Quasi-normal modes of black holes in Horndeski gravity

Abstract

We study the perturbations to General Relativistic black holes (i.e. those without scalar hair) in Horndeski scalar-tensor gravity. First, we derive the equations of odd and even parity perturbations of both the metric and scalar field in the case of a Schwarzschild black hole, and show that the gravitational waves emitted from such a system contain a mixture of quasi-normal mode frequencies from the usual General Relativistic spectrum and those from the new scalar field spectrum, with the new scalar spectrum characterised by just two free parameters. We then specialise to the sub-family of Horndeski theories in which gravitational waves propagate at the speed of light $c$ on cosmological backgrounds; the scalar quasi-normal mode spectrum of such theories is characterised by just a single parameter $\mu$ acting as an effective mass of the scalar field. Analytical expressions for the quasi-normal mode frequencies of the scalar spectrum in this sub-family of theories are provided for both static and slowly rotating black holes. In both regimes comparisons to quasi-normal modes calculated numerically show good agreement with those calculated analytically in this work.

10 pages, 4 figures. Submitted to PRD

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