Higher-order Skyrme hair of black holes

TL;DR

This paper explores higher-order derivative models in Einstein-Skyrme theory, identifying new stable black hole solutions with scalar hair supported by 8 and 12 derivative terms, including stability analysis and multiple solution branches.

Contribution

It introduces two novel higher-derivative models supporting stable black hole hair without the Skyrme term, expanding understanding of stability conditions in Einstein-Skyrme-like theories.

Findings

Two new models support stable black hole hair with 8 and 12 derivatives.

Lower branches become stable in the 12th-order model when Skyrme term is absent.

Multiple solutions exist for certain parameters, with the lowest branch being stable.

Abstract

Higher-order derivative terms are considered as replacement for the Skyrme term in an Einstein-Skyrme-like model in order to pinpoint which properties are necessary for a black hole to possess stable static scalar hair. We find two new models able to support stable black hole hair in the limit of the Skyrme term being turned off. They contain 8 and 12 derivatives, respectively, and are roughly the Skyrme-term squared and the so-called BPS-Skyrme-term squared. In the twelfth-order model we find that the lower branches, which are normally unstable, become stable in the limit where the Skyrme term is turned off. We check this claim with a linear stability analysis. Finally, we find for a certain range of the gravitational coupling and horizon radius, that the twelfth-order model contains 4 solutions as opposed to 2. More surprisingly, the lowest part of the would-be unstable branch turns out to be the stable one of the 4 solutions.