In order to derive the new, “flex-free” TDI combinations we will start by taking specific combinations of the one-way data entering in each of the expressions derived in the previous Section 4. Note, however, that now the expressions for the -measurements assume the following form38*).
The new TDI combinations are chosen in such a way so as to retain only one of the three noises , , if possible. In this way we can then implement an iterative procedure based on the use of these basic combinations and of time-delay operators, to cancel the laser noises after dropping terms that are quadratic in or linear in the accelerations. This iterative time-delay method, to first order in the velocity, is illustrated abstractly as follows. Given a function of time , time delay by is now denoted either with the standard comma notation [2*] or by applying the delay operator introduced in the previous Section 4,[45*], to remind us that we consider moving arrays. When the sum of these corrections to the terms of a data combination vanishes, the combination is called flex-free.
Also, note that each delay operator has a unique inverse , whose expression can be derived by requiring that , and neglecting quadratic and higher order velocity terms. Its action on a time series is
The unequal-arm Michelson combination relies on the four measurements , , , and . Note that the two combinations , represent the two synthesized two-way data measured onboard spacecraft 1, and can be written in the following form (see Eqs. (47*), (48*), and (49*) for deriving the following synthesized two-way measurements)54*) and (55*) it is easy to derive the following expression for the unequal-arm interferometric combination which eliminates : 56*): 57*) and (58*), we can use the delay technique again to finally derive the following expression for the new unequal-arm Michelson combination that accounts for the flexing effect:
In the above Section 5.1, we have used the same symbol for the unequal-arm 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 time-of-flight variations in the clockwise and counter-clockwise senses (primed and unprimed delays).
In the case of the Sagnac variables , however, this is not the case as it is easy to understand on simple physical grounds. In the case of for instance, light originating from spacecraft 1 is simultaneously sent around the array on clockwise and counter-clockwise 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 from canceling in the combination.
In order to find the solution to this problem let us first rewrite in such a way to explicitly emphasize what it does: attempts to remove the same fluctuations affecting two beams that have been made to propagated clockwise and counter-clockwise around the array,60*) we find that they are equal to 62*) we finally get the expression for the Sagnac combination that is unaffected by laser noise in presence of rotation, same circuit. For LISA, the remaining laser phase noises in , , are several orders of magnitude below the secondary noises.
In the case of , however, the rotation of the array breaks the symmetry and therefore its uniqueness. However, there still exist three generalized TDI laser-noise-free data combinations that have properties very similar to , and which can be used for the same scientific purposes . These combinations, which we call , can be derived by applying again our time-delay operator approach.
If the delay-times also change with time, the perfect cancellation of the laser noises is no longer achieved in the 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 time-delay iterative procedure highlighted in this section. The interested reader is referred to that paper for details.
In this subsection we present a mathematical formulation of the “second-generation” 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 first-generation TDI, in which only the Sagnac effect is considered [7, 44]. In this case the up-down links are unequal while the delay-times remain constant. The mathematical formulation of Section 4 can be extended in a straight-forward way where now the six time-delays and 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 .
When the arms are allowed to flex, that is, the operators themselves are functions of time, the operators no longer commute. One must then resort to non-commutative algebra. We outline the procedure below. Since lot of the discussion has been covered in the previous subsections we just describe the algebraic formulation. The equation Eq. (26*) generalizes in two ways: (1) we need to consider now six operators and , and (2) we need to take into account the non-commutativity of the operators – the order of the operators is important. Accordingly Eq. (26*) generalizes to,left by any polynomial in , then it is also a solution to the Eqs. (68*) and, therefore, in the module – the module of noise-free TDI observables. For details see .
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 . Although we may be able to apply non-commutative 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 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.
In general, a commutator of two operators is defined as the operator . In our situation and are strings of operators built up of the operators . For example, in the Sagnac combination the following commutator [12*] occurs:64*) and which happens to be small. In the context of the reasonably optimized model of LISA say given in [12*], the residual noise term is small. For this model, we have and thus, even if one considers say 20 successive optical paths, that is, about of light travel time, . This is well below few meters and thus can be neglected in the residual laser noise computation. Moreover, terms (and higher order) can be dropped since they are of the order of (they come with a factor ), which is much smaller than 1 part in , which is the level at which the laser frequency noise must be canceled. Thus, we keep terms only to the first degree in and also neglect higher time derivative terms in .
The result for the Sagnac combination can be generalized. In order to simplify notation we write or for the time-delay operators, where and , that is, or are any of the operators . Then a commutator is:
Note that the LHS acts on , while the right-hand side multiplies at an appropriately delayed time. For the Sagnac combination this is readily seen from Eq. (64*). Also the notation on the RHS is obvious: if for some , we have, say, then and so on; the same holds for for a given . From this equation it immediately follows that if the operators are a permutation of the operators , then the commutator,same links, except that the nodes (spacecraft) of the links are taken in different orders. If the armlengths were constant, the pathlengths would be identical and the commutator would be zero. But here, by neglecting terms and those of higher orders, we have effectively assumed that s are constant, so the increments also cancel out, resulting in a vanishing commutator.
These vanishing commutators (in the approximation we are working in) can be used to simplify the algebra. We first construct the ideal generated by the commutators such as those given by Eq. (73*). Then we quotient the ring by , thereby constructing a smaller ring . 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.
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 . 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.
We arbitrarily choose the non-functional arm to be the one connecting S/C 2 and S/C 3. This means from our labeling that the polynomials are now restricted to only the four operators and we can set the polynomials . This simplifies the Eqs. (68*) considerably because the last two equations reduce to and . Substituting these in the first equation gives just one equation:
If we can solve this equation for then the full polynomial vector can be obtained because and . It is clear that solutions are of the Michelson type. Also notice that the coefficients of this equation has the operators and occurring in them. So the solutions too will be in terms of and only. Physically, the operators and correspond to round trips.75*), , which is a commutator that vanishes since is a permutation of . Thus, it is an element of and Eq. (75*) is a solution (over the quotient ring).
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.
We briefly mention some results given in [11*]. We start with the solutions of Eq. (75*). Note that these are of degree 3 in and . The commutator corresponding to this solution is and is of degree 4. There is only one such commutator at degree 4 and therefore one solution. The next higher degree solutions are found when the commutators have degree 8. The solutions are of degree 7. There are three such commutators at degree 8. We list the solutions and the commutators below. One solution is:
The second solution is:
The third solution corresponds to the commutator . This solution is given by:
Higher degree solutions can be constructed. An iterative algorithm has been described in  for this purpose. The degrees of the commutators are in multiples of 4. If we call the degree of the commutators as where , then the solutions are of degree . The cases mentioned above, correspond to and . The general formula for the number of commutators of degree is . So at we have 10 commutators and so as many solutions of degree 11.
These are the degrees of polynomials of in the operators . But for the full polynomial vector, which has and , we need to go over to the operators expressed in terms of . Then the degree of each of the is doubled to , while and are each of degree . Thus, for a general value of , the solution contains polynomials of maximum degree in the time-delay 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 and 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 . We now investigate this limit and make a very rough estimate of it. As mentioned earlier, the LISA model in  gives . From we compute the error in , namely, . If we allow the error to be no more than say 10 meters, then we find . Since each time-delay is about 16.7 s for LISA, the number of successive time-delays is about 270. This is the maximum degree of the polynomials. This means one can go up to . If we set the limit more stringently at , then the highest degree of the polynomial reduces to about 80, which means one can go up to . Thus, there are a large number of TDI observables available to do the physics.
Some remarks are in order:
- A geometric combinatorial approach was adopted in  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 second-generation TDI observables can be found in .