Zooming in on radio relics -- II. How relic morphology probes density fluctuations at the edge of galaxy clusters

Gas properties in the outer intracluster medium (ICM) are not well-constrained, as traditional probes lose sensitivity at Mpc distances. We show that the morphology of radio relics effectively encodes the power spectrum of the surrounding density fluctuations, and that they hence represent a new observational window. To demonstrate this, we use cosmologically motivated shock-tube simulations in which we systematically vary the coherence length, amplitude, and power-law slope of the upstream density power spectra. We then post-process our simulations with the cosmic ray electron spectral solver, Crest, thereby producing a suite of mock radio relics. We find that the downstream morphology of our simulated relics is independently sensitive to each of the aforementioned parameters. Specifically, we show that observed 'double strand' features can be formed by curved shock fronts in projection, and that the scale of these features maps directly to the fluctuation coherence length. Increasing the fluctuation amplitude, meanwhile, progressively lengthens the downstream extent of the relic, thus explaining why relics are observed to be broader than the idealised expectation. It also broadens the Mach number distribution across the shock, which simultaneously increases the integrated radio flux density and produces patchier emission. Finally, steepening the power-law slope makes 'double strand' features more likely, and additionally increases both the number of radio filaments oriented parallel to the shock front and their spacing. At higher Mach numbers, steepening the power-law slope can further lead to the production of curved radio filaments, which trace large eddies. We apply our analysis to the Toothbrush and Sausage relics, and find evidence for a typical fluctuation coherence length of ~500 kpc, a non-uniform amplitude, and power spectra that are steeper than a Kolmogorov-like scaling.