Novel black-hole solutions in Einstein-scalar-Gauss-Bonnet theories with a cosmological constant
We consider the Einstein-scalar-Gauss-Bonnet theory in the presence of a cosmological constant $\Lambda$, either positive or negative, and look for novel, regular black-hole solutions with a non-trivial scalar hair. We first perform an analytic study in the near-horizon asymptotic regime, and demonstrate that a regular black-hole horizon with a non-trivial hair may be always formed, for either sign of $\Lambda$ and for arbitrary choices of the coupling function between the scalar field and the Gauss-Bonnet term. At the far-away regime, the sign of $\Lambda$ determines the form of the asymptotic gravitational background leading either to a Schwarzschild-Anti-de Sitter-type background ($\Lambda 0$), with a non-trivial scalar field in both cases. We demonstrate that families of novel black-hole solutions with scalar hair emerge for $\Lambda 0$, no such solutions may be found. In the former case, we perform a comprehensive study of the physical properties of the solutions found such as the temperature, entropy, horizon area and asymptotic behaviour of the scalar field.