Zooming-in on cluster radio relics -- I. How density fluctuations explain the Mach number discrepancy, microgauss magnetic fields, and spectral index variations

TL;DR

The paper investigates three key puzzles of cluster radio relics—radio–X-ray Mach-number discrepancies, μG-scale magnetic fields, and spectral-index variations—by pairing cosmological simulations with high-resolution shock-tube experiments. Upstream density fluctuations produce a distribution of Mach numbers at the shock front and drive a Rayleigh–Taylor instability downstream, which amplifies magnetic fields beyond simple compression and disrupts laminar cooling assumptions. Using CREST to model CR-electron spectra and CRAYON+ for synchrotron emission, the study shows that the combination of Mach-number dispersion and downstream turbulence naturally flattens spectra and biases radio-inferred Mach numbers higher than X-ray values, while RTI-driven magnetic amplification yields μG fields and emission features consistent with observations. These results offer a coherent framework for interpreting relic observations, highlight projection biases in magnetic-field inferences, and set the stage for refined diagnostics in future relic studies.

Abstract

It is generally accepted that radio relics are the result of synchrotron emission from shock-accelerated electrons. Current models, however, are still unable to explain several aspects of their formation. In this paper, we focus on three outstanding problems: i) Mach number estimates derived from radio data do not agree with those derived from X-ray data, ii) cooling length arguments imply a magnetic field that is at least an order of magnitude larger than the surrounding intracluster medium (ICM), and iii) spectral index variations do not agree with standard cooling models. To solve these problems, we first identify typical shock conditions in cosmological simulations, using the results to inform significantly higher resolution shock-tube simulations. We apply the cosmic ray electron spectra code CREST and the emission code CRAYON+ to these, thereby generating mock observables ab-initio. We identify that upon running into an accretion shock, merger shocks generate a shock-compressed sheet, which, in turn, runs into upstream density fluctuations in pressure equilibrium. This mechanism directly gives rise to solutions to the three problems: it creates a distribution of Mach numbers at the shock-front, which flattens cosmic ray electron spectra, thereby biasing radio-derived Mach number estimates to higher values. We show that this effect is particularly strong in weaker shocks. Secondly, the density sheet becomes Rayleigh-Taylor unstable at the contact discontinuity, causing turbulence and additional compression downstream. This amplifies the magnetic field from ICM-like conditions up to microgauss levels. We show that synchrotron-based measurements are strongly biased by the tail of the distribution here too. Finally, the same instability also breaks the common assumption that matter is advected at the post-shock velocity downstream, thus invalidating laminar-flow based cooling models.

Figures (22)

• Figure 1: Top: Mean pressure along the $x$ axis at $t=0$ Myr (dashed) and $t=100$ Myr (solid) in our Mach 3 'Flat' simulation (see Table \ref{['tab:sims']}). Dotted lines indicate the edges of various regions in the shock tube (see Sect. \ref{['subsec:shock-tube-setup-resolution']}). We greyed out panels that we do not analyse. Bottom: As above, but lines show the mean density. The initial pressure discontinuity causes a shock to propagate into region III. As a result of shock compression, a dense sheet moves rightwards, bounded by the density jumps at the contact discontinuity and the shock front. This feature is critical to the mechanism we analyse in this paper.
• Figure 2: Cosmological simulation of a galaxy cluster undergoing a major merger at $z=0.14$. Four left-most panels, clockwise from top left: Projected shock-dissipated energy rate, gas pressure, gas density, and dissipation-weighted Mach number, respectively. Projections have a depth of $\pm 7.5$ Mpc from the cluster centre. Right-most panels: enlarged cut-outs showing slices through the midplane of the projection. The white, dashed box indicates the size of the upstream region in our idealised shock tube simulations. The merger drives a shock wave out to the cluster outskirts. The collision of this shock wave with an accretion shock leads to the production of a thin shell of compressed gas (see white arrow). A movie of this process can be found https://www.youtube.com/watch?v=jKhqhaCTCL4.
• Figure 3: Mach 3 shock is driven into a turbulent upstream density field. The initial pressure and density values for the shock tube correspond to those shown in Fig. \ref{['figure:cosmological']}. Panels: i) gas pressure, ii) gas density, iii) dissipated-energy-weighted Mach number, iv) the fraction of the gas speed in the $x$-component, v) the $x$-component of gas velocity, vi) gas speed divided by sound speed. Data are shown in slices except for iii), which is a thin projection of 35 kpc. All values are shown in the shock rest frame at $t=180$ Myr. Upstream density turbulence directly leads to shock corrugation, a distribution of Mach numbers at the shock front, and downstream velocity turbulence. An animated version of this figure can be found https://www.youtube.com/watch?v=ERBftXpMqgs.
• Figure 4: Schematic showing how upstream density turbulence causes velocity turbulence to be generated downstream. The corrugation of the shock front naturally leads to a misalignment of pressure and density gradients (shown here by red and green arrows, respectively). This results in a baroclinic term, which in turn induces vorticity, causing the contact discontinuity to become Rayleigh-Taylor unstable.
• Figure 5: Mach number distributions for all models, where each cell has been weighted by its normalised contribution to the shock surface. Lines indicate the median taken over all snapshots, whilst the shaded values indicate the interquartile range. The addition of magnetic turbulence (Flat) to a homogeneous density distribution (Flat-ConstB) broadens the distribution only very mildly. By contrast, the addition of upstream density fluctuations in the simulation (Turb and Turb-ConstB) turns an extremely narrow distribution into a much broader one. The width of the distribution is proportional to the peak Mach number.