Criticality in the collapse of spherically symmetric massless scalar fields in semi-classical loop quantum gravity

In a recent paper we showed that the collapse to a black hole in one-parameter families of initial data for massless, minimally coupled scalar fields in spherically symmetric semi-classical loop quantum gravity exhibited a universal mass scaling similar to the one in classical general relativity. In particular, no evidence of a mass gap appeared as had been suggested by previous studies. The lack of a mass gap indicated the possible existence of a self-similar critical solution as in general relativity. Here we provide further evidence for its existence. Using an adaptive mesh refinement code, we show that "echoes" arise as a result of the discrete self-similarity in space-time. We also show the existence of "wiggles" in the mass scaling relation, as in the classical theory. The results from the semi-classical theory agree well with those of classical general relativity unless one takes unrealistically large values for the polymerization parameter.