Inverting No-Hair Theorems: How Requiring General Relativity Solutions Restricts Scalar-Tensor Theories

Black hole solutions in general scalar-tensor theories are known to permit hair, i.e. non-trivial scalar profiles and/or metric solutions different from the ones of General Relativity (GR). Imposing that some such solutions—e.g. Schwarzschild or de Sitter solutions motivated in the context of black hole physics or cosmology—should exist, the space of scalar-tensor theories is strongly restricted. Here we investigate precisely what these restrictions are within general quadratic/cubic higher-order scalar-tensor theories for stealth solutions, whose metric is given by that in GR, supporting time-dependent scalar hair with a constant kinetic term. We derive, in a fully covariant approach, the conditions under which the Euler-Lagrange equations admit all (or a specific set of) exact GR solutions, as the first step toward our understanding of a wider class of theories that admit approximately stealth solutions. Focusing on static and spherically symmetric black hole spacetimes, we study the dynamics of linear odd-parity perturbations and discuss possible deviations from GR. Importantly, we find that requiring the existence of all stealth solutions prevents any deviations from GR in the odd-parity sector. In less restrictive scenarios, in particular for theories only requiring the existence of Schwarzschild(-de Sitter) black holes, we identify allowed deviations from GR, derive the stability conditions for the odd modes, and investigate the generic deviation of a non-trivial speed of gravitational waves. All calculations performed in this paper are reproducible via companion Mathematica notebooks[1].