I'm not sure why you're bringing up synchrotron radiation in particular. Although it's important for some sources, it's not driving the majority of what is being seen here. Comptonization is important for active black holes. The radiation from the milky way has strong emission lines - it's not a synchrotron spectrum. Mostly thermal emission is emitted from clusters of galaxies. There are a lot of different emission processes to consider in astrophysics.