Approximately stealth black hole in higher-order scalar-tensor theories

We investigate a generic quadratic higher-order scalar-tensor theory with a scordatura term, which is expected to provide a consistent perturbative description of stealth solutions with a timelike scalar field profile. In the DHOST subclass, exactly stealth solutions are known to yield perturbations infinitely strongly coupled and thus cannot be trusted. Beyond DHOST theories with the scordatura term, such as in ghost condensation and U-DHOST, we show that stealth configurations cannot be realized as exact solutions but those theories instead admit approximately stealth solutions where the deviation from the exactly stealth configuration is controlled by the mass scale $M$ of derivative expansion. The approximately stealth solution is time-dependent, which can be interpreted as the black hole mass growth due to the accretion of the scalar field. From observed astrophysical black holes, we put an upper bound on $M$ as $\hat{c}_{\rm D1}^{1/2} M\lesssim 2\times 10^{11}$ GeV, where $\hat{c}_{\rm D1}$ is a dimensionless parameter of order unity that characterizes the scordatura term. As far as $M$ is sufficiently below the upper bound, the accretion is slow and the approximately stealth solutions can be considered as stealth at astrophysical scales for all practical purposes while perturbations are weakly coupled all the way up to the cutoff $M$ and the apparent ghost is as heavy as or heavier than $M$.