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Gravitational Waves in Viable f(R) Models

We study gravitational waves in viable $f(R)$ theories under a non-zero background curvature. In general, an $f(R)$ theory contains an extra scalar degree of freedom corresponding to a massive scalar mode of gravitational wave. For viable $f(R)$ models, since there always exits a de-Sitter point where the background curvature in vacuum is non-zero, the mass squared of the scalar mode of gravitational wave is about the de-Sitter point curvature $R_{d}\sim10^{-66}eV^{2}$. We illustrate our results in two types of viable $f(R)$ models: the exponential gravity and Starobinsky models. In both cases, the mass will be in the order of $10^{-33}eV$ when it propagates in vacuum. However, in the presence of matter density in galaxy, the scalar mode can be heavy. Explicitly, in the exponential gravity model, the mass becomes almost infinity, implying the disappearance of the scalar mode of gravitational wave, while the Starobinsky model gives the lowest mass around $10^{-24}eV$, corresponding to the lowest frequency of $10^{-9}$ Hz, which may be detected by the current and future gravitational wave probes, such as LISA and ASTROD-GW.