TimeDelay Interferometry
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Abstract
Equalarm detectors of gravitational radiation allow phase measurements many orders of magnitude below the intrinsic phase stability of the laser injecting light into their arms. This is because the noise in the laser light is common to both arms, experiencing exactly the same delay, and thus cancels when it is differenced at the photo detector. In this situation, much lower level secondary noises then set the overall performance. If, however, the two arms have different lengths (as will necessarily be the case with spaceborne interferometers), the laser noise experiences different delays in the two arms and will hence not directly cancel at the detector. In order to solve this problem, a technique involving heterodyne interferometry with unequal arm lengths and independent phasedifference readouts has been proposed. It relies on properly timeshifting and linearly combining independent Doppler measurements, and for this reason it has been called timedelay interferometry (TDI).
This article provides an overview of the theory, mathematical foundations, and experimental aspects associated with the implementation of TDI. Although emphasis on the application of TDI to the Laser Interferometer Space Antenna (LISA) mission appears throughout this article, TDI can be incorporated into the design of any future spacebased mission aiming to search for gravitational waves via interferometric measurements. We have purposely left out all theoretical aspects that data analysts will need to account for when analyzing the TDI data combinations.
Keywords
Interferometry Gravitationalwave detectors1 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 groundbased 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 cutoff 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 supermassive blackhole 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 spacebased 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 5 × 10^{6} km to observe and detect low frequency cosmic GWs. Ground and spacebased detectors will complement each other in the observation of GWs in an essential way, analogous to the way optical, radio, Xray, γray, 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, extragalactic supermassive blackhole 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 supermassive blackhole 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 blackhole 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 supermassive blackhole binaries are strong emitters of GWs and these spectacular events can be detectable beyond redshift of z = 10. 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 redshifted version of the spectrum they formed with, which would throw light on the physical conditions at the epoch of the early Universe.
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frequency variations of the source of the electromagnetic signal about ν_{0},
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relative motions of the electromagnetic source and the mirrors (or amplifying transponders) that do the folding,
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temporal variations of the index of refraction along the beams, and
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according to general relativity, to any timevariable gravitational fields present, such as the transversetraceless metric curvature of a passing plane gravitationalwave train.
In the present singlespacecraft 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 (timekeeping) 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 onearm interferometers in space (which have one Doppler readout and a “3pulse” 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 2L/c to spacecraft en route to the outer solar system (for example L ≃ 5−8 AU for the Cassini spacecraft) [52].
Lowfrequency interferometric gravitationalwave 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 spacebased 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 timevarying Doppler data must be recorded and postprocessed to allow for armlength differences [53]. The data streams will have temporal structure, which can be described as due to manypulse responses to δfunction excitations, depending on timeofflight delays in the response functions of the instrumental Doppler noises and in the response to incident planeparallel, transverse, and traceless gravitational waves.
Although the theory of TDI can be used by any future spacebased 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 firstorder 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 onearm 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 Earthbased interferometers, which operate in the longwavelength limit (LWL) (arm lengths ≪ gravitational wavelength ∼ c/f_{0}, where f_{0} 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 interconvertible.
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 onearm 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 noninertial motions of the spacecraft, and (iii) beampointing 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 oneway Doppler data measured by LISA. The actual procedure can easily be understood in terms of properly defined timedelay operators that act on the oneway Doppler measurements. In Section 4 we develop a formalism involving the algebra of the timedelay 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 timedelay 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 (α, β, γ, ζ) are first obtained; in particular, the symmetric Sagnac combination ζ, 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 unequalarm Michelson combinations (X, Y, Z) in terms of the α, β, γ, and ζ 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 interspacecraft 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 interspacecraft Doppler measurements.
TDI so formulated presumes the spacecrafttospacecraft lighttraveltimes to be constant in time, and independent from being up or downlinks. Reduction of data from moving interferometric laser arrays in solar orbit will in fact encounter nonsymmetric 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 noncommuting timedelay operators, there exists a quite general procedure for deriving generalized TDI combinations that account for the effects of timedependence of the arms. Using this approach it is possible to derive “flexfree” expression for the unequalarm Michelson combinations X_{1}, 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 noncommutativity of operators the polynomial ring is noncommutative. 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 signaltonoiseratio (SNR) to any gravitationalwave 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 \(\sqrt 2\) in the lowpart of the frequency band, and by more than \(\sqrt 3\) 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 [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.
2 Physical and Historical Motivations of TDI
Equalarm interferometer detectors of gravitational waves can observe gravitational radiation by canceling the laser frequency fluctuations affecting the light injected into their arms. This is done by comparing phases of split beams propagated along the equal (but nonparallel) arms of the detector. The laser frequency fluctuations affecting the two beams experience the same delay within the two equallength arms and cancel out at the photodetector where relative phases are measured. This way gravitationalwave signals of dimensionless amplitude less than 10^{−20} can be observed when using lasers whose frequency stability can be as large as roughly a few parts in 10^{−13}.
In the following sections we will further elaborate and generalize TDI to the realistic LISA configuration.
3 TimeDelay Interferometry
Several notations have been used in this context. The double index notation recently employed in [45], where six quantities are involved, is selfevident. However, when algebraic manipulations are involved the following notation seems more convenient to use. The spacecraft are labeled 1, 2, 3 and their separating distances are denoted L_{1}, L_{2}, L_{3}, with L_{ i } being opposite spacecraft i. We orient the vertices 1, 2, 3 clockwise in Figure 3. Unit vectors between spacecraft are \({\hat n_i}\), oriented as indicated in Figure 3. We index the phase difference data to be analyzed as follows: The beam arriving at spacecraft has subscript i and is primed or unprimed depending on whether the beam is traveling clockwise or counterclockwise (the sense defined here with reference to Figure 3) around the LISA triangle, respectively. Thus, as seen from the figure, s_{1} is the phase difference time series measured at reception at spacecraft 1 with transmission from spacecraft 2 (along L_{3}).
We extend the cyclic terminology so that at vertex i, i = 1, 2, 3, the random displacement vectors of the two proof masses are respectively denoted by \({\vec \delta _i}(t),{\vec \delta ^{\prime}_i}(t)\), and the random displacements (perhaps several orders of magnitude greater) of their optical benches are correspondingly denoted by \({\vec \Delta _i}(t),\;{\vec \Delta ^{\prime}_i}(t)\) where the primed and unprimed indices correspond to the right and left optical benches, respectively. As pointed out in [15], the analysis does not assume that pairs of optical benches are rigidly connected, i.e., \({\vec \Delta _i} \neq {\vec \Delta ^{\prime}_i}\), in general. The present LISA design shows optical fibers transmitting signals both ways between adjacent benches. We ignore timedelay effects for these signals and will simply denote by μ_{ i }(t) the phase fluctuations upon transmission through the fibers of the laser beams with frequencies ν_{ i }, and ν′_{ i }. The μ_{ i }(t) phase shifts within a given spacecraft might not be the same for large frequency differences νi−ν′_{ i } For the envisioned frequency differences (a few hundred MHz), however, the remaining fluctuations due to the optical fiber can be neglected [15]. It is also assumed that the phase noise added by the fibers is independent of the direction of light propagation through them. For ease of presentation, in what follows we will assume the center frequencies of the lasers to be the same, and denote this frequency by ν_{0}.
The laser phase noise in s′_{3} is therefore equal to \({{\mathcal D}_1}{p_2}(t)  {p^{\prime}_3}(t)\). Similarly, since s_{2} is the phase shift measured on arrival at spacecraft 2 along arm 1 of a signal transmitted from spacecraft 3, the laser phase noises enter into it with the following time signature: \({{\mathcal D}_1}{p^{\prime}_3}(t)  {p_2}(t)\). Figure 4 endeavors to make the detailed light paths for these observations clear. An outgoing light beam transmitted to a distant spacecraft is routed from the laser on the local optical bench using mirrors and beam splitters; this beam does not interact with the local proof mass. Conversely, an incoming light beam from a distant spacecraft is bounced off the local proof mass before being reflected onto the photo receiver where it is mixed with light from the laser on that same optical bench. The interspacecraft phase data are denoted s_{1} and s′_{1} in Figure 4.
Beams between adjacent optical benches within a single spacecraft are bounced off proof masses in the opposite way. Light to be transmitted from the laser on an optical bench is first bounced off the proof mass it encloses and then directed to the other optical bench. Upon reception it does not interact with the proof mass there, but is directly mixed with local laser light, and again down converted. These data are denoted τ_{1} and τ′_{1} in Figure 4.
The expressions for the s_{ i }, s′_{ i } and τ_{ i }, τ′_{1} phase measurements can now be developed from Figures 3 and 4, and they are for the particular LISA configuration in which all the lasers have the same nominal frequency ν_{0}, and the spacecraft are stationary with respect to each other.^{1} Consider the s′_{1}(t) process (Eq. (14) below). The photo receiver on the right bench of spacecraft 1, which (in the spacecraft frame) experiences a timevarying displacement \({\vec \Delta ^{\prime}_1}\), measures the phase difference s′_{1} by first mixing the beam from the distant optical bench 3 in direction \({\hat n_2}\), and laser phase noise p_{3} and optical bench motion \({\vec \Delta _3}\) that have been delayed by propagation along L_{2}, after one bounce off the proof mass \(({\vec \delta ^{\prime}_1})\), with the local laser light (with phase noise p′_{1}). Since for this simplified configuration no frequency offsets are present, there is of course no need for any heterodyne conversion [57].
The gravitationalwave phase signal components \(s^{gw}_{i},s\prime^{gw}_{i},i=1,2,3\) in Eqs. (12) and (14) are given by integrating with respect to time the Eqs. (1) and (2) of reference [2], which relate metric perturbations to optical frequency shifts. The optical path phase noise contributions \(s_{i}^{optical\;path},s\prime_{i}^{optical\;path}\) which include shot noise from the low SNR in the links between the distant spacecraft, can be derived from the corresponding term given in [15]. The τ_{ i }, τ′_{ i } measurements will be made with high SNR so that for them the shot noise is negligible.
4 Algebraic Approach for Canceling Laser and Optical Bench Noises
In groundbased detectors the arms are chosen to be of equal length so that the laser light experiences identical delay in each arm of the interferometer. This arrangement precisely cancels the laser frequency/phase noise at the photodetector. The required sensitivity of the instrument can thus only be achieved by near exact cancellation of the laser frequency noise. However, in LISA it is impossible to achieve equal distances between spacecraft, and the laser noise cannot be canceled in this way. It is possible to combine the recorded data linearly with suitable timedelays corresponding to the three arm lengths of the giant triangular interferometer so that the laser phase noise is canceled. Here we present a systematic method based on modules over polynomial rings which guarantees all the data combinations that cancel both the laser phase and the optical bench motion noises.
We first consider the simpler case, where we ignore the opticalbench motion noise and consider only the laser phase noise. We do this because the algebra is somewhat simpler and the method is easy to apply. The simplification amounts to physically considering each spacecraft rigidly carrying the assembly of lasers, beamsplitters, and photodetectors. The two lasers on each spacecraft could be considered to be locked, so effectively there would be only one laser on each spacecraft. This mathematically amounts to setting \({\vec \Delta _i} = {\vec \Delta ^{\prime}_i} = 0\) and p_{ i } = p′_{ i }. The scheme we describe here for laser phase noise can be extended in a straightforward way to include optical bench motion noise, which we address in the last part of this section.
The data combinations, when only the laser phase noise is considered, consist of the six suitably delayed data streams (interspacecraft), the delays being integer multiples of the light travel times between spacecraft, which can be conveniently expressed in terms of polynomials in the three delay operators \({{\mathcal D}_1},{{\mathcal D}_2},{{\mathcal D}_3}\). The laser noise cancellation condition puts three constraints on the six polynomials of the delay operators corresponding to the six data streams. The problem, therefore, consists of finding sixtuples of polynomials which satisfy the laser noise cancellation constraints. These polynomial tuples form a module^{2} called the module of syzygies. There exist standard methods for obtaining the module, by which we mean methods for obtaining the generators of the module so that the linear combinations of the generators generate the entire module. The procedure first consists of obtaining a Gröbner basis for the ideal generated by the coefficients appearing in the constraints. This ideal is in the polynomial ring in the variables \({{\mathcal D}_1},{{\mathcal D}_2},{{\mathcal D}_3}\) over the domain of rational numbers (or integers if one gets rid of the denominators). To obtain the Gröbner basis for the ideal, one may use the Buchberger algorithm or use an application such as Mathematica [65]. From the Gröbner basis there is a standard way to obtain a generating set for the required module. This procedure has been described in the literature [3, 29]. We thus obtain seven generators for the module. However, the method does not guarantee a minimal set and we find that a generating set of 4 polynomial sixtuples suffice to generate the required module. Alternatively, we can obtain generating sets by using the software Macaulay 2.
The importance of obtaining more data combinations is evident: They provide the necessary redundancy — different data combinations produce different transfer functions for GWs and the system noises so specific data combinations could be optimal for given astrophysical source parameters in the context of maximizing SNR, detection probability, improving parameter estimates, etc.
4.1 Cancellation of laser phase noise
Note that we have intentionally excluded from the data additional phase fluctuations due to the GW signal, and noises such as the opticalpath noise, proofmass noise, etc. Since our immediate goal is to cancel the laser frequency noise we have only kept the relevant terms. Combining the streams for canceling the laser frequency noise will introduce transfer functions for the other noises and the GW signal. This is important and will be discussed subsequently in the article.
4.2 Cancellation of laser phase noise in the unequalarm interferometer
The use of commutative algebra is very conveniently illustrated with the help of the simpler example of the unequalarm interferometer. Here there are only two arms instead of three as we have for LISA, and the mathematics is much simpler and so it easy to see both physically and mathematically how commutative algebra can be applied to this problem of laser phase noise cancellation. The procedure is well known for the unequalarm interferometer, but here we will describe the same method but in terms of the delay operators that we have introduced.
The notions of commutativity of polynomials, L.C.M., etc. belong to the field of commutative algebra. In fact we will be using the notion of a Groöbner basis which is in a sense the generalization of the notion of the greatest common divisor (gcd). Since LISA has three spacecraft and six interspacecraft beams, the problem of the unequalarm interferometer only gets technically more complex; in principle the problem is the same as in this simpler case. Thus, the simple operations which were performed here to obtain a laser noise free combination X(t) are not sufficient and more sophisticated methods need to be adopted from the field of commutative algebra. We address this problem in the forthcoming text.
4.3 The module of syzygies
Equation (21) has nontrivial solutions. Several solutions have been exhibited in [2, 15]. We merely mention these solutions here; in the forthcoming text we will discuss them in detail. The solution ζ is given by \( {{\rm{q}}^{\rm{T}}} = {{\rm{q^{\prime}}}^T} = ({{\mathcal D}_1},{{\mathcal D}_2},{{\mathcal D}_3})\). The solution α is described by \({{\rm{q}}^T} =  (1,{{\mathcal D}_3},{{\mathcal D}_1},{{\mathcal D}_3})\) and \({{\rm{q^{\prime}}}^T} = (1,{{\mathcal D}_1},{{\mathcal D}_2},{{\mathcal D}_2})\). The solutions β and γ are obtained from α by cyclically permuting the indices of \({{\mathcal D}_k},{\rm{q}}\) and q′. These solutions are important, because they consist of polynomials with lowest possible degrees and thus are simple. Other solutions containing higher degree polynomials can be generated conveniently from these solutions. Since the system of equations is linear, linear combinations of these solutions are also solutions to Eq. (21).
However, it is important to realize that we do not have a vector space here. Three independent constraints on a sixtuple do not produce a space which is necessarily generated by three basis elements. This conclusion would follow if the solutions formed a vector space but they do not. The polynomial sixtuple q, q′ can be multiplied by polynomials in \({{\mathcal D}_1},{{\mathcal D}_2},{{\mathcal D}_3}\) (scalars) which do not form a field. Thus, the inverse in general does not exist within the ring of polynomials. We, therefore, have a module over the ring of polynomials in the three variables \({{\mathcal D}_1},{{\mathcal D}_2},{{\mathcal D}_3}\). First we present the general methodology for obtaining the solutions to Eq. (21) and then apply it to Eq. (21).
There are three linear constraints on the polynomials given by Eq. (21). Since the equations are linear, the solutions space is a submodule of the module of sixtuples of polynomials. The module of sixtuples is a free module, i.e., it has six basis elements that not only generate the module but are linearly independent. A natural choice of the basis is f_{ m } = (0, …, 1, …, 0) with 1 in the mth place and 0 everywhere else; m runs from 1 to 6. The definitions of generation (spanning) and linear independence are the same as that for vector spaces. A free module is essentially like a vector space. But our interest lies in its submodule which need not be free and need not have just three generators as it would seem if we were dealing with vector spaces.
We will assume that the polynomials have rational coefficients, i.e., the coefficients belong to \({\mathcal Q}\), the field of the rational numbers. The set of polynomials form a ring — the polynomial ring in three variables, which we denote by \({\mathcal R} = {\mathcal Q}[{{\mathcal D}_1},{{\mathcal D}_2},{{\mathcal D}_3}]\). The polynomial vector \(({q_3},{q^{\prime}_1},{q^{\prime}_2},{q^{\prime}_3})\;\;\in \;{{\mathcal R}^4}\). The set of solutions to Eq. (28) is just the kernel of the homomorphism \(\varphi \;:{{\mathcal R}^4}\; \rightarrow {\mathcal R}\), where the polynomial vector (q_{3}, q′_{1}, q′_{2}, q′_{3}) is mapped to the polynomial \((1  {{\mathcal D}_1}{{\mathcal D}_2}{{\mathcal D}_3}){q_3}\; + ({{\mathcal D}_1}{{\mathcal D}_3}  {{\mathcal D}_2}){q^{\prime}_1} + {{\mathcal D}_1}(1  {\mathcal D}_3^2){q^{\prime}_2}\; + (1  {\mathcal D}_1^2){q^{\prime}_3}\). Thus, the solution space ker φ is a submodule of \({{\mathcal R}^4}\). It is called the module of syzygies. The generators of this module can be obtained from standard methods available in the literature. We briefly outline the method given in the books by Becker et al. [3], and Kreuzer and Robbiano [29] below. The details have been included in Appendix A.
4.4 Gröbner basis
There are several ways to look at the theory of Gröbner basis. One way is the following: Suppose we are given polynomials g_{1}, g_{2}, …, g_{ m } in one variable over say \({\mathcal Q}\) and we would like to know whether another polynomial f belongs to the ideal generated by the g’s. A good way to decide the issue would be to first compute the gcd g of g_{1}, g_{2}, …, g_{ m } and check whether f is a multiple of g. One can achieve this by doing the long division of f by g and checking whether the remainder is zero. All this is possible because \({\mathcal Q}[x]\) is a Euclidean domain and also a principle ideal domain (PID) wherein any ideal is generated by a single element. Therefore we have essentially just one polynomial — the gcd — which generates the ideal generated by g_{1}, g_{2}, …, g_{ m }. The ring of integers or the ring of polynomials in one variable over any field are examples of PIDs whose ideals are generated by single elements. However, when we consider more general rings (not PIDs) like the one we are dealing with here, we do not have a single gcd but a set of several polynomials which generates an ideal in general. A Gröbner basis of an ideal can be thought of as a generalization of the gcd. In the univariate case, the Gröbner basis reduces to the gcd.
Gröbner basis theory generalizes these ideas to multivariate polynomials which are neither Euclidean rings nor PIDs. Since there is in general not a single generator for an ideal, Gröbner basis theory comes up with the idea of dividing a polynomial with a set of polynomials, the set of generators of the ideal, so that by successive divisions by the polynomials in this generating set of the given polynomial, the remainder becomes zero. Clearly, every generating set of polynomials need not possess this property. Those special generating sets that do possess this property (and they exist!) are called Gröbner bases. In order for a division to be carried out in a sensible manner, an order must be put on the ring of polynomials, so that the final remainder after every division is strictly smaller than each of the divisors in the generating set. A natural order exists on the ring of integers or on the polynomial ring \({\mathcal Q}(x)\); the degree of the polynomial decides the order in \({\mathcal Q}(x)\). However, even for polynomials in two variables there is no natural order a priori (is x^{2} + y greater or smaller than x + y^{2}?). But one can, by hand as it were, put an order on such a ring by saying x ≫ y, where ≫ is an order, called the lexicographical order. We follow this type of order, \({{\mathcal D}_1} \gg {{\mathcal D}_2} \gg {{\mathcal D}_3}\) and ordering polynomials by considering their highest degree terms. It is possible to put different orderings on a given ring which then produce different Gröbner bases. Clearly, a Gröbner basis must have ‘small’ elements so that division is possible and every element of the ideal when divided by the Gröbner basis elements leaves zero remainder, i.e., every element modulo the Gröbner basis reduces to zero.
This Gröbner basis of the ideal \({\mathcal U}\) is then used to obtain the generators for the module of syzygies. Note that although the Gröbner basis depends on the order we choose among the \({{\mathcal D}_k}\), the module itself is independent of the order [3].
4.5 Generating set for the module of syzygies
The generating set for the module is obtained by further following the procedure in the literature [3, 29]. The details are given in Appendix A, specifically for our case. We obtain seven generators for the module. These generators do not form a minimal set and there are relations between them; in fact this method does not guarantee a minimum set of generators. These generators can be expressed as linear combinations of α, β, γ, ζ and also in terms of X^{(1)}, X^{(2)}, X^{(3)}, X^{(4)} given below in Eq. (31). The importance in obtaining the seven generators is that the standard theorems guarantee that these seven generators do in fact generate the required module. Therefore, from this proven set of generators we can check whether a particular set is in fact a generating set. We present several generating sets below.
Another set of generators are just α, β, γ, and ζ. This can be checked using Macaulay 2, or one can relate α, β, γ, and ζ to the generators X^{(A)}, A = 1, 2, 3, 4, by polynomial matrices. In Appendix B, we express the seven generators we obtained following the literature, in terms of α, β, γ, and ζ. Also we express α, β, γ, and ζ in terms of X^{(A)}. This proves that all these sets generate the required module of syzygies.
The question now arises as to which set of generators we should choose which facilitates further analysis. The analysis is simplified if we choose a smaller number of generators. Also we would prefer low degree polynomials to appear in the generators so as to avoid cancellation of leading terms in the polynomials. By these two criteria we may choose X^{(A)} or α, β, γ, ζ. However, α, β, γ, ζ possess the additional property that this set is left invariant under a cyclic permutation of indices 1, 2, 3. It is found that this set is more convenient to use because of this symmetry.
4.6 Canceling optical bench motion noise
4.7 Physical interpretation of the TDI combinations
By using the four generators, it is possible to construct several other interferometric combinations, such as the unequalarm Michelson (X, Y, Z), the Beacons (P, Q, R), the Monitors (E, F, G), and the Relays (U, V, W). Contrary to the Sagnac combinations, these only use four of the six data combinations η_{ i }, η′_{ i }. For this reason they have obvious utility in the event of selected subsystem failures [15].
5 TimeDelay Interferometry with Moving Spacecraft
The rotational motion of the LISA array results in a difference of the light travel times in the two directions around a Sagnac circuit [44, 7]. Two time delays along each arm must be used, say L′_{ i } and L_{ i } for clockwise or counterclockwise propagation as they enter in any of the TDI combinations. Furthermore, since L_{ i } and L′_{ i } not only differ from one another but can be time dependent (they “flex”), it was shown that the “first generation” TDI combinations do not completely cancel the laser phase noise (at least with present laser stability requirements), which can enter at a level above the secondary noises. For LISA, and assuming \({\dot L_i} \simeq \;10{\rm{m/s}}\) [21], the estimated magnitude of the remaining frequency fluctuations from the laser can be about 30 times larger than the level set by the secondary noise sources in the center of the frequency band. In order to solve this potential problem, it has been shown that there exist new TDI combinations that are immune to first order shearing (flexing, or constant rate of change of delay times). These combinations can be derived by using the timedelay operators formalism introduced in the previous Section 4, although one has to keep in mind that now these operators no longer commute [58].
5.1 The unequalarm Michelson
5.2 The Sagnac combinations
In the above Section 5.1, we have used the same symbol X for the unequalarm Michelson combination for both the rotating (i.e., constant delay times) and stationary cases. This emphasizes that, for this TDI combination (and, as we will see below, also for all the combinations including only four links) the forms of the equations do not change going from systems at rest to the rotating case. One needs only distinguish between the timeofflight variations in the clockwise and counterclockwise senses (primed and unprimed delays).
In the case of the Sagnac variables (α, β, γ, ζ), however, this is not the case as it is easy to understand on simple physical grounds. In the case of α for instance, light originating from spacecraft 1 is simultaneously sent around the array on clockwise and counterclockwise loops, and the two returning beams are then recombined. If the array is rotating, the two beams experience a different delay (the Sagnac effect), preventing the noise ϕ_{1} from canceling in the α combination.
In the case of ζ, however, the rotation of the array breaks the symmetry and therefore its uniqueness. However, there still exist three generalized TDI lasernoisefree data combinations that have properties very similar to ζ, and which can be used for the same scientific purposes [54]. These combinations, which we call (ζ_{1}, ζ_{2}, ζ_{3}), can be derived by applying again our timedelay operator approach.
If the delaytimes also change with time, the perfect cancellation of the laser noises is no longer achieved in the (ζ_{1}, ζ_{2}, ζ_{3}) combinations. However, it has been shown in [58] that the magnitude of the residual laser noises in these combinations are significantly smaller than the LISA secondary system noises, making their effects entirely negligible.
The expressions for the Monitor, Beacon, and Relay combinations, accounting for the rotation and flexing of the LISA array, have been derived in the literature [58] by applying the timedelay iterative procedure highlighted in this section. The interested reader is referred to that paper for details.
5.3 Algebraic approach to secondgeneration TDI
In this subsection we present a mathematical formulation of the “secondgeneration” TDI, which generalizes the one presented in Section 4 for stationary LISA. Although a full solution as in the case of stationary LISA seems difficult to obtain, significant progress can be made.
There is, however, a case in between the 1st and 2nd generation TDI, called modified firstgeneration TDI, in which only the Sagnac effect is considered [7, 44]. In this case the updown links are unequal while the delaytimes remain constant. The mathematical formulation of Section 4 can be extended in a straightforward way where now the six timedelays \({{\mathcal D}_i}\) and \({{\mathcal D}_{{i^\prime}}}\) must be taken into account. The polynomial ring still remains commutative but it is now in six variables. The corresponding module of syzygies can be constructed over this larger polynomial ring [41].
When the operators do not commute, the algebraic problem is far more complex. If we follow on the lines of the commutative case, the first step would be to find a Gröbner basis for the ideal generated by the coefficients appearing in Eq. (69), namely, the set of polynomials \(\{1  {{\mathcal D}_2}{{\mathcal D}_3}{{\mathcal D}_1},{{\mathcal D}_{{2^\prime}}}  {{\mathcal D}_3}{{\mathcal D}_1},({{\mathcal D}_{{3^\prime}}}{{\mathcal D}_3}  1){{\mathcal D}_1},{{\mathcal D}_{{1^\prime}}}{{\mathcal D}_1}  1\} \). Although we may be able to apply noncommutative Gröbner basis methods, the general solution seems quite difficult. However, simplifications are possible because of the inherent symmetries in the problem and so the ring \({\mathcal K}\) can be quotiented by a certain ideal, simplifying the algebraic problem. One then needs to deal with a ‘smaller’ ring, which may be easier to deal with. We describe below how this can achieved with the help of certain commutators.
These vanishing commutators (in the approximation we are working in) can be used to simplify the algebra. We first construct the ideal \({\mathcal U}\) generated by the commutators such as those given by Eq. (73). Then we quotient the ring \({\mathcal K}\) by \({\mathcal U}\), thereby constructing a smaller ring \({\mathcal K}/{\mathcal U} \equiv \bar {\mathcal K}\). This ring is smaller because it has fewer distinct terms in a polynomial. Although, this reduces the complexity of the problem, a full solution to the TDI problem is still lacking.
In the following Section 5.4, we will consider the case where we have only two arms of LISA in operation, that is one arm is nonfunctional. The algebraic problem simplifies considerably and it turns out to be tractable.
5.4 Solutions with one arm nonfunctional
We must envisage the possibility that not all optical links of LISA can be operating at all times for various reasons like technical failure for instance or even the operating costs. An analysis covering the scientific capabilities achievable by LISA in the eventuality of loosing one and two links has been discussed in [62]. Here we obtain the TDI combinations when one entire arm becomes dysfunctional. See [11] for a full discussion. The results of this section are directly usable by the eLISA/NGO mission.
If we can solve this equation for q_{1}, q′_{1} then the full polynomial vector can be obtained because \(q_2^\prime = {q_1}{{\mathcal D}_3}\) and \({q_3} = q_1^\prime{{\mathcal D}_{{2^\prime}}}\). It is clear that solutions are of the Michelson type. Also notice that the coefficients of this equation has the operators \(a = {{\mathcal D}_3}{{\mathcal D}_{{3^\prime}}}\) and \(b = {{\mathcal D}_{{2^\prime}}}{{\mathcal D}_2}\) occurring in them. So the solutions q_{1}, q′_{1} too will be in terms of a and b only. Physically, the operators a and b correspond to round trips.
What we would like to emphasize is that there are more solutions of this type — in fact there are infinite number of such solutions. The solutions are based on vanishing of commutators. In [11] such commutators are enumerated and for each such commutator there is a corresponding solution. Further an algorithm is given to construct such solutions.
Higher degree solutions can be constructed. An iterative algorithm has been described in [11] for this purpose. The degrees of the commutators are in multiples of 4. If we call the degree of the commutators as 4 where n = 1, 2, …, then the solutions q_{1}, q′_{1} are of degree 4n − 1. The cases mentioned above, correspond to n = 1 and n = 2. The general formula for the number of commutators of degree 4n is ^{2n−1}C_{n−1}. So at n = 3 we have 10 commutators and so as many solutions q_{1}, q′_{1} of degree 11.
These are the degrees of polynomials of q_{1}, q′_{1} in the operators a, b. But for the full polynomial vector, which has q′_{2} and q_{3}, we need to go over to the operators expressed in terms of \({{\mathcal D}_i},{{\mathcal D}_{{i^\prime}}}\). Then the degree of each of the q_{1}, q′_{1} is doubled to 8n − 2, while q′_{2} and q_{3} are each of degree 8n − 1. Thus, for a general value of, the solution contains polynomials of maximum degree 8n − 1 in the timedelay operators.
From the mathematical point of view there is an infinite family of solutions. Note that no claim is made on exhaustive listing of solutions. However the family of solutions is sufficiently rich, because we can form linear combinations of these solutions and they also are solutions.
From the physical point of view, since terms in \(\ddot L\) and \({\dot L^2}\) and higher orders have been neglected, a limit on the degree of the polynomial solutions arises. That is up to certain degree of the polynomials, we can safely assume the commutators to vanish. But as the degree of the polynomials increases it is not possible to neglect these higher order terms any longer and then such a limit becomes important. The limit is essentially set by \(\ddot L\). We now investigate this limit and make a very rough estimate of it. As mentioned earlier, the LISA model in [12] gives \(\ddot L \sim {10^{ 6}}{\rm{m/}}{{\rm{s}}^2}\). From \(\ddot L\) we compute the error in L, namely, \(\Delta L \sim {1 \over 2}\Delta {t^2}\ddot L\). If we allow the error to be no more than say 10 meters, then we find Δt ∼ 4500 s. Since each timedelay is about 16.7 s for LISA, the number of successive timedelays is about 270. This is the maximum degree of the polynomials. This means one can go up to n ≲ 30. If we set the limit more stringently at ΔL ∼ 1 m, then the highest degree of the polynomial reduces to about 80, which means one can go up to n = 10. Thus, there are a large number of TDI observables available to do the physics.

A geometric combinatorial approach was adopted in [61] where several solutions were presented. Our approach is algebraic where the operations are algebraic operations on strings of operators. The algebraic approach has the advantage of easy manipulation of data streams, although some geometrical insight could be at a premium.

Another important aspect is the GW response of such TDI observables. The GW response to a TDI observable may be calculated in the simplest way by assuming equal arms (the possible differences in lengths would be sensitive to frequencies outside the LISA bandwidth). This leads in the Fourier domain to polynomials in the same phase factor from which the signal to noise ratio can be found. A comprehensive and generic treatment of the responses of secondgeneration TDI observables can be found in [30].
6 Optimal LISA Sensitivity
All the above interferometric combinations have been shown to individually have rather different sensitivities [15], as a consequence of their different responses to gravitational radiation and system noises. Since LISA has the capability of simultaneously observing a gravitationalwave signal with many different interferometric combinations (all having different antenna patterns and noises), we should no longer regard LISA as a single detector system but rather as an array of gravitationalwave detectors working in coincidence. This suggests that the LISA sensitivity could be improved by optimally combining elements of the TDI space.
In order now to identify the eigenvalues of the matrix C^{−1} · A, we first notice that the 3 × 3 matrix A has rank 1. This implies that the matrix C^{−1} · A has also rank 1, as it is easy to verify. Therefore two of its three eigenvalues are equal to zero, while the remaining nonzero eigenvalue represents the solution we are looking for.
 1.
Among all possible interferometric combinations LISA will be able to synthesize with its four generators α, β, γ, ζ, the particular combination giving maximum signaltonoise ratio can be obtained by using only three of them, namely (α, β, γ).
 2.
The expression of the optimal signaltonoise ratio given by Eq. (89) implies that LISA should be regarded as a network of three interferometer detectors of gravitational radiation (of responses (α, β, γ)) working in coincidence [20, 40].
6.1 General application
In order to calculate the sensitivity corresponding to the expression of the optimal signaltonoise ratio, we have proceeded similarly to what was done in [2, 15], and described in more detail in [56]. We assume an equalarm LISA (L = 16.67 s), and take the onesided spectra of proof mass and aggregate opticalpathnoises (on a single link), expressed as fractional frequency fluctuation spectra, to be \(S_y^{{\rm{proof}}\,{\rm{mass}}} = 2.5 \times {10^{ 48}}{[f/1\,{\rm{Hz}}]^{ 2}}\,{\rm{H}}{{\rm{z}}^{ 1}}\) and \(S_y^{{\rm{optical}}\,{\rm{path}}} = 1.8 \times {10^{ 37}}{[f/1\,{\rm{Hz}}]^2}\,{\rm{H}}{{\rm{z}}^{ 1}}\), respectively (see [15, 5]). We also assume that aggregate optical path noise has the same transfer function as shot noise.
The optimum SNR is the square root of the sum of the squares of the SNRs of the three “orthogonal modes” (A, E, T). To compare with previous sensitivity curves of a single LISA Michelson interferometer, we construct the SNRs as a function of Fourier frequency for sinusoidal waves from sources uniformly distributed on the celestial sphere. To produce the SNR of each of the (A, E, T) modes we need the gravitationalwave response and the noise response as a function of Fourier frequency. We build up the gravitationalwave responses of the three modes (A, E, T) from the gravitationalwave responses of (α, β, γ). For 7000 Fourier frequencies in the ∼ 10^{−4} Hz to ∼ 1 Hz LISA band, we produce the Fourier transforms of the gravitationalwave response of (α, β, γ) from the formulas in [2, 56]. The averaging over source directions (uniformly distributed on the celestial sphere) and polarization states (uniformly distributed on the Poincaré sphere) is performed via a Monte Carlo method. From the Fourier transforms of the (α, β, γ) responses at each frequency, we construct the Fourier transforms of (A, E, T). We then square and average to compute the meansquared responses of (A, E, T) at that frequency from 10^{4} realizations of (source position, polarization state) pairs.
We adopt the following terminology: We refer to a single element of the module as a data combination, while a function of the elements of the module, such as taking the maximum over several data combinations in the module or squaring and adding data combinations belonging to the module, is called as an observable. The important point to note is that the laser frequency noise is also suppressed for the observable although it may not be an element of the module.
6.2 Optimization of SNR for binaries with known direction but with unknown orientation of the orbital plane
Binaries are important sources for LISA and therefore the analysis of such sources is of major importance. One such class is of massive or supermassive binaries whose individual masses could range from 10^{3}M_{⊙} to 10^{8}M_{⊙} and which could be up to a few Gpc away. Another class of interest are known binaries within our own galaxy whose individual masses are of the order of a solar mass but are just at a distance of a few kpc or less. Here the focus will be on this latter class of binaries. It is assumed that the direction of the source is known, which is so for known binaries in our galaxy. However, even for such binaries, the inclination angle of the plane of the orbit of the binary is either poorly estimated or unknown. The optimization problem is now posed differently: The SNR is optimized after averaging over the polarizations of the binary signals, so the results obtained are optimal on the average, that is, the source is tracked with an observable which is optimal on the average [40]. For computing the average, a uniform distribution for the direction of the orbital angular momentum of the binary is assumed.
When the binary masses are of the order of a solar mass and the signal typically has a frequency of a few mHz, the GW frequency of the binary may be taken to be constant over the period of observation, which is typically taken to be of the order of an year. A complete calculation of the signal matrix and the optimization procedure of SNR is given in [39]. Here we briefly mention the main points and the final results.
 1.
The Michelson combination X (faint solid curve).
 2.
The observable obtained by taking the maximum sensitivity among X, Y, and Z for each direction, where Y and Z are the Michelson observables corresponding to the remaining two pairs of arms of LISA [2]. This maximum is denoted by max[X, Y, Z] (dashdotted curve) and is operationally given by switching the combinations X, Y, Z so that the best sensitivity is achieved.
 3.
The eigencombination \({\vec \upsilon _ +}\) which has the best sensitivity among all data combinations (dashed curve).
 4.
The network observable (solid curve).
7 Experimental Aspects of TDI
It is clear that the suppression of the laser phase fluctuations by more than nine orders of magnitude with the use of TDI is a very challenging experimental task. It requires developing and building subsystems capable of unprecedented accuracy and precision levels, and test their endtoend performance in a laboratory environment that naturally precludes the availability of 5 × 10^{6} km delay lines! In what follows we will address some aspects related to the experimental implementation of TDI, and derive the performance specifications for the subsystems involved. We will not address, however, any of the experimental aspects related to the verification of TDI in a laboratory environment. For that, we refer the interested reader to de Vine et al. [8, 32], Spero et al. [48], and Mitryk et al. [33].
 1.
accurate knowledge of the time shifts, L′_{ i }(t), L_{ i }(t) i = 1, 2, 3, to be applied to the heterodyne measurements s_{ i }(t), s′_{ i }(t), τ_{ i }(t), τ′_{ i }(t) i = 1, 2, 3;
 2.
accurate synchronization among the three clocks onboard the three spacecraft as these are used for timestamping the recorded heterodyne phase measurements;
 3.
sampling time stability (i.e., clock stability) for successfully suppressing the laser noise to the desired level;
 4.
an accurate reconstruction algorithm of the phase measurements corresponding to the required time delays as these in general will not be equal to integer multiples of the sampling time;
 5.
a phase meter capable of a very large dynamic range in order to suppress the laser noise to the required level while still preserving the phase fluctuations induced by a gravitationalwave signal in the TDI combinations.
7.1 Timedelays accuracies
Armlength accuracies at the centimeters level have already been demonstrated in the laboratory [16, 50, 64, 26], making us confident that the required level of timedelays accuracy will be available.
In relation to the accuracies derived above, it is interesting to calculate the time scales during which the armlengths will change by an amount equal to the accuracies themselves. This identifies the minimum time required before updating the armlength values in the TDI combinations.
One way to address this problem is to treat the delays in the TDI combinations as parameters to be determined by a nonlinear leastsquares procedure, in which the minimum of the minimizer is achieved at the correct delays since that the laser noise will exactly cancel there in the TDI combinations. Such a technique, which was named timedelay interferometric ranging (TDIR) [60], requires a starting point in the timedelays space in order to implement the minimization, and it will work quite effectively jointly with the ranging data available onboard.
7.2 Clocks synchronization
The effectiveness of the TDI data combinations requires the clocks onboard the three spacecraft to be synchronized. In what follows we will identify the minimum level of offsynchronization among the clocks that can be tolerated. In order to proceed with our analysis we will treat one of the three clocks (say the clock onboard spacecraft 1) as the master clock defining the time for LISA, while the other two to be synchronized to it.
The relativistic (Sagnac) timedelay effect due to the fact that the LISA trajectory is a combination of two rotations, each with a period of one year, will have to be accounted for in the synchronization procedure and, as has already been discussed earlier, will be accounted for within the secondgeneration formulation of TDI.
We find that the righthand side of the inequality given by Eq. (110) reaches its minimum of about 47 nanoseconds at the Fourier frequency f_{min} = 1.0 × 10^{−4} Hz. This means that clocks synchronized at a level of accuracy significantly better than 47 nanoseconds will result into a residual laser noise that is much smaller than the secondary noise sources entering into the ζ combination.
As a final note, a required clock synchronizations of about 40 ns derived in this section translates into a ranging accuracy of 12 meters, which has been experimentally shown to be easily achievable [16, 50, 64, 26].
7.3 Clocks timing jitter
The sampling times of all the measurements needed for synthesizing the TDI combinations will not be constant, due to the intrinsic timing jitters of the onboard measuring system. Among all the subsystems involved in the data measuring process, the onboard clock is expected to be the dominant source of time jitter in the sampled data. Presently existing space qualified clocks can achieve an Allan standard deviation of about 10^{−13} for integration times from 1 to 10 000 seconds. This timing stability translates into a time jitter of about 10^{−13} seconds over a period of 1 second. A perturbation analysis including the three sampling time jitters due to the three clocks shows that any laser phase fluctuations remaining in the four TDI generators will also be proportional to the sampling time jitters. Since the latter are approximately four orders of magnitude smaller than the armlength and clocks synchronization accuracies derived earlier, we conclude that the magnitude of laser noise residual into the TDI combinations due to the sampling time jitters is negligible.
7.4 Sampling reconstruction algorithm
The derivations of the timedelays and clocks synchronization accuracies highlighted earlier presumed the availability of the phase measurement samples at the required timedelays. Since this condition will not be true in general, as the timedelays used by the TDI combinations will not be equal to integermultiples of the sampling time, with a sampling rate of, let us say, 10 Hz, the time delays could be off their correct values by a tenth of a second, way more than the 10 nanoseconds timedelays and clocks synchronization accuracies estimated above.
Earlier suggestions [27] for addressing this problem envisioned sampling the data at veryhigh rates (perhaps of the order of hundreds of MHz), so reducing the additional error to the estimated timedelays to a few nanoseconds. Although in principle such a solution would allow us to suppress the residual laser noise to the required level, it would create an insurmountable problem for transmitting the science data to the ground due to the limited spacetoground data rates.
An alternate scheme for obtaining the phase measurement points needed by TDI [59] envisioned sampling the phase measurements at the required delayed times. This scheme naturally requires knowledge of the timedelays and synchronization of the clocks at the required accuracy levels during data acquisition. Although such a procedure could be feasible in principle, it would still leave open the possibility of irreversible corruption of the TDI combinations in the eventuality of performance degradation in the ranging and clock synchronization procedures.
Given that the data will need to be sampled at a rate of 10 Hz, an alternative options is to implement an interpolation scheme for reconstructing the required data points from the sampled measurements. An analysis [59] based on the implementation of the truncated Shannon [47] formula, however, showed that several months of data were required in order to reconstruct the phase samples at the estimated timedelays with a sufficiently high accuracy. This conclusion implied that several months (at the beginning and end) of the entire data records measured by LISA would be of no use, resulting into a significant mission science degradation.
Although the truncated Shannon formula was proved to be impracticable [59] for reconstructing phase samples at the required timedelays, it was then recognized that [46] a more efficient and accurate interpolation technique [31] could be adopted. In what follows, we provide a brief account of this data processing technique, which is known as “fractionaldelay filtering” (FDF).
If, however, we give up on the requirement of minimizing the error in the leastsquares sense and replace it with a minimax criterion error applied to the absolute value of the difference between the ideal transfer function (i.e., \({e^{2\pi ifD/{f_s}}}\)) and a modified sincfunction, we will be able to achieve a rapid convergence while suppressing the ringing effects associated with the sinc function.
7.5 Data digitization and bitaccuracy requirement
It has been shown [59] that the maximum of the ratio between the amplitudes of the laser and the secondary phase fluctuations occurs at the lower end of the LISA bandwidth (i.e., 0.1 mHz) and it is equal to about 10^{10}. This corresponds to the minimum dynamic range for the phasemeters to correctly measure the laser fluctuations and the weaker (gravitationalwave) signals simultaneously. An additional safety factor of ≈ 10 should be sufficient to avoid saturation if the noises are well described by Gaussian statistics. In terms of requirements on the digital signal processing subsystem, this dynamic range implies that approximately 36 bits are needed when combining the signals in TDI, only to bridge the gap between laser frequency noise and the other noises and gravitationalwave signals. More bits might be necessary to provide enough information to efficiently filter the data when extracting weak gravitationalwave signals embedded into noise.
The phasemeters will be the onboard instrument that will perform the phase measurements containing the gravitational signals. They will also need to simultaneously measure the timedelays to be applied to the TDI combinations via ranging tones overimposed on the laser beams exchanged by the spacecraft. And they will need to have the capability of simultaneously measure additional sideband tones that are required for the calibration of the onboard UltraStable Oscillator used in the downconversion of heterodyned carrier signal [57, 27].
Work toward the realization of a phasemeter capable of meeting these very stringent performance and operational requirements has aggressively been performed both in the United States and in Europe [43, 23, 22, 6, 63], and we refer the reader interested in the technical details associated with the development studies of such device to the above references and those therein.
8 Concluding Remarks
In this article we have summarized the use of TDI for canceling the laser phase noise from heterodyne phase measurements performed by a constellation of three spacecraft tracking each other along arms of unequal length. Underlying the TDI technique is the mathematical structure of the theory of Gröbner basis and the algebra of modules over polynomial rings. These methods have been motivated and illustrated with the simple example of an unequalarm interferometer in order to give a physical insight of TDI. Here, these methods have been rigorously applied to the idealized case of a stationary interferometer such as the LISA mission. This allowed us to derive the generators of the module from which the entire TDI data set can be obtained; they can be extended in a straightforward way to more than three spacecraft for possible future mission concepts. The stationary LISA case was used as a propaedeutical introduction to the physical motivation of TDI, and for further extending it to the realistic LISA configuration of freefalling spacecraft orbiting around the Sun. The TDI data combinations canceling laser phase noise in this general case are referred to as secondgeneration TDI, and they contain twice as many terms as their corresponding firstgeneration combinations valid for the stationary configuration.
As a data analysis application we have shown that it is possible to identify specific TDI combinations that will allow LISA to achieve optimal sensitivity to gravitational radiation [38, 40, 39]. The resulting improvement in sensitivity over that of an unequalarm Michelson interferometer, in the case of monochromatic signals randomly distributed over the celestial sphere and of random polarization, is nonnegligible. We have found this to be equal to a factor of \(\sqrt 2 \) in the lowpart of the frequency band, and slightly more than \(\sqrt 3 \) in the highpart of the LISA band. The SNR for binaries whose location in the sky is known, but their polarization is not, can also be optimized, and the degree of improvement depends on the location of the source in the sky.
We also addressed several experimental aspects of TDI, and emphasized that it has already been successfully tested experimentally [8, 32, 48, 33].
As of the writing of this second edition of our living review article, it is very gratifying to see how much TDI has matured since the publishing of its first version. The purpose of this second edition review of TDI was to provide the basic mathematical tools and knowledge of the current experimental results needed for working on future TDI projects. We hope to have accomplished this goal, and that others will be stimulated to work in this fascinating field of research.
Footnotes
 1.
It should be noticed that the optical bench design shown in Figure 4 is one the earlier ones proposed for the LISA mission, and it represents one of the possible configurations for integrating the onboard dragfree system with the TDI measurements. Although other optical bench designs will result into different interproofmass phase measurements, they can be accommodated within TDI [37].
 2.
A module is an Abelian group over a ring as contrasted with a vector space which is an Abelian group over a field. The scalars form a ring and just like in a vector space, scalar multiplication is defined. However, in a ring the multiplicative inverses do not exist in general for the elements, which makes all the difference!
Notes
Acknowledgement
S.V.D. acknowledges support from IFCPAR, Delhi, India under which the work was carried out in collaboration with J.Y. Vinet. S.V.D. also thanks IUCAA for a visiting professorship during which this article was updated. This research was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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