Stability of hairy black holes in shift-symmetric scalar-tensor theories via the effective field theory approach

Shift-symmetric Horndeski theories admit an interesting class of Schwarzschild-de Sitter black hole solutions exhibiting time-dependent scalar hair. The properties of these solutions may be studied via a bottom-up effective field theory (EFT) based on the back-ground symmetries. This is in part possible by making use of a convenient coordinate choice - Lemaitre-type coordinates - in which the profile of the Horndeski scalar field is linear in the relevant time coordinate. We construct this EFT, and use it to understand the stability of hairy black holes in shift-symmetric Horndeski theories, providing a set of constraints that the otherwise-free functions appearing in the Horndeski Lagrangian must satisfy in order to admit stable black hole solutions. The EFT is analyzed in the decoupling limit to under-stand potential sources of instability. We also perform a complete analysis of the EFT with odd-parity linear perturbations around general spherically symmetric space-time.

Automated summary

Shift-symmetric Horndeski theories exhibit a class of Schwarzschild-de Sitter black hole solutions with time-dependent scalar hair. Lemaitre-type coordinates are used to study the properties of these solutions, where the profile of the Horndeski scalar field is linear in the relevant time coordinate. An effective field theory (EFT) is constructed to understand the stability of hairy black holes in shift-symmetric Horndeski theories and provide constraints for stable black hole solutions. The EFT is analyzed in the decoupling limit to identify potential sources of instability and a complete analysis is performed for odd-parity linear perturbations in spherically symmetric spacetime.