We point out that there are only three polarizations for gravitational waves in f (R) gravity, and the polarization due to the massive scalar mode is a mix of the pure longitudinal and transverse breathing polarization. The classification of the six polarizations by the Newman-Penrose quantities is based on weak, plane and null gravitational waves, so it is not applicable to the massive mode.

In this work, we discuss the polarization contents of Einstein-aether theory and the generalized tensor-vector-scalar (TeVeS) theory, as both theories have a normalized timelike vector field. We derive the linearized equations of motion around the flat spacetime background using the gaugeinvariant variables to easily separate physical degrees of freedom. We find the plane wave solutions are then found, and identify the polarizations by examining the geodesic deviation equations.We find that there are five polarizations in Einstein-aether theory and six polarizations in the generalized TeVeS theory. In particular, the transverse breathing mode is mixed with the pure longitudinal mode. We also discuss the experimental tests of the extra polarizations in Einsteinaether theory using pulsar timing arrays combined with the gravitational-wave speed bound derived from the observations on GW 170817 and GRB 170817A. It turns out that it might be difficult to use pulsar timing arrays to distinguish different polarizations in Einstein-aether theory. The same speed bound also forces one of the propagating modes in the generalized TeVeS theory to travel much faster than the speed of light. Since the strong coupling problem does not exist in some parameter subspaces, the generalized TeVeS theory is excluded in these parameter subspaces. *

Gravitational waves are perturbations of the metric of space-time. Six polarizations are possible, although general relativity predicts that only two such polarizations, tensor plus and tensor cross are present for gravitational waves. We give the analytical formulas for the antenna response functions for the six polarizations which are valid for any equal-arm interferometric gravitationalwave detectors without optical cavities in the arms. The response function averaged over the source direction and polarization angle decreases at high frequencies which deteriorates the signal-to-noise ratio registered in the detector. At high frequencies, the averaged response functions for the tensor and breathing modes fall of as 1/f 2 , the averaged response function for the longitudinal mode falls off as 1/f and the averaged response function for the vector mode falls off as ln(f )/f 2 . *

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