Geometric time delay interferometry

The space-based gravitational-wave observatory LISA, a NASA-ESA mission to be launched after 2012, will achieve its optimal sensitivity using time delay interferometry (TDI), a LISA-specific technique needed to cancel the otherwise overwhelming laser noise in the interspacecraft phase measurements. The TDI observables of the Michelson and Sagnac types have been interpreted physically as the virtual measurements of a synthesized interferometer. In this paper, I present Geometric TDI, a new and intuitive approach to extend this interpretation to all TDI observables. Unlike the standard algebraic formalism, Geometric TDI provides a combinatorial algorithm to explore exhaustively the space of second-generation TDI observables (i.e., those that cancel laser noise in LISA-like interferometers with time-dependent arm lengths). Using this algorithm, I survey the space of second-generation TDI observables of length (i.e., number of component phase measurements) up to 24, and I identify alternative, improved forms of the standard second-generation TDI observables. The alternative forms have improved high-frequency gravitational-wave sensitivity in realistic noise conditions (because they have fewer nulls in the gravitational-wave and noise response functions), and are less susceptible to instrumental gaps and glitches (because their component phase measurements span shorter time periods).

Figure 1

In this idealization of the basic time-transport observable used with LISA, two ideal clocks travel along geodesics, with clock 1 continuously transferring its proper time to clock 2 by way of pulsed light signals. GWs are measured as the fluctuations in the time of flight between the clocks (see main text).

Figure 2

These time delayed sums and differences of two $y_{ij}$ measurements cancel laser phase noise at time $t$. In all of them, two laser pulses arrive at, or depart from, the same spacecraft at time $t$.

Figure 3

Left.—The arrows of this closed loop reproduce the paths of light in a standard equal-arm Michelson interferometer, and the corresponding time-ordered sum of phase measurements [Eq. (7)] reproduces the phase-difference output of the interferometer. Right.—For unequal arm lengths, laser phase noise cancellation can be recovered by having both interfering beams travel along each arm once, building up the same light-travel time. Compare with Fig. 1 of Ref. 10.

Figure 4

Mapping first-generation TDI observables into closed arrow loops. The Sagnac ($\alpha$), unequal-arm Michelson ($X$), and Relay ($U$) observables of first-generation TDI have a straightforward interpretation as synthesized two-beam interferometers. More interestingly, the beacon ($P$) and monitor ($E$) observables can be seen as four-beam interferometers, with the four beams combining into different pairs at the events of initial emission and final reception. To interpret the symmetrized Sagnac ($\zeta$) observables as six-beam interferometers, three different beam pairings must be invoked to explain the cancellation of laser noise at emission and reception and the relative insensitivity of $\zeta$ to GWs (see main text).

Figure 5

TDI response of the second-generation $X$-type variables to the fundamental secondary noises, according to Eq. (20), assuming equal arms and proof-mass optical-path noise spectral densities given by $S_i^{\mathrm{pm}}=2.5\times {10}^{-48}[f/\mathrm{Hz}{]}^{-2}{\mathrm{Hz}}^{-1}$ and $S_i^{\mathrm{op}}=1.8\times {10}^{-37}[f/\mathrm{Hz}{]}^2{\mathrm{Hz}}^{-1}$ (following Ref. 20). The top panel shows the noise response of the standard 16-link $X_1$ observable, as compared to the response of the alternative 16-link forms of Eq. (22), which have half as many nulls. The middle panel shows the noise response of the two-beam, 20-link $X_1$ observables of Eq. (30). The bottom panel shows the three new noise responses found for 24-link $X_1$ observables (solid and dashed: self- and non-self-splicings of 12-link observables; dashed-dotted: self-splicings of 8-link observables, with inclusions). To avoid visual clutter, two of the 24-link $X_1$ curves are not plotted beyond $f=1/(2L)$ (shaded region).

Figure 6

The time-reversal plus bridge symmetry is exploited by certain $\dot{L}$-closed non-self-splicings at length 20 and above, such as the link string of Eq. (37).