Zooming in on cluster radio relics

I. How density fluctuations explain the Mach number discrepancy, microgauss magnetic fields, and spectral index variations

¹ Leibniz-Institut für Astrophysik Potsdam (AIP) An der Sternwarte 16 D-14482 Potsdam, Germany
² Institut für Physik und Astronomie, Universität Potsdam Karl-Liebknecht-Str. 24/25 14476 Potsdam, Germany
³ Max-Planck-Institut für Astrophysik Karl-Schwarzschild-Str. 1 85748 Garching, Germany
⁴ Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik Albert-Ueberle-Str. 2 69120 Heidelberg, Germany

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 14 November 2024
Accepted: 27 October 2025

Abstract

It is generally accepted that radio relics are the result of synchrotron emission from shock-accelerated electrons. However, current models 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. We used a hybrid approach to solve these problems; we first identified typical shock conditions in cosmological simulations and then used these to inform idealised shock-tube simulations, which can be run with a substantially higher resolution. We post-processed our simulations with the cosmic ray electron spectra code CREST and the emission code CRAYON+, which allowed us to generate mock observables ab-initio. We observed that, upon running into an accretion shock, merger shocks generate a dense, shock-compressed sheet, which in turn runs into upstream density fluctuations. This mechanism directly gives rise to solutions to the three aforementioned problems: density fluctuations lead to a distribution of Mach numbers forming at the shock front. This flattens cosmic ray electron spectra, thereby biasing radio-derived Mach number estimates to higher values. We show that such estimates are particularly inaccurate in weaker shocks (ℳ ≲ 2). Secondly, the density sheet becomes Rayleigh-Taylor unstable at the contact discontinuity, which causes turbulence and additional compression downstream. This amplifies the magnetic field from ICM-like conditions up to μG 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.