Mergers of hairy black holes: Constraining topological couplings from entropy

Hairy black-holes are a unique prediction of certain theories that extend General Relativity (GR) with a scalar field. The presence of scalar hair is reflected non-trivially in the entropy of the black hole along with any topological coupling that may be present in the action. Demanding that a system of two merging black holes obeys the global second law of thermodynamics imposes a bound on this topological coupling coefficient. In this work we study how this bound is pushed from its GR value by the presence of scalar hair by considering estimates of binary black-hole merger parameters through inference studies of mock gravitational-wave (GW) events. We find that the scalar charge may produce a statistically significant deviation of the change in entropy over the GR prediction. However, we also find the latter is susceptible to biases arising out of GW inferences which ends up being two orders of magnitude larger, therefore overwhelming any change induced by the scalar hair.