Gravitational Wave Polarizations in $f(R)$ Gravity and Scalar-Tensor
  Theory
  The detection of gravitational waves by the Laser Interferometer
Gravitational-Wave Observatory opens a new era to use gravitational waves to
test alternative theories of gravity. We investigate the polarizations of
gravitational waves in $f(R)$ gravity and Horndeski theory, both containing
scalar modes. These theories predict that in addition to the familiar $+$ and
$\times$ polarizations, there are transverse breathing and longitudinal
polarizations excited by the massive scalar mode and the new polarization is a
single mixed state. It would be very difficult to detect the longitudinal
polarization by interferometers, while pulsar timing array may be the better
tool to detect the longitudinal polarization.
