We first consider the simpler case, where we ignore the optical-bench 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, beam-splitters, 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 and . The scheme we describe here for laser phase noise can be extended in a straight-forward 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 (inter-spacecraft), 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 , , . 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 six-tuples of polynomials which satisfy the laser noise cancellation constraints. These polynomial tuples form a module2 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 , , 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 . 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 six-tuples 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.
We now only have six data streams and , where . These can be regarded as 3 component vectors and , respectively. The six data streams with terms containing only the laser frequency noise are
Note that we have intentionally excluded from the data additional phase fluctuations due to the GW signal, and noises such as the optical-path noise, proof-mass 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.
The goal of the analysis is to add suitably delayed beams together so that the laser frequency noise terms add up to zero. This amounts to seeking data combinations that cancel the laser frequency noise. In the notation/formalism that we have invoked, the delay is obtained by applying the operators to the beams and . A delay of is represented by the operator acting on the data, where , , and are integers. In general, a polynomial in , which is a polynomial in three variables, applied to, say, combines the same data stream with different time-delays of the form . This notation conveniently rephrases the problem. One must find six polynomials say , , , such that
It is useful to express Eq. (16*) in matrix form. This allows us to obtain a matrix operator equation whose solutions are and , where and are written as column vectors. We can similarly express , , as column vectors , , , respectively. In matrix form Eq. (16*) becomes17*) becomes
The use of commutative algebra is very conveniently illustrated with the help of the simpler example of the unequal-arm 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 unequal-arm interferometer, but here we will describe the same method but in terms of the delay operators that we have introduced.
Let denote the laser phase noise entering the laser cavity as shown in Figure 5*. Consider this light making a round trip around arm 1 whose length we take to be . If we interfere this phase with the incoming light we get the phase , where2. It was first proposed by Tinto and Armstrong in [53*].
The cancellation of laser frequency noise becomes obvious from the operator algebra in the following way. In the operator notation,5*.
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 Gröbner basis which is in a sense the generalization of the notion of the greatest common divisor (gcd). Since LISA has three spacecraft and six inter-spacecraft beams, the problem of the unequal-arm 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 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.
Equation (21*) has non-trivial 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 is given by . The solution is described by and . The solutions and are obtained from by cyclically permuting the indices of , , and . 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 six-tuple 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 six-tuple , can be multiplied by polynomials in , , (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 , , . 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 six-tuples of polynomials. The module of six-tuples 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 with 1 in the -th place and 0 everywhere else; 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.
The problem at hand is of finding the generators of this submodule, i.e., any element of the submodule should be expressible as a linear combination of the generating set. In this way the generators are capable of spanning the full submodule or generating the submodule. In order to achieve our goal, we rewrite Eq. (21*) explicitly component-wise:
The first step is to use Gaussian elimination to obtain and in terms of ,28*) amounts to solving the problem since the remaining polynomials , have been expressed in terms of , , , in Eq. (27*). Note that we cannot carry on the Gaussian elimination process any further, because none of the polynomial coefficients appearing in Eq. (28*) have an inverse in the ring.
We will assume that the polynomials have rational coefficients, i.e., the coefficients belong to , the field of the rational numbers. The set of polynomials form a ring – the polynomial ring in three variables, which we denote by . The polynomial vector . The set of solutions to Eq. (28*) is just the kernel of the homomorphism , where the polynomial vector is mapped to the polynomial . Thus, the solution space is a submodule of . 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.
There are several ways to look at the theory of Gröbner basis. One way is the following: Suppose we are given polynomials in one variable over say and we would like to know whether another polynomial belongs to the ideal generated by the ’s. A good way to decide the issue would be to first compute the gcd of , , …, and check whether is a multiple of . One can achieve this by doing the long division of by and checking whether the remainder is zero. All this is possible because 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 . 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 ; the degree of the polynomial decides the order in . However, even for polynomials in two variables there is no natural order a priori (is greater or smaller than ?). But one can, by hand as it were, put an order on such a ring by saying , where is an order, called the lexicographical order. We follow this type of order, 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.
In the literature, there exists a well-known algorithm called the Buchberger algorithm, which may be used to obtain the Gröbner basis for a given set of polynomials in the ring. So a Gröbner basis of can be obtained from the generators given in Eq. (29*) using this algorithm. It is essentially again a generalization of the usual long division that we perform on univariate polynomials. More conveniently, we prefer to use the well known application Mathematica. Mathematica yields a 3-element Gröbner basis for :29*) are linear combinations of the polynomials in and hence generates . One also observes that the elements look ‘small’ in the order mentioned above. However, one can satisfy oneself that is a Gröbner basis by using the standard methods available in the literature. One method consists of computing the S-polynomials (see Appendix A) for all the pairs of the Gröbner basis elements and checking whether these reduce to zero modulo .
This Gröbner basis of the ideal 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 , the module itself is independent of the order [3*].
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 , , , and also in terms of , , , 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.
Alternatively, we may use a software package called Macaulay 2 which directly calculates the generators given the Eqs. (26*). Using Macaulay 2, we obtain six generators. Again, Macaulay’s algorithm does not yield a minimal set; we can express the last two generators in terms of the first four. Below we list this smaller set of four generators in the order :
Another set of generators which may be useful for further work is a Gröbner basis of a module. The concept of a Gröbner basis of an ideal can be extended to that of a Gröbner basis of a submodule of where is a field, since a module over the polynomial ring can be considered as generalization of an ideal in a polynomial ring. Just as in the case of an ideal, a Gröbner basis for a module is a generating set with special properties. For the module under consideration we obtain a Gröbner basis using Macaulay 2:
Another set of generators are just , , , and . This can be checked using Macaulay 2, or one can relate , , , and to the generators , , by polynomial matrices. In Appendix B, we express the seven generators we obtained following the literature, in terms of , , , and . Also we express , , , and in terms of . 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 or , , , . However, , , , possess the additional property that this set is left invariant under a cyclic permutation of indices . It is found that this set is more convenient to use because of this symmetry.
There are now twelve Doppler data streams which have to be combined in an appropriate manner in order to cancel the noise from the laser as well as from the motion of the optical benches. As in the previous case of canceling laser phase noise, here too, we keep the relevant terms only, namely those terms containing laser phase noise and optical bench motion noise. We then have the following expressions for the four data streams on spacecraft 1:36*) from Eq. (35*), we can rewriting the resulting expression (and those obtained from it by permutation of the spacecraft indices) in the following form: 16*) for the simpler configuration with only three lasers, analyzed in the previous Sections 4.1 to 4.4, we see that they are identical in form.
It is important to notice that the four interferometric combinations , which can be used as a basis for generating the entire TDI space, are actually synthesized Sagnac interferometers. This can be seen by rewriting the expression for , for instance, in the following form,43*) contains a combination of one-way measurements describing a light beam propagating clockwise around the array, while the other terms in the second square-bracket give the equivalent of another beam propagating counter-clockwise around the constellation.
Contrary to , , and , can not be visualized as the difference (or interference) of two synthesized beams. However, it should still be regarded as a Sagnac combination since there exists a time-delay relationship between it and , , and [2*]:symmetrized Sagnac combination.
By using the four generators, it is possible to construct several other interferometric combinations, such as the unequal-arm Michelson , the Beacons , the Monitors , and the Relays . Contrary to the Sagnac combinations, these only use four of the six data combinations , . For this reason they have obvious utility in the event of selected subsystem failures [15*].
These observables can be written in terms of the Sagnac observables in the following way,6*.  and [45*]) that this combination can be visualized as the difference of two sums of phase measurements, each corresponding to a specific light path from a laser onboard spacecraft 1 having phase noise . The first square-bracket term in Eq. (46*) represents a synthesized light-beam transmitted from spacecraft 1 and made to bounce once at spacecraft 2 and 3, respectively. The second square-bracket term instead corresponds to another beam also originating from the same laser, experiencing the same overall delay as the first beam, but bouncing off spacecraft 3 first and then spacecraft 2. When they are recombined they will cancel the laser phase fluctuations exactly, having both experienced the same total delay (assuming stationary spacecraft). The combinations should therefore be regarded as the response of a zero-area Sagnac interferometer.