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We study the quasi-normal modes (QNMs) of static, spherically symmetric blackholes in $f(R)$ theories. We show how these modes in theories with non-trivial$f(R)$ are fundamentally different from those in General Relativity. In thespecial case of $f(R) = \alpha R^2$ theories, it has been recently argued thatiso-spectrality between scalar and vector modes breaks down. Here, we show thatsuch a break down is quite general across all $f(R)$ theories, as long as theysatisfy $f''(0)/(1+f''(0)) \neq 0$, where a prime denotes derivative of thefunction with respect to its argument. We specifically discuss the origin ofthe breaking of isospectrality. We also show that along with this breaking theQNMs receive a correction that arises when $f''(0)/(1+f'(0)) \neq 0$ owing tothe inhomogeneous term that it introduces in the mode equation. We discuss howthese differences affect the "ringdown" phase of binary black hole mergers andthe possibility of constraining $f(R)$ models with gravitational-waveobservations. We also find that even though the iso-spectrality is broken in$f(R)$ theories, in general, nevertheless in the corresponding scalar-tensortheories in the Einstein frame it is unbroken.

Quasi-normal Modes of Static Spherically Symmetric Black Holes in $f(R)$ Theory