LISA time-delay interferometry zero-signal solution: Geometrical properties

Time-delay interferometry (TDI) is the data processing technique needed for generating interferometric combinations of data measured by the multiple Doppler readouts available onboard the three Laser Interferometer Space Antenna (LISA) spacecraft. Within the space of all possible interferometric combinations TDI can generate, we have derived a specific combination that has zero response to the gravitational wave signal, and called it the zero-signal solution (ZSS). This is a two-parameter family of linear combinations of the generators of the TDI space, and its response to a gravitational wave becomes null when these two parameters coincide with the values of the angles of the source location in the sky. Remarkably, the ZSS does not rely on any assumptions about the gravitational waveform, and in fact it works for waveforms of any kind. Our approach is analogous to the data analysis method introduced by Gürsel and Tinto in the context of networks of Earth-based, wideband, interferometric gravitational wave detectors observing in coincidence a gravitational wave burst. The ZSS should be regarded as an application of the Gürsel and Tinto method to the LISA data.

Figure 1

Schematic LISA configuration. Each spacecraft is equidistant from point $o$, with unit vectors ${\widehat{p}}_i$ indicating directions to the three spacecraft. Unit vectors ${\widehat{n}}_i$ point between spacecraft pairs with the indicated orientation.

Figure 2

Schematic diagram of LISA configurations involving six laser beams. Optical path delays taken in the counterclockwise sense are denoted with a prime, while unprimed delays are in the clockwise sense.

Figure 3

The array and the wave coordinates. The $x$, $y$, and $z$ axes are the LISA coordinate axes with the origin at the point $o$, and the array lies in the $x$, $y$ plane. The $x^{\prime }$, $y^{\prime }$, $z^{\prime }$ axes are the coordinate axes for the incoming gravitational wave. The $z^{\prime }$ axis is parallel to the direction of propagation of the wave. The $x^{\prime }$, $y^{\prime }$ axes are those with respect to which $h_{+}(t)$ and $h_{\times }(t)$ are defined. The angles $\theta$, $\varphi$ describe the location of the source in the sky.

Figure 4

(a) Contours of constant level of the modulus squared of the ZSS constructed for a sinusoidal signal of frequency $f_0={10}^{-2}\mathrm{H}\mathrm{z}$, incoming from a source located at ${\theta }_s=32°$, ${\varphi }_s=107°$. The plot is symmetric with respect to the LISA plane ($\theta =90°$); (b) two cross sections along the $\theta$ axis of the modulus squared of the ZSS for the same signal of (a); one curve corresponds to $\varphi ={\varphi }_s=107°$, and the other to $\varphi =207°\ne {\varphi }_s$. (c) Two cross sections along the $\varphi$ axis, with $\theta ={\theta }_s=32°$, and $\theta =59°$.

Figure 5

The responses of the three Sagnac combinations, $\alpha$ (a), $\beta$ (b), and $\gamma$ (c), to a gravitational wave burst of unit amplitudes. The source has been assumed to be located at ${\theta }_s=32°$, ${\varphi }_s=107°$.

Figure 6

The time-domain responses of the inverse-Fourier transforms of the three functions, $a_1\tilde{\alpha }$ (a), $a_2\tilde{\beta }$ (b), $a_3\tilde{\gamma }$ (c), which determine the ZSS. The angles ($\theta ,\varphi$), over which the search is performed have been fixed to be equal to $\theta =31.5°$, $\varphi =106.5°$; (d) the ZSS constructed from the functions shown in (a)–(c).

Figure 7

Plots (a)–(d) are as in Fig. 6, but now with ($\theta ={\theta }_s$, $\varphi ={\varphi }_s$).