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We present a systematic exploration of the loss of predictivity inEinstein-scalar-Gauss-Bonnet (ESGB) gravity. We first formulate a gaugecovariant method of characterizing the breakdown of the hyperbolicity of theequations of motion in the theory. With this formalism, we show that stronggeodesic focusing leads to the breakdown of hyperbolicity, and the latter isunrelated to the violation of the null convergence condition. We thennumerically study the hyperbolicity of the equations during gravitationalcollapse for two specific ESGB gravity theories: "shift symmetric Gauss-Bonnetgravity" and a version of the theory that admits "spontaneously scalarized"black holes. We devise a "phase space" model to describe the end states for agiven class of initial data. Using our phase space picture, we demonstrate thatthe two theories we consider remain predictive (hyperbolic) for a range of GBcouplings. The range of couplings, however, is small, and thus, the presence of"spontaneously scalarized" solutions requires fine-tuning of initial data. Ourresults, therefore, cast doubt as to whether scalarized black hole solutionscan be realistically realized in Nature even if ESGB gravity happened to be thecorrect gravitational description.

Where and why does Einstein-Scalar-Gauss-Bonnet theory break down?