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 for 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 noncoaligned 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 70 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 70 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.

Figure 1

Curves for ${\delta }_{23}=0$ in the sky, where $L=2$ and $V=3$ are assumed.

Figure 2

Contour map of the coefficients in the null stream as Eq. (14), where $\theta$ and $\varphi$ denote the latitude and longitude, respectively. The detectors are labeled as $H=1$, $L=2$, $V=3$, and $K=4.$ In this figure, white means zero, red is positive, and blue denotes negative. From top to bottom, the panels show $P_a{F}_a^S$, $P_b{F}_b^V$, and $P_c{F}_c^W$.

Figure 3

Contour map of the coefficients in the null stream as Eq. (15). This corresponds to Fig. 2. From top to bottom, the panels show $Q_a{F}_a^S$, $Q_b{F}_b^V$, and $Q_c{F}_c^W$.

Figure 4

The 70 sky positions that satisfy simultaneously $P_a{F}_a^S=0$ and $Q_a{F}_a^S=0$, where we assume $H=1$, $L=2$, $V=3$ and $K=4$.