Hugoniot jump conditions yield shock strength (X-ray Mach number $\mathcal{M}_X$) from X-ray data, while DSA theory provides shock strength (radio Mach number $\mathcal{M}_r$) from relic spectral indices. Comparing these tests whether relic electron acceleration follows DSA theory. However, observations typically yield $\mathcal{M}_r > \mathcal{M}_X$ ($\mathcal{M}_X \approx 1.5 \sim 2.5$, $\mathcal{M}_r \approx 2 \sim 5$) [61, 62]. If this discrepancy is real, standard DSA theory alone cannot fully explain relic formation. Proposed solutions include: (1) shock acceleration of not only ICM thermal electrons but also fossil non-thermal electrons from radio galaxies [63]; (2) multiple shock models where relic electrons experience repeated shock acceleration over time [64]. Inchingolo et al. [64] used MHD simulations of a massive merging cluster with two relics (A and B), employing tracer particles to track non-thermal electron trajectories from $z=1$ to $z=0$ and identify shocks using temperature jump conditions. While most electrons in relic A experienced only the final shock, $\sim 70\%$ in relic B underwent multiple shocks. Relic B's radio emission was significantly stronger with better spectral agreement with observations. This model reduces reliance on fossil electrons from radio galaxies. Zimbardo and Perri [65, 66] extended classical DSA to superdiffusive shock acceleration (SSA) to explain $\mathcal{M}_r > \mathcal{M}_X$. They assumed superdiffusive rather than normal transport for radiating electrons, caused by non-Gaussian random walks of high-energy particles leading to nonlinear mean squared displacement growth $\langle \Delta x^2(t) \rangle \propto t^\beta$ with $1 < \beta < 2$ (recovering normal diffusion as $\beta \to 1$). In the SSA framework, the relation between post-shock non-thermal particle spectral index and shock Mach number becomes:

$$
\delta = \frac{2(\mathcal{M}_s^2 + 3)}{(2-\beta)(3-\beta)\mathcal{M}_s^2} + 1.
$$

Physical interpretations for superdiffusive transport include Richardson diffusion and mechanisms from their solar wind studies, though further numerical investigation is needed.

After detaching from driving subclusters, shocks propagating in uniform media would weaken with distance. However, moving into cluster peripheries, shocks travel along steep density gradients into low-density regions, helping maintain shock strength over large distances—more pronounced for steeper gradients [67]. Typical ICM density profiles $\rho_{\rm gas}(r)$ are relatively flat in central regions ($\sim r^{-1}$) and steepen outward ($\sim r^{-2.5}$ near $R_{500}$). Thus, shock detachment location significantly affects subsequent evolution; shocks separating closer to cluster centers have shorter lifetimes, perhaps explaining why relics are found primarily in cluster peripheries.

In some clusters, radio relics connect or overlap with halos (e.g., Toothbrush cluster [68]), or halos cover regions between double relics (e.g., Sausage cluster [55]). In these cases, relic fluxes significantly exceed halo fluxes, possibly reflecting a transition from first-order Fermi acceleration at shocks to second-order Fermi acceleration by turbulence.

5 Summary and Outlook

Galaxy clusters are the largest self-gravitationally bound systems, serving as laboratories for galaxy formation and evolution while precisely tracing cosmic large-scale structure. Clusters exhibit diverse, unique features across X-ray, radio, and other wavebands. Diffuse radio emission reveals the existence of cosmic rays and magnetic fields in the ICM, indicating high-energy physical processes that dissipate gravitational potential energy into non-thermal components. These non-thermal components subsequently affect ICM (micro)physical properties and potentially influence cluster evolution [2]. Individual member galaxies can supply non-thermal electrons through star formation and supermassive black hole feedback, but the enormous sizes of diffuse sources ($\sim 50$ kpc $\sim 10$ Mpc) and short non-thermal electron propagation scales limited by cooling timescales require in situ (re)acceleration mechanisms on cluster scales. In the ICM, high-energy non-thermal electrons lose energy through multiple mechanisms beyond synchrotron radiation, potentially producing observable inverse Compton scattering of CMB photons in hard X-ray and gamma-ray bands. Furthermore, ICM magnetic fields influence both non-thermal electron acceleration and synchrotron emission, yet our understanding of magnetic field structure remains limited.

Recently, a new class of even more extended diffuse sources—radio megahalos—has been reported [69]. LOFAR survey data revealed four clusters where known radio halos are embedded within newly discovered megahalos $\sim 30$ times larger in volume but with much lower surface brightness. Distinguishable through differences in radial surface brightness profiles and spectral indices, this discovery indicates non-thermal electrons and magnetic fields extend far beyond previous results, requiring electron acceleration mechanisms effective on larger scales (awaiting verification). The diversity of cluster diffuse radio emission opens new avenues for low-frequency radio studies.

Different diffuse radio source types encode different production mechanisms from turbulence to shocks and connect closely to cluster merger histories and dynamical processes. Their study advances understanding of ICM structure and properties, particularly the important but poorly constrained magnetic field. Given the steep spectra of cluster diffuse radio emission, low-frequency radio bands are ideal for deep studies. Upcoming facilities like LOFAR 2.0, with sub-arcsecond resolution and extremely low frequency windows, will powerfully expand studies of high-redshift clusters. SKA's low-frequency array will dramatically increase the number of diffuse cluster radio sources, extending samples to medium-high redshifts, facilitating cluster formation and evolution studies while significantly improving observational precision for individual sources.

A key SKA science goal is detecting neutral hydrogen signals from the cosmic dawn and reionization epochs. In this endeavor, foreground removal—particularly of diffuse sources—represents the most important technical challenge. Cluster large-scale diffuse radio emission constitutes a major extragalactic diffuse foreground component. Therefore, theoretical modeling of cluster diffuse radio emission will directly support foreground subtraction efforts. Building upon previous halo modeling [32, 70], we have recently performed numerical modeling of radio relics. Sampling cluster populations from IllustrisTNG cosmological simulations [71] and combining particle acceleration mechanisms including DSA and post-shock turbulent reacceleration [72] with magnetic fields from the simulations, we generated simulated relic sky maps shown in Figure 4 [FIGURE:4].

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Diffuse Radio Emission in Galaxy Clusters

WU Wenyu¹,², XIE Fuguo¹
(1. Astrophysics Division, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China)

Abstract: Galaxy clusters are the largest self-gravitationally bound systems in the Universe. Their diffuse radio emission is synchrotron radiation generated by non-thermal electrons moving in magnetic fields, and can be mainly classified into radio halos, radio mini-halos, and radio relics. Radio halos occur mostly in merging clusters, and their origin is thought to be described by the merger-induced turbulent reacceleration model. Radio mini-halos are of small number and mainly found in relaxed cool-core clusters. Their origin can also be described by the turbulent reacceleration model, except that the turbulence is induced by gas sloshing in the cluster's core. Radio relics are polarized sources mostly found in clusters' periphery. They trace merger-induced shock waves. Finally, a larger and deeper survey of diffuse cluster radio sources in the era of LOFAR 2.0 and SKA is discussed.

Key words: galaxy cluster; radio halo; radio mini-halo; radio relic; synchrotron emission