Angular and frequency response of the gravitational wave interferometers in the metric theories of gravity

We analyze detector responses of gravitational wave detectors for gravitational waves with arbitrary polarizations predicted in the metric theories of gravity. We present the general formulas for the frequency responses valid in various interferometric arrangements including Michelson, Delay-Line and Fabry-Perot detectors. We analyze the angular and frequency behavior and the sensitivity patterns of the responses for each polarization mode.

Figure 1

The full trajectory of the photon relevant in definition of the response (2.1). The following abbreviations are used: ’em’—emitter, ’bm’—beam splitter, ’f’—front mirror, ’e’—end mirror.

Figure 2

Convention used in the definition of the polar angle $\psi$ and the source frame $\{x^{\prime },y^{\prime },z^{\prime }\}$. In that frame gravitational waves take their canonical forms discussed in Sec. 3. The dashed unit vectors are tangent to the lines of constant $\theta$ and $\varphi$; $z^{\prime }$ axis lies along the direction of the wave propagation. It is assumed that the origins of the frames $\{x,y,z\}$ and $\{x^{\prime },y^{\prime },z^{\prime }\}$ coincide. The unit vector ${\mathbf{n}}_{a,b}$ indicates the orientation of the detector’s arm.

Figure 3

Angular dependence of the magnitude of the transfer functions $\mathcal{T}$ (upper left) and ${\mathcal{T}}_1$ (upper right) for $\omega L_{a,b}$ equal 0.1 (dotted line), 2 (dashed line), 5 (thick line) and 20 (thin line). Lower plot shows the dependence on the frequency of $\mathcal{T}$ (dotted curve), ${\mathcal{T}}_1$ (dashed curve), ${\mathcal{T}}_{20}$ (thin curve) and ${\mathcal{T}}_{\mathrm{FP}}$ for $\rho =0.9$ (thick curve) at the point $\vartheta =-\pi$.

Figure 4

The configuration and orientation of the two arms of the Michelson, Delay-Line and Fabry-Perot interferometers discussed in Sec. 5b.

Figure 5

Shapes of the frequency response functions of the equal-arm Michelson interferometer for different polarization mode. From above: scalar longitudinal, scalar transversal, vectorial and tensorial polarization. We use the normalized frequencies $\omega L$, where $L$ is the arm length. From the left: $\omega L=0.1$, 2, 5 and 10. The scale is not preserved.

Figure 6

Frequency response $|H_M(\omega )|$ of the equal-arm Michelson interferometer as the function of the azimuthal angle $\varphi$, fixed polar angle $\theta$ and $\psi =0$ for scalar longitudinal ($\theta =\pi /2$) (a), scalar transversal ($\theta =\pi /2$) (b), vectorial $x$ ($\theta =\pi /2$) (c), vectorial $y$ ($\theta =\pi /4$) (d), tensorial $+$ ($\theta =\pi /2$) (e) and tensorial $\times$ ($\theta =\pi /4$) (f) polarization. Dotted, dashed, thick and thin curves correspond to the normalize angular frequencies $x$ equal 0.1, 2, 5 and 10, respectively. The frequency response has units [$L$].

Figure 7

The frequency responses $|H_M|$ of the equal-arm Michelson interferometer averaged over the polarizations and sky location of sources as a function of $\omega L$: (a) scalar longitudinal, (b) scalar transversal, (c) vectorial and (d) tensorial polarization. The frequency response has units [$L$]. To obtain plots of $H_{\mathrm{DLM}}$ and $H_{\mathrm{FPM}}$ one should multiply $|H_M(\omega )|$ by $|\frac{\sin N\omega L}{\sin \omega L}|$ and $|\frac{1+\rho }{1-\rho e^{-2i\omega L}}|$ respectively.

Figure 8

Triangular configuration of spacecraft.

Figure 9

Self-correlations of the optimal $A_M$ observable as a function of the frequency for $l=0$, 2, 4 and 6 for the tensorial (thick black curve), vectorial (thick gray curve), scalar longitudinal (thin curve) and scalar transversal (thin dotted curve) polarization. The angular power has units [$L^2$].

Figure 10

Cross-correlations of the optimal $A_M$ and $E_M$ observables as a function of the frequency for $l=2$, 3, 4 5 and 6. Different polarization modes are marked as in Fig. 9.