Novel black-hole solutions in Einstein-scalar-Gauss-Bonnet theories with a cosmological constant

Abstract

We consider the Einstein-scalar-Gauss-Bonnet theory in the presence of a cosmological constant Λ, 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 Λ 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 Λ determines the form of the asymptotic gravitational background leading either to a Schwarzschild-Anti-de Sitter-type background (Λ < 0) or a regular cosmological horizon (Λ > 0), with a non-trivial scalar field in both cases. We demonstrate that families of novel black-hole solutions with scalar hair emerge for Λ < 0, for every choice of the coupling function between the scalar field and the Gauss-Bonnet term, whereas for Λ > 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. 1

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

“…for all forms of the coupling function f (φ), with D being the scalar charge, as found in [39]. For Λ < 0 and V (φ) = 1, black-hole solutions with an asymptotically Anti-de Sitter behaviour are expected to emerge as in [60,160]. These solutions do not possess a scalar charge since the asymptotic behaviour of the scalar field is given by the expression…”

“…To this end, we assume that, as r → r h , the metric function e A(r) should vanish (and e B(r) should diverge) whereas the scalar field must remain finite. As was explicitly shown in previous constructions [39,160], this amounts to working in the limit A (r) → ∞ while keeping φ (r) and φ (r) finite as the black-hole horizon is approached. Working in these limits, Eq.…”

“…For V (φ) = 1, the expression of C may take the form of a second-order polynomial foṙ f 2 h , which then leads to two branches of black-hole solutions (or four, if negative values oḟ f h are also allowed) describing solutions having either a minimum or a maximum horizon radius [60,160]. In reality, only solutions with a minimum horizon radius are found while the second branch is plagued by instabilities.…”

“…where F = I E /β is the Helmholtz free-energy of the system given in terms of the Euclidean version of the action I E , and β = 1/(k B T ). However, as was discussed in [160,177,178], this formula needs to be appropriately modified when the spacetime asymptotic solution deviates from the asymptotically-flat limit. Equivalently, as was demonstrated in [178], one may employ an alternative method developed in [179,180] in which the entropy of the black hole is identified with the Noether charge on the horizon under diffeomorphisms.…”

“…In a previous work [160], we studied the EsGB theory in the presence of a negative cosmological constant Λ, and demonstrated that black-hole solutions with an asymptotically Anti-de Sitter behaviour and a scalar hair arise as easily as their asymptotically-flat counterparts. In the present work, we will address the case where this constant energy density is replaced by a non-trivial potential for the scalar field.…”