Probing gravitational wave polarizations with Advanced LIGO, Advanced Virgo and KAGRA

Assuming that, for a given source of gravitational waves (GWs), we know its sky position, as is the case of GW events with an electromagnetic counterpart such as GW170817, we discuss a null stream method to probe GW polarizations including spin-0 (scalar) GW modes and spin-1 (vector) modes, especially with an expected network of Advanced LIGO, Advanced Virgo and KAGRA. For two independent null streams for four non-co-aligned GW detectors, we study a location on the sky, exactly at which the spin-0 modes of GWs vanish in any null stream for the GW detector network, though the strain output at a detector may contain the spin-0 modes. Our numerical calculations show that there exist seventy sky positions that satisfy this condition of killing the spin-0 modes in the null streams. If a GW source with an electromagnetic counterpart is found in one of the seventy sky positions, the spin-1 modes will be testable separately from the spin-0 modes by the null stream method. In addition, we study a superposition of the two null streams to show that any one of the three modes (one combined spin-0 and two spin-1 modes) can be eliminated by suitably adjusting a weighted superposition of the null streams and thereby a set of the remaining polarization modes can be experimentally tested.