Large and ultracompact Gauss-Bonnet black holes with a self-interacting scalar field

We consider the Einstein-scalar-Gauss-Bonnet theory, and study the case where a negative cosmological constant is replaced by a more realistic, negative scalar-field potential. We study different forms of the coupling function between the scalar field and the Gauss-Bonnet term as well as of the scalar potential. In all cases, we obtain asymptotically flat, regular black-hole solutions with a nontrivial scalar field which naturally dies out at large distances. For a quadratic negative potential, two distinct subgroups of solutions emerge: the first comprises light black holes with a large horizon radius, and the second includes massive, ultracompact black holes. The most ultracompact solutions, having approximately $1/20$ of the horizon radius of the Schwarzschild solution with the same mass, emerge for the exponential and linear coupling functions. For other polynomial forms of the scalar potential, the subgroup of ultracompact solutions disappears, and the black holes obtained may have a horizon radius larger or smaller than the Schwarzschild solution depending on the particular value of their mass.

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