Cosmological implications for hairy black holes via spontaneous symmetry breaking: Are Hairy Black Holes Primordial?

Abstract: We investigate whether hairy black holes generated through spontaneous symmetry breaking in Einstein-Scalar-Gauss-Bonnet (ESGB) theory, involving a complex scalar field with a global $U(1)$ symmetry, can be compatible with cosmological evolution. To this end, we introduce the ESGB theory with a scalar self-interaction that becomes relevant on cosmological scales while remaining negligible near the black hole. Owing to the time dependence of the GB term on cosmological scales, the scalar field dynamics in the evolving FLRW background differ qualitatively from those in the nearly static black hole background. In particular, for scalar-GB couplings compatible with hairy black hole formation, the effective potential supports a symmetry-broken vacuum throughout inflation. However, after inflation, a decelerated expansion changes the sign of the GB term, temporarily making the effective potential unbounded from below. As the GB contribution subsequently decreases, the scalar self-interaction eventually dominates and restores the symmetry. Within this schematic framework, we derive stringent constraints on the coupling strengths, the cutoff scale, and the black hole mass, which primarily arise for avoiding efficient tachyonic amplification of the scalar field perturbations during the unbounded phase. For cutoff scales compatible with both cosmological evolution and scalar hair formation, we find that only ultralight black holes with masses of the order of a few grams can develop scalar hair, identifying them as hairy primordial black holes.