Correlation analysis for isotropic stochastic gravitational wave backgrounds with maximally allowed polarization degrees

We study correlation analysis for monopole components of stochastic gravitational wave backgrounds, including the maximally allowed polarization degrees. We show that, for typical detector networks, the correlation analysis can probe virtually five spectra: three for the intensities of the tensor, vector, and scalar modes and two for the chiral asymmetries of the tensor and vector modes. The chiral asymmetric signal for the vector modes has been left untouched so far. In this paper, we derive the overlap reduction function for this signal and thus complete the basic ingredients required for widely dealing with polarization degrees. We comprehensively analyze the geometrical properties of all the five overlap reduction functions. In particular, we point out the importance of reflection transformations for configuring preferable networks in the future.

Figure 1

The parity odd ORFs for the VIRGO-LIGO Hanford network. The black solid and red dashed lines correspond to the tensor and vector modes, respectively. The distance between two detectors is $d\sim 8.2\times {10}^3\mathrm{km}$.

Figure 2

The configuration of the detector network under consideration. We fix the positions of the two detectors and examine their detector tensors with respect to the reflection (mirror transformation) at the $yz$ plane.

Figure 3

The two arm directions for the flipped detector tensor [see Eq. (68)]. The arms are mirror symmetric with respect to the $yz$ plane. We define ${\theta }_0$ as the angle between their bisector and the $z$ axis.

Figure 4

Left: the configuration of the type I invariant detector tensor [see Eq. (70)]. The vector ${\mathit{u}}_B$ is on the $x$ axis and ${\mathit{v}}_B$ is on the $yz$ plane with the angle ${\theta }_1$ relative to the $y$ axis. Right: the configuration of the type II invariant detector tensor [see Eq. (71)]. The two vectors are on the $yz$ plane with the bisecting angle ${\theta }_2$.

Figure 5

Left: the type I network formed by combining one flipped detector and one type I invariant detector. This network is characterized by two angles ${\theta }_0$ and ${\theta }_1$ (see Figs. 3 and 4 for their definitions). Right: the type II network formed by combining one flipped detector and one type II invariant detector. This network is characterized by the angular parameters ${\theta }_0$ and ${\theta }_2$ (see Figs. 3 and 4 for their definitions).

Figure 6

The highly symmetric configurations of the type I network resulting in Eq. (76). All the five ORFs vanish for these networks. Left: the configuration with the angular parameters $({\theta }_0,{\theta }_1)=(0,0)$. The upper detector (red lines) is on the $xz$ plane and the lower detector (blue lines) is on the $xy$ plane. Right: the configuration with the angular parameters $({\theta }_0,{\theta }_1)=(0,\pi /2)$. Both upper detector (red lines) and lower detector (blue lines) are on the $xz$ plane.

Figure 7

Left: the contour plot of the function $F_{\mathrm{I}}^{W_T}({\theta }_0,{\theta }_1)$ defined in Eq. (78). The minimum value is 0 and the maximum value is 0.840. Right: the contour plot of the function $F_{\mathrm{I}}^{W_V}({\theta }_0,{\theta }_1)$ ranging from 0 to 0.520.

Figure 8

Highly symmetric network configuration with $({\theta }_0,{\theta }_1)=(\pi /2,0)$ corresponding to the maximums in Fig. 7. Both upper detector (red lines) and lower detector (blue lines) are parallel to the $xy$ plane. This network is parity odd for the reflections at the $xz$ and $yz$ planes.

Figure 9

The highly symmetric configurations of the type II network resulting in Eq. (84). All the five ORFs vanish for these networks. Left: the configuration with the angular parameters $({\theta }_0,{\theta }_2)=(0,\pi /4)$. The upper detector (red lines) is on the $xz$ plane and the lower detector (blue lines) is on the $yz$ plane. Right: the configuration with the angular parameters $({\theta }_0,{\theta }_2)=(\pi /2,\pi /2)$. The upper detector (red lines) is parallel to the $xy$ plane and the lower detector (blue lines) is on the $yz$ plane with bisector of the two arms on the $z$ axis.

Figure 10

Left: the contour plot of the function $F_{\mathrm{II}}^{W_T}({\theta }_0,{\theta }_2)$ defined in Eq. (86). The minimum value is 0 and the maximum value is 0.520. Right: the contour plot of the function $F_{\mathrm{II}}^{W_V}({\theta }_0,{\theta }_2)$ ranging from 0 to 0.508.

Figure 11

Highly symmetric network configuration with $({\theta }_0,{\theta }_2)=(0,0)$ corresponding to the maximums in Fig. 10. The upper detector (red lines) is on the $xz$ plane and lower detector (blue lines) is on the $yz$ plane with bisector of two arms on the $y$ axis. This network is parity odd for the reflections at the $xz$ and $yz$ planes.

Figure 12

The relative configuration of two ground based detectors A and B. The two detectors are on the blue planes which are tangential to the Earth sphere. The angle $\beta$ shows the separation of two detectors, measured from the center of the Earth. The two angles ${\sigma }_A$ and ${\sigma }_B$ describe the orientations of bisectors of the two detector arms (dotted lines) in a counterclockwise manner, relative to the great circle connecting the two detectors. The angular parameters $\mathrm{\Delta }$ and $\delta$ are defined by $\mathrm{\Delta }\equiv ({\sigma }_A+{\sigma }_B)/2$ and $\delta \equiv ({\sigma }_A-{\sigma }_B)/2$.