## 1 Introduction

Breakthroughs in modern technology have made possible the construction of extremely large interferometers both on the ground and in space for the detection and observation of gravitational waves (GWs). Several ground-based detectors around the globe have been operational for several years, and are now in the process of being upgraded to achieve even higher sensitivities. 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 upgraded 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 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 1 Hz. Thus the ground based interferometers will not be sensitive below this limiting frequency. 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. The Laser Interferometer Space Antenna (LISA) mission, and more recent variations of its design [13*, 35*], is the typical example of a space-based interferometer aiming to detect and study gravitational radiation in the millihertz band. In order to make such observations LISA relied on coherent laser beams exchanged between three identical spacecraft forming a giant (almost) equilateral triangle of side to observe and detect low frequency cosmic GWs. Ground- and space-based detectors will complement each other in the observation of GWs in an essential way, analogous to the way optical, radio, X-ray, -ray, etc. observations do for the electromagnetic spectrum. As these detectors begin to make their observations, 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 observable in the mHz band 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 this frequency region. 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, 34]. In the lower frequency range ( 1 mHz) there is a large number of such unresolvable sources in each of the frequency bins. 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 . 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 ( being respectively the lower and upper frequency cut-offs characterizing the detector’s operational bandwidth) use a coherent train of electromagnetic waves (of nominal frequency ) 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 ,
- 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) [55]. 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 [52*], 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 . This is the reason why these one-arm interferometers in space (which have one Doppler readout and a “3-pulse” response to gravitational waves [14]) are most sensitive to mHz gravitational waves. This integration time is also comparable to the microwave propagation (or “storage”) time to spacecraft en route to the outer solar system (for example for the Cassini spacecraft) [52].

Low-frequency interferometric gravitational-wave detectors in solar orbits, such as the LISA mission and the currently considered eLISA/NGO mission [5*, 13*, 35*], 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 [53*]. The data streams will have temporal structure, which can be described as due to many-pulse responses to -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.

Although the theory of TDI can be used by any future space-based interferometer aiming to detect gravitational radiation, this article will focus on its implementation by the LISA mission [5*].

The LISA design envisioned a constellation of three spacecraft orbiting the Sun. Each spacecraft was to 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 [57*], giving six residual Doppler time series as the raw data of a stationary time delay space interferometer. Following [51*, 2*, 10*], 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. [51, 2*, 10*]).

In contrast to Earth-based interferometers, which operate in the long-wavelength limit (LWL) (arm lengths gravitational wavelength , where is a characteristic frequency of the GW), LISA does 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 inter-convertible.

This second 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 [15*]. 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 measured by LISA. The actual procedure can easily be understood in terms of properly defined time-delay operators that act on the one-way Doppler measurements. In Section 4 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 canceling the laser frequency noise is a module over the polynomial ring in which the time-delay operators play the role of the indeterminates [10]. In the literature, the module is called the module of syzygies [3*, 29*]. We show that the module can be generated from four generators, so that any data combination canceling 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 for LISA.

Also in Section 4 specific interferometric combinations are derived, and their physical interpretations are discussed. The expressions for the Sagnac interferometric combinations are first obtained; in particular, the symmetric Sagnac combination , 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 in terms of the , , , and 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 , , and ) 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 [44*, 7*, 45*, 41*, 9*]. 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 , and obtain the generalized expressions for all the TDI combinations [58*]. Alternatively, a rigorous mathematical formulation can be given in terms of rings and modules. But because of the non-commutativity of operators the polynomial ring is non-commutative. Thus the algebraic problem becomes extremely complex and a general solution seems difficult to obtain [9*]. But we show that for the special case when one arm of LISA is dysfunctional a plethora of solutions can be found [11*]. Such a possibility must be envisaged because of reasons such as technical failure or even operating costs.

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 [15*, 38*, 40*] can be improved by a factor of in the low-part of the frequency band, and by more than 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 [38*, 39*]. As an example of such type of source, we consider known binaries within our own galaxy.

In Section 7, we finally address aspects of TDI of more practical and experimental nature, and provide a list of references where more details about these topics can be found. It is worth mentioning that, as of today, TDI has already gone through several successful experimental tests [8*, 32*, 48*, 33*, 25] and that it has been endorsed by the eLISA/NGO [13*, 35*] project as its baseline technique for achieving its required sensitivity to gravitational radiation.

We emphasize that, although this article will use as baseline mission reference the LISA mission, the results here presented can easily be extended to other space mission concepts.