Probing nontensorial polarizations of stochastic gravitational-wave backgrounds with ground-based laser interferometers

In a general metric theory of gravitation in four dimensions, six polarizations of a gravitational wave are allowed: two scalar and two vector modes, in addition to two tensor modes in general relativity. Such additional polarization modes appear due to additional degrees of freedom in modified theories of gravitation or theories with extra dimensions. Thus, observations of gravitational waves can be utilized to constrain the extended models of gravitation. In this paper, we investigate detectability of additional polarization modes of gravitational waves, particularly focusing on a stochastic gravitational-wave background, with laser-interferometric detectors on the Earth. We found that more than three detectors can separate the mixture of polarization modes in detector outputs, and that they have almost the same sensitivity to each polarization mode of stochastic gravitational-wave background.

Figure 1

Six GW polarizations in a general metric theory of gravitation. The two ellipses (or circles) show the effect of a GW with each polarization on test masses arranged on a circle at the moments of different phases by $\pi$. The circled dot and the arrow represent the propagating direction of the GW.

Figure 2

Coordinate systems.

Figure 3

Angular pattern functions of a detector for each polarization. (a) Plus mode $|F_{+}|$, (b) cross mode $|F_{\times }|$, (c) $x$ mode $|F_x|$, (d) $y$ mode $|F_y|$, and (e) longitudinal mode $|F_{\ell }|$. The angular pattern function of the breathing mode is the same as that of the longitudinal mode except for an overall factor $1/\sqrt{2}$. At the center of the figure, the arms of an interferometer are shown.

Figure 4

Angular pattern functions of a detector for the tensor, vector, and scalar modes. The plots from the left are $\sqrt{F_{+}^2+F_{\times }^2}$, $\sqrt{F_x^2+F_y^2}$, and $\sqrt{F_b^2+F_{\ell }^2}$, respectively.

Figure 5

Coordinate system on the Earth for a detector pair.

Figure 6

Relative positions and orientations of a specific detector pair. The left panel shows the combinations $(\cos 4{\sigma }_{-},\cos 4{\sigma }_{+})$. The right panel shows the combinations $(\beta ,\cos 4{\sigma }_{+})$. The possible optimal configuration, types (i) and (ii), are also shown in the left panel.

Figure 7

Overlap reduction functions for real-detector pairs on the Earth. Each curve shows the tensor mode (solid lines, red), the vector mode (dotted lines, green), and the scalar mode (dashed lines, blue).

Figure 8

Overlap reduction functions for real-detector pairs on the Earth. Each curve shows the tensor mode (solid lines, red), the vector mode (dotted lines, green), and the scalar mode (dashed lines, blue).

Figure 9

Optimal configurations of a detector pair. For a fixed orientation of the interferometer 1, all possible orientations of the optimal interferometer 2 for a fixed separation $\beta$ are illustrated.

Figure 10

Normalized SNRs of the detector pair of type (i) (solid lines) and type (ii) (dotted lines) as a function of $\beta$. The tensor, vector, and scalar modes are represented by red, green, and blue curves, respectively, as indicated on the right side of the figure.

Figure 11

Normalized SNRs of the optimal and real-detector pairs as a function of $\beta$. Each curve shows $\max \{\mathrm{SNR}{|}_{\mathrm{type}(\mathrm{i})},\mathrm{SNR}{|}_{\mathrm{type}(\mathrm{ii})}\}$ for the tensor mode (solid line, red), the vector mode (dotted line, green), and the scalar mode (dashed line, blue). The squares (red), diamonds (green), and triangles (blue) are the normalized SNRs of real-detector pairs for the tensor, vector, and scalar modes, respectively. At the arrow, four points are overlapped: $(\mathrm{\text{detector pair}},\mathrm{\text{mode}})=(2,\mathrm{\text{tensor}})$, (2, vector), (3, vector), and (3, scalar). The numbers in the figure represent the real-detector pairs: $1=\mathrm{HL}$, $2=\mathrm{AC}$, $3=\mathrm{CH}$, $4=\mathrm{LV}$, $5=\mathrm{HV}$, $6=\mathrm{CV}$, $7=\mathrm{CL}$, $8=\mathrm{AV}$, $9=\mathrm{AH}$, and $10=\mathrm{AL}$.

Figure 12

Normalized SNR of an antipodal detector pair as a function of the separation $D$. The tensor, vector, and scalar modes are represented by solid (red), dotted (green), and dashed (blue) curves, respectively. In the case of the (real) Earth, the separation is $D\approx 1.27\times {10}^4\mathrm{km}$.

Figure 13

Plot of a squared effective overlap reduction function, $W_M(f)$, for the H-L-V detector set. No divergence of the function occurs in an observational frequency band.