Stable massless scalar polarization of f(R) gravity

Polarization is a prominent feature of gravitational wave observations and can be used to distinguish between different modified gravity theories. Compared to General Relativity, f(R) gravity exhibits an additional polarization originating from a scalar field, which is a combination of the longitudinal and breathing modes. When the scalar mass of f(R) is zero, the mixed mode will reduce to a pure breathing mode with the disappearance of the longitudinal mode. However, this reducing seems to be disallowed because a positive scalar mass is often required to maintain the stability of the cosmological perturbation. In fact, the massless scalar case can provide a stable perturbation, but more detailed constraints need to be considered. For the completeness of the polarization analysis, we explore the possibility that there are stable massless scalar polarizations in viable dark energy f(R) models. We find that the existence of stable massless scalar polarization depends on the structure of f(R) model and can be used to distinguish different models in f(R) gravity.