We study black holes in a modified gravity scenario involving a scalar field quadratically coupled to the Gauss-Bonnet invariant. The scalar is assumed to be in a spontaneously broken phase at spatial infinity due to a bare Higgs-like potential. For a proper choice of sign, the nonminimal coupling to gravity leads to symmetry restoration near the black hole horizon, prompting the development of the scalar wall in its vicinity. The wall thickness depends on the bare mass of the scalar and can be much smaller than the Schwarzschild radius. In a weakly coupled regime, the quadratic coupling to the Gauss-Bonnet invariant effectively becomes linear, and no walls are formed. We find approximate analytical solutions for the scalar field in the test field regime, and obtain numerically static black hole solutions within this setup. We discuss cosmological implications of the model and show that it is fully consistent with the existence of an inflationary stage, unlike the spontaneous scalarization scenario assuming the opposite sign of the nonminimal coupling to gravity. Our model predicts the speed of gravitational waves to be extremely close to unity—in comfortable agreement with the observation of the GW170817 event and its electromagnetic counterpart.

• Received 15 July 2022
• Accepted 7 September 2022

DOI:https://doi.org/10.1103/PhysRevD.106.063524