1 Introduction

Breakthroughs in modern technology have made possible the construction of extremely large interferometers both on ground and in space for the detection and observation of gravitational waves (GWs). Several ground based detectors are being constructed or are already operational around the globe. These are the LIGO and VIRGO interferometers, which have arm lengths of 4 km and 3 km, respectively, and the GEO and TAMA interferometers with arm lengths of 600 m and 300 m, respectively. These detectors will operate in the high frequency range of GWs of $~ 1 Hz$ to a few kHz. A natural limit occurs on decreasing the lower frequency cut-off of 10 Hz because it is not practical to increase the arm lengths on ground and also because of the gravity gradient noise which is difficult to eliminate below 10 Hz. However, VIRGO and future detectors such as the advanced LIGO, the proposed LCGT in Japan, and the large European detector plan to go to substantially below 10 Hz. Thus, in any case, the ground based interferometers will not be sensitive below the limiting frequency of 1 Hz. But on the other hand, in the cosmos there exist interesting astrophysical GW sources which emit GWs below this frequency such as the galactic binaries, massive and super-massive black-hole binaries, etc. If we wish to observe these sources, we need to go to lower frequencies. The solution is to build an interferometer in space, where such noises will be absent and allow the detection of GWs in the low frequency regime. LISA is a proposed mission which will use coherent laser beams exchanged between three identical spacecraft forming a giant (almost) equilateral triangle of side $5× 106 km$ to observe and detect low frequency cosmic GWs. The ground based detectors and LISA complement each other in the observation of GWs in an essential way, analogous to the way optical, radio, X-ray, $g$ -ray, etc. observations do for the electromagnetic spectrum. As these detectors begin to operate, a new era of gravitational astronomy is on the horizon and a radically different view of the universe is expected to emerge.

The astrophysical sources that LISA could observe include galactic binaries, extra-galactic super-massive black-hole binaries and coalescences, and stochastic GW background from the early universe. Coalescing binaries are one of the important sources in the LISA frequency band. These include galactic and extra galactic stellar mass binaries, and massive and super-massive black-hole binaries. The frequency of the GWs emitted by such a system is twice its orbital frequency. Population synthesis studies indicate a large number of stellar mass binaries in the frequency range below 2-3 mHz [4 , 17 ]. In the lower frequency range ( $< 1 mHz$ ) there is a large number of such sources in each of the frequency bins. Since GW detectors are omni-directional, it is impossible to resolve an individual source. These sources effectively form a stochastic GW background referred to as binary confusion noise.

Massive black-hole binaries are interesting both from the astrophysical and theoretical points of view. Coalescences of massive black holes from different galaxies after their merger during growth of the present galaxies would provide unique new information on galaxy formation. Coalescence of binaries involving intermediate mass black holes could help to understand the formation and growth of massive black holes. The super-massive black-hole binaries are strong emitters of GWs and these spectacular events can be detectable beyond red-shift of $z = 1$ . These systems would help to determine the cosmological parameters independently. And, just as the cosmic microwave background is left over from the Big Bang, so too should there be a background of gravitational waves. Unlike electromagnetic waves, gravitational waves do not interact with matter after a few Planck times after the Big Bang, so they do not thermalize. Their spectrum today, therefore, is simply a red-shifted version of the spectrum they formed with, which would throw light on the physical conditions at the epoch of the early universe.

Interferometric non-resonant detectors of gravitational radiation (with frequency content $0 < f < fu$ ) use a coherent train of electromagnetic waves (of nominal frequency $n0 » fu$ ) folded into several beams, and at one or more points where these intersect, monitor relative fluctuations of frequency or phase (homodyne detection). The observed low frequency fluctuations are due to several causes:

frequency variations of the source of the electromagnetic signal about $n0$ ,
relative motions of the electromagnetic source and the mirrors (or amplifying transponders) that do the folding,
temporal variations of the index of refraction along the beams, and
according to general relativity, to any time-variable gravitational fields present, such as the transverse-traceless metric curvature of a passing plane gravitational wave train.

To observe gravitational waves in this way, it is thus necessary to control, or monitor, the other sources of relative frequency fluctuations, and, in the data analysis, to use optimal algorithms based on the different characteristic interferometer responses to gravitational waves (the signal) and to the other sources (the noise) [31 ]. By comparing phases of electromagnetic beams referenced to the same frequency generator and propagated along non-parallel equal-length arms, frequency fluctuations of the frequency reference can be removed, and gravitational wave signals at levels many orders of magnitude lower can be detected.

In the present single-spacecraft Doppler tracking observations, for instance, many of the noise sources can be either reduced or calibrated by implementing appropriate microwave frequency links and by using specialized electronics [28 ], so the fundamental limitation is imposed by the frequency (time-keeping) fluctuations inherent to the reference clock that controls the microwave system. Hydrogen maser clocks, currently used in Doppler tracking experiments, achieve their best performance at about 1000 s integration time, with a fractional frequency stability of a few parts in $10-16$ . This is the reason why these one-arm interferometers in space (which have one Doppler readout and a ”3-pulse” response to gravitational waves [8 ]) are most sensitive to mHz gravitational waves. This integration time is also comparable to the microwave propagation (or ”storage”) time $2L/c$ to spacecraft en route to the outer solar system (for example $L -~ 5- 8 AU$ for the Cassini spacecraft) [28 ].

Next-generation low-frequency interferometric gravitational wave detectors in solar orbits, such as the LISA mission [3 ], have been proposed to achieve greater sensitivity to mHz gravitational waves. However, since the armlengths of these space-based interferometers can differ by a few percent, the direct recombination of the two beams at a photo detector will not effectively remove the laser frequency noise. This is because the frequency fluctuations of the laser will be delayed by different amounts within the two arms of unequal length. In order to cancel the laser frequency noise, the time-varying Doppler data must be recorded and post-processed to allow for arm-length differences [29 ]. The data streams will have temporal structure, which can be described as due to many-pulse responses to $d$ -function excitations, depending on time-of-flight delays in the response functions of the instrumental Doppler noises and in the response to incident plane-parallel, transverse, and traceless gravitational waves.

LISA will consists of three spacecraft orbiting the sun. Each spacecraft will be equipped with two lasers sending beams to the other two ( $~ 0.03 AU$ away) while simultaneously measuring the beat frequencies between the local laser and the laser beams received from the other two spacecraft. The analysis of TDI presented in this article will assume a successful prior removal of any first-order Doppler beat notes due to relative motions [33 ], giving six residual Doppler time series as the raw data of a stationary time delay space interferometer. Following [27 , 1 , 6 ], we will regard LISA not as constituting one or more conventional Michelson interferometers, but rather, in a symmetrical way, a closed array of six one-arm delay lines between the test masses. In this way, during the course of the article, we will show that it is possible to synthesize new data combinations that cancel laser frequency noises, and estimate achievable sensitivities of these combinations in terms of the separate and relatively simple single arm responses both to gravitational wave and instrumental noise (cf. [27 , 1 , 6 ]).

In contrast to Earth-based interferometers, which operate in the long-wavelength limit (LWL) (arm lengths $«$ gravitational wavelength $~ c/f 0$ , where $f 0$ is a characteristic frequency of the GW), LISA will not operate in the LWL over much of its frequency band. When the physical scale of a free mass optical interferometer intended to detect gravitational waves is comparable to or larger than the GW wavelength, time delays in the response of the instrument to the waves, and travel times along beams in the instrument, cannot be ignored and must be allowed for in computing the detector response used for data interpretation. It is convenient to formulate the instrumental responses in terms of observed differential frequency shifts - for short, Doppler shifts - rather than in terms of phase shifts usually used in interferometry, although of course these data, as functions of time, are interconvertible.

This first review article on TDI is organized as follows. In Section 2 we provide an overview of the physical and historical motivations of TDI. In Section 3 we summarize the one-arm Doppler transfer functions of an optical beam between two carefully shielded test masses inside each spacecraft resulting from (i) frequency fluctuations of the lasers used in transmission and reception, (ii) fluctuations due to non-inertial motions of the spacecraft, and (iii) beam-pointing fluctuations and shot noise [7 ]. Among these, the dominant noise is from the frequency fluctuations of the lasers and is several orders of magnitude (perhaps 7 or 8) above the other noises. This noise must be very precisely removed from the data in order to achieve the GW sensitivity at the level set by the remaining Doppler noise sources which are at a much lower level and which constitute the noise floor after the laser frequency noise is suppressed. We show that this can be accomplished by shifting and linearly combining the twelve one-way Doppler data LISA will measure. The actual procedure can easily be understood in terms of properly defined time-delay operators that act on the one-way Doppler measurements. We develop a formalism involving the algebra of the time-delay operators which is based on the theory of rings and modules and computational commutative algebra. We show that the space of all possible interferometric combinations cancelling the laser frequency noise is a module over the polynomial ring in which the time-delay operators play the role of the indeterminates. In the literature, the module is called the module of syzygies [6 ]. We show that the module can be generated from four generators, so that any data combination cancelling the laser frequency noise is simply a linear combination formed from these generators. We would like to emphasize that this is the mathematical structure underlying TDI in LISA.

In Section 4 specific interferometric combinations are then derived, and their physical interpretations are discussed. The expressions for the Sagnac interferometric combinations $(a,b, g,z)$ are first obtained; in particular, the symmetric Sagnac combination $z$ , for which each raw data set needs to be delayed by only a $single$ arm transit time, distinguishes itself against all the other TDI combinations by having a higher order response to gravitational radiation in the LWL when the spacecraft separations are equal. We then express the unequal-arm Michelson combinations $(X, Y,Z)$ in terms of the $a$ , $b$ , $g$ , and $z$ combinations with further transit time delays. One of these interferometric data combinations would still be available if the links between one pair of spacecraft were lost. Other TDI combinations, which rely on only four of the possible six inter-spacecraft Doppler measurements (denoted $P$ , $E$ , and $U$ ) are also presented. They would of course be quite useful in case of potential loss of any two inter-spacecraft Doppler measurements.

TDI so formulated presumes the spacecraft-to-spacecraft light-travel-times to be constant in time, and independent from being up- or down-links. Reduction of data from moving interferometric laser arrays in solar orbit will in fact encounter non-symmetric up- and downlink light time differences that are significant, and need to be accounted for in order to exactly cancel the laser frequency fluctuations [24 , 5 , 25 ]. In Section 5 we show that, by introducing a set of non-commuting time-delay operators, there exists a quite general procedure for deriving generalized TDI combinations that account for the effects of time-dependence of the arms. Using this approach it is possible to derive “flex-free” expression for the unequal-arm Michelson combinations $X1$ , and obtain the generalized expressions for all the TDI combinations [34 ].

In Section 6 we address the question of maximization of the LISA signal-to-noise-ratio (SNR) to any gravitational wave signal present in its data. This is done by treating the SNR as a functional over the space of all possible TDI combinations. As a simple application of the general formula we have derived, we apply our results to the case of sinusoidal signals randomly polarized and randomly distributed on the celestial sphere. We find that the standard LISA sensitivity figure derived for a single Michelson interferometer [7 , 19 , 21 ] can be improved by a factor of $V~ 2-$ in the low-part of the frequency band, and by more than $V~ 3-$ in the remaining part of the accessible band. Further, we also show that if the location of the GW source is known, then as the source appears to move in the LISA reference frame, it is possible to optimally track the source, by appropriately changing the data combinations during the course of its trajectory [19 , 20 ]. As an example of such type of source, we consider known binaries within our own galaxy.

This first version of our “Living Review” article on TDI does not include all the results of more practical and experimental nature, as well as all the aspects of TDI that the data analysts will need to account for when analyzing the LISA TDI data combinations. Our forthcoming “second edition” of this review paper will include these topics. It is worth mentioning that, as of today, the LISA project has endorsed TDI as its baseline technique for achieving the desired sensitivity to gravitational radiation. Several experimental verifications and tests of TDI are being, and will be, performed at the NASA and ESA LISA laboratories. Although significant theoretical and experimental work has already been done for understanding and overcoming practical problems related to the implementation of TDI, more work on both sides of the Atlantic is still needed. Results of this undergoing effort will be included in the second edition of this living document.