Sensitivity functions of spaceborne gravitational wave detectors for arbitrary time-delay interferometry combinations

The principal aim of the space-based gravitational wave detectors is to explore the gravitational waves in the 0.1 mHz-1 Hz frequency band. To maximize the potential capability of the experimental apparatus regarding the instrument performance, one needs to acquire accurate information on its sensitivity limit. The sensitivity curve in question, by definition, depends on the amplitudes of signal and noise involved in the measurement. In this work, we explicitly derive, under rather universal assumptions irrelevant to the detailed form of the time-delay interferometry combination, general results of the sensitivity functions. The key feature of the present approach is that both the all-sky and polarization average can be factorized and henceforth evaluated analytically. The resultant expressions are then applied to a variety of time-delay interferometry combinations, inclusively for the optimal channels. In particular, the asymptotical forms of the sensitivity functions are obtained at the high and low frequency limits, and the subsequential implications are analyzed. When compared with the approaches in terms of numerical integration, the obtained formulism furnishes a more straightforward as well as efficient access to the relevant signal noise ratios for the spaceborne gravitational wave detectors.

Figure 1

The experimental layout of the space-based laser interferometry consists of laser sources and links.

Figure 2

The detector frame [38] defined in terms of $({\widehat{e}}_x,{\widehat{e}}_y,{\widehat{e}}_z)$.

Figure 3

The calculated individual contributions from each terms given in Eq. (33), $|\frac{f(u)}{u^2}|$, as functions of $u$.

Figure 4

Sensitivity curves of the spaceborne GW detectors LISA and TianQin, for the Michelson $X_1$ combination. The results obtained by analytic and numerical integration are shown in solid and dashed curves.

Figure 5

The sensitivity curves of the Michelson and Sagnac combinations for the LISA mission ($\mathrm{SNR}=5$). The inset shows a zoom for the lower frequency region.

Figure 6

The sensitivity curves of the Michelson and Sagnac combinations for the TianQin mission ($\mathrm{SNR}=5$). The inset shows a zoom for the lower frequency region.

Figure 7

The sensitivity curve of $A$, $E$, $T$, as well as the optimal combinations of the Michelson combination for the LISA detector ($\mathrm{SNR}=5$). The integration time is one year.

Figure 8

The sensitivity curve of $A$, $E$, $T$, as well as the optimal combinations of the Michelson combination for the TianQin detector ($\mathrm{SNR}=5$). The integration time is one year.

Figure 9

The optimal SNR divided by the SNR of Michelson combination for LISA and TianQin detector, as a function of the Fourier frequency $f$. The sensitivity gain in the low-frequency band is equal to $\sqrt{2}$, while it can get larger than 2 at selected frequencies in the high-frequency region of the accessible band. The integration time has been assumed to be one year.