Frequency response of space-based interferometric gravitational-wave detectors

Gravitational waves are perturbations of the metric of space-time. Six polarizations are possible, although general relativity predicts that only two such polarizations, tensor plus and tensor cross are present for gravitational waves. We give the analytical formulas for the antenna response functions for the six polarizations which are valid for any equal-arm interferometric gravitational-wave detectors without optical cavities in the arms. The response function averaged over the source direction and polarization angle decreases at high frequencies which deteriorates the signal-to-noise ratio registered in the detector. At high frequencies, the averaged response functions for the tensor and breathing modes fall of as $1/f^2$, the averaged response function for the longitudinal mode falls off as $1/f$ and the averaged response function for the vector mode falls off as $\ln (f)/f^2$.

Figure 1

The normal vectors of the detector planes for TianQin and LISA in ecliptic coordinates. TianQin points to the direction of J0806.3+1527 located at (${\theta }_{tq}=94.7°$, ${\varphi }_{tq}=120.5°$).

Figure 2

The evolution of the angular response function for the combined tensor mode in ecliptic coordinates during the orbital motion of the detector. The left panel is for TianQin and the right panel is LISA. From the top to the bottom, the time interval is one third of the orbital period of the detector (1.22 day for TianQin and 4 months for LISA). TianQin always points to the same direction, the most sensitive regions do not change. For LISA, the most sensitive regions move along the latitude of $\pm 30°$. For the animations of the plots, please see http://doi.org/10.5281/zenodo.2574620.

Figure 3

The evolution of the angular response function for the combined vector mode in ecliptic coordinates during the orbital motion of the detector. The left panel is for TianQin and the right panel is LISA. From the top to the bottom, the time interval is one third of the orbital period of the detector (1.22 day for TianQin and 4 months for LISA).

Figure 4

The evolution of the angular response function for the breathing/longitudinal mode in ecliptic coordinates during the orbital motion of the detector. The left panel is for TianQin and the right panel is LISA. From the top to the bottom, the time interval is one third of the orbital period of the detector (1.22 day for TianQin and 4 months for LISA).

Figure 5

The frequency dependence of the TianQin’s response function for the breathing and longitudinal modes. The left panel is for the breathing mode and the right panel is for the longitudinal mode. From the top to the bottom, three different frequencies of 0.01 Hz, 0.1 Hz and 0.5 Hz are chosen, respectively.

Figure 6

The averaged antenna response functions for different polarizations. In the lower $x$ axis, we use the variable $a=2\pi fL/c$ and choose the opening angle $\gamma =\pi /3$. The dashed curves show the asymptotic behaviors at high frequencies. Similar results were obtained in [37] by performing a Monte Carlo simulation, here we use the analytical expressions (27)–(30). In the upper $x$ axis, we show TianQin’s frequency dependent antenna response function for different polarizations. For TianQin, the arm length is $L=\sqrt{3}\times {10}^8\mathrm{m}$ and the opening angle $\gamma =\pi /3$.

Figure 7

The effective strain sensitivity curves for TianQin.

Figure 8

The power-law integrated curves characterizing the sensitivity of TianQin to a stochastic background for different polarizations.We choose $\mathrm{SNR}=1$ and the observational time of one year.

Figure 9

The wave coordinate system [30]. GWs propagate along the $z$ direction. One arm of the detector ($\widehat{u}$) locates on the $x–z$ plane with the inclination angle ${\theta }_1$ and the opening angle between the two arms $\widehat{u}$ and $\widehat{v}$ is $\gamma$. The angle between the polarization axis $\widehat{m}$ and the $x$ axis is ${\psi }_1$, and $\epsilon$ is the angle between the plane containing the vectors $\widehat{u}$ and $\widehat{w}$ and the plane of the interferometer.

Figure 10

The detector coordinate system [29, 65]. The two arms of the detector lie along the direction $(\cos (\gamma /2),-\sin (\gamma /2),0)$ and $(\cos (\gamma /2),\sin (\gamma /2),0)$, and the source is located at the direction $-\widehat{w}=(\sin {\theta }_d\cos {\varphi }_d,\sin {\theta }_d\sin {\varphi }_d,\cos {\theta }_d)$.