5 Time-Delay 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 Li for clockwise or counter-clockwise propagation as they enter in any of the TDI combinations. Furthermore, since Li 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 LΛ™i ≃ 10 mβˆ•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 time-delay operators formalism introduced in the previous Section 4, although one has to keep in mind that now these operators no longer commute [58*].

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 η-measurements assume the following form

η1 = π’Ÿ3Ο•2 − Ο•1 , η1′ = π’Ÿ2′Ο•3 − Ο•1 , (47 ) η2 = π’Ÿ1Ο•3 − Ο•2 , η2′ = π’Ÿ3′Ο•1 − Ο•2 , (48 ) η3 = π’Ÿ2Ο•1 − Ο•3 , η3′ = π’Ÿ1′Ο•2 − Ο•3 , (49 )
where the Ο•i, i = 1,2,3 measurements are as given in Eq. (38*).

The new TDI combinations are chosen in such a way so as to retain only one of the three noises Ο•i, i = 1,2,3, 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 LΛ™βˆ•c 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 Ψ = Ψ(t), time delay by Li is now denoted either with the standard comma notation [2*] or by applying the delay operator π’Ÿi introduced in the previous Section 4,

π’Ÿi Ψ = Ψ,i ≡ Ψ (t − Li (t)). (50 )
We then impose a second time delay Lj(t):
π’Ÿjπ’Ÿi Ψ = Ψ;ij ≡ Ψ (t − Lj (t) − Li(t − Lj(t))) ≃ Ψ (t − Lj (t) − Li(t) + Λ™Li(t)Lj ) Λ™ Λ™ ≃ Ψ,ij + Ψ,ijLiLj . (51 )
A third time delay Lk (t) gives
π’Ÿ π’Ÿ π’Ÿ Ψ = Ψ = Ψ (t − L (t) − L (t − L (t)) − L (t − L (t) − L (t − L (t)))) k j i ;ijk k [ j k i ] k j k ≃ Ψ,ijk + Λ™Ψ,ijk LΛ™i (Lj + Lk) + Λ™LjLk , (52 )
and so on, recursively; each delay generates a first-order correction proportional to its rate of change times the sum of all delays coming after it in the subscripts. Commas have now been replaced with semicolons [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 π’Ÿi has a unique inverse D −1 i, whose expression can be derived by requiring that −1 Di π’Ÿi = I, and neglecting quadratic and higher order velocity terms. Its action on a time series Ψ (t) is

D − 1Ψ(t) ≡ Ψ (t + Li(t + Li)). (53 ) i
Note that this is not like an advance operator one might expect, since it advances not by Li(t) but rather Li(t + Li).

5.1 The unequal-arm Michelson

The unequal-arm Michelson combination relies on the four measurements η 1, η ′ 1, η ′ 2, and η 3. Note that the two combinations η1 + η2′,3, η1′ + η3,2′ 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)

η1 + η2′,3 = (π’Ÿ3π’Ÿ3 ′ − I )Ο•1, (54 ) η ′ + η ′ = (π’Ÿ ′π’Ÿ − I )Ο• , (55 ) 1 3,2 2 2 1
where I is the identity operator. Since in the stationary case any pairs of these operators commute, i.e., π’Ÿi π’Ÿj′ − π’Ÿj′π’Ÿi = 0, from Eqs. (54*) and (55*) it is easy to derive the following expression for the unequal-arm interferometric combination X which eliminates Ο•1:
X = [π’Ÿ2 ′π’Ÿ2 − I](η1 + η2′,3) − [(π’Ÿ3π’Ÿ3 ′ − I )](η1′ + η3,2′) . (56 )
If, on the other hand, the time-delays depend on time, the expression of the unequal-arm Michelson combination above no longer cancels Ο•1. In order to derive the new expression for the unequal-arm interferometer that accounts for “flexing”, let us first consider the following two combinations of the one-way measurements entering into the X observable given in Eq. (56*):
[(η1′ + η3;2′) + (η1 + η2′;3);22′] = [D2 ′D2D3D3 ′ − I]Ο•1 , (57 ) [(η1 + η2′;3) + (η1′ + η3;2′);3′3] = [D3D3 ′D2 ′D2 − I]Ο•1 . (58 )
Using Eqs. (57*) and (58*), we can use the delay technique again to finally derive the following expression for the new unequal-arm Michelson combination X1 that accounts for the flexing effect:
X1 = [D2 ′D2D3D3 ′ − I][(η1 + η2′;3′) + (η1′ + η3;2′);3′3] − [D D ′D ′D − I][(η′ + η ′) + (η + η ′ ) ′]. (59 ) 3 3 2 2 1 3;2 1 2;3 ;22
As usual, X2 and X3 are obtained by cyclic permutation of the spacecraft indices. This expression is readily shown to be laser-noise-free to first order of spacecraft separation velocities Λ™Li: it is “flex-free”.

5.2 The Sagnac combinations

In the above Section 5.1, we have used the same symbol X 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 (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 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 Ο• 1 from canceling in the α combination.

In order to find the solution to this problem let us first rewrite α 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,

α = [η ′ + π’Ÿ ′η ′ + π’Ÿ ′π’Ÿ ′η′] − [η + π’Ÿ η + π’Ÿ π’Ÿ η ], (60 ) 1 2 3 1 2 2 1 3 2 1 3 2
where we have accounted for clockwise and counter-clockwise light delays. It is straight-forward to verify that this combination no longer cancels the laser and optical bench noises. If, however, we expand the two terms inside the square-brackets on the right-hand side of Eq. (60*) we find that they are equal to
[η1′ + π’Ÿ2 ′η3′ + π’Ÿ1′π’Ÿ2 ′η2′] = [π’Ÿ2 ′π’Ÿ1 ′π’Ÿ3 ′ − I ]Ο•1 (61 ) [η + π’Ÿ η + π’Ÿ π’Ÿ η ] = [π’Ÿ π’Ÿ π’Ÿ − I]Ο• . (62 ) 1 3 2 1 3 2 3 1 2 1
If we now apply our iterative scheme to the combinations given in Eq. (62*) we finally get the expression for the Sagnac combination α1 that is unaffected by laser noise in presence of rotation,
α1 = [π’Ÿ3π’Ÿ1 π’Ÿ2 − I][η1′ + π’Ÿ2′η3′ + π’Ÿ1 ′π’Ÿ2 ′η2′] − [π’Ÿ2 ′π’Ÿ1 ′π’Ÿ3 ′ − I][η1 + π’Ÿ3 η2 + π’Ÿ1 π’Ÿ3 η2]. (63 )
If the delay-times are also time-dependent, we find that the residual laser noise remaining into the combination α1 is actually equal to
Λ™ [( Λ™ Λ™ Λ™ ) ( ′ ′ ′) ( Λ™′ Λ™′ Λ™′) ] Ο•1,1231′2′3′ L1 + L2 + L3 L 1 + L 2 + L3 − L1 + L2 + L3 (L1 + L2 + L3 ) . (64 )
Fortunately, although first order in the relative velocities, the residual is small, as it involves the difference of the clockwise and counter-clockwise rates of change of the propagation delays on the same circuit. For LISA, the remaining laser phase noises in αi, i = 1,2,3, are several orders of magnitude below the secondary noises.

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 laser-noise-free 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 (ζ1,ζ2,ζ3), can be derived by applying again our time-delay operator approach.

Let us consider the following combination of the η i, η ′ i measurements, each being delayed only once [2*]:

η3,3 − η3′,3 + η1,1′ = [D3D2 − D1 ′]Ο•1 , (65 ) η1′,1 − η2,2′ + η2′,2′ = [D3′D2 ′ − D1 ]Ο•1 , (66 )
where we have used the commutativity property of the delay operators in order to cancel the Ο•2 and Ο•3 terms. Since both sides of the two equations above contain only the Ο•1 noise, ζ1 is found by the following expression:
ζ1 = [D3 ′D2 ′ − D1 ](η3,3 − η3′,3 + η1,1′) − [D2D3 − D1 ′](η1′,1 − η2,2′ + η2′,2′). (67 )
If the light-times in the arms are equal in the clockwise and counter-clockwise senses (e.g., no rotation) there is no distinction between primed and unprimed delay times. In this case, ζ 1 is related to our original symmetric Sagnac ζ by ζ1 = ζ,23 − ζ,1. Thusm for the LISA case (arm length difference < 1%), the SNR of ζ1 will be the same as the SNR of ζ.

If the delay-times also change with time, the perfect cancellation of the laser noises is no longer achieved in the (ζ1,ζ2,ζ3) 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.

5.3 Algebraic approach to second-generation TDI

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 π’Ÿi and π’Ÿi ′ 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 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 π’Ÿi and π’Ÿi ′, 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,

′ ′ q1 + q1 − q2π’Ÿ3′ − q3π’Ÿ2 = 0 , q2 + q′2 − q′3π’Ÿ1′ − q1π’Ÿ3 = 0 , (68 ) ′ ′ q3 + q3 − q1π’Ÿ2′ − q2π’Ÿ1 = 0 .

Eliminating q1 and q2 from the three Eqs. (68*) while respecting the order of the variables we get:

q3(1 − π’Ÿ2 π’Ÿ3π’Ÿ1 ) + q′(π’Ÿ2 ′ − π’Ÿ3 π’Ÿ1 ) + q′(π’Ÿ3 ′π’Ÿ3 − 1)π’Ÿ1 + q′(π’Ÿ1 ′π’Ÿ1 − 1) = 0. (69 ) 1 2 3
The polynomial ring 𝒬 (π’Ÿi,π’Ÿi ′) ≡ 𝒦, is non-commutative, of polynomials in the six variables π’Ÿi, π’Ÿi′ and coefficients in the rational field 𝒬. The polynomial vectors (qi,q′i) satisfying the above equations form a left module over 𝒦. A left module means that one can multiply a solution (q,q′) i i from the left by any polynomial in 𝒦, 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 [9].

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 − π’Ÿ2 π’Ÿ3 π’Ÿ1,π’Ÿ2 ′ − π’Ÿ3 π’Ÿ1, (π’Ÿ3 ′π’Ÿ3 − 1)π’Ÿ1, π’Ÿ1′π’Ÿ1 − 1 }. 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 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.

In general, a commutator of two operators X, Y is defined as the operator [X, Y ] ≡ XY − Y X. In our situation X and Y are strings of operators built up of the operators D1, D2, D3, D1′,D2 ′,D3 ′. For example, in the Sagnac combination the following commutator [12*] occurs:

[D D D ,D ′D ′D ′] ≡ D D D D ′D ′D ′ − D ′D ′D ′D D D . (70 ) 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Here X = D1D2D3 and Y = D1 ′D2 ′D3′. This commutator leads to the residual noise term given in Eq. (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 ¨L ∼ 10−6 m βˆ•s2 and thus, even if one considers say 20 successive optical paths, that is, about Δt ∼ 330 s of light travel time, 2 Δt ¨L ∼ 0.1 m. This is well below few meters and thus can be neglected in the residual laser noise computation. Moreover, LΛ™2 terms (and higher order) can be dropped since they are of the order of ≲ 10 −15 (they come with a factor 1βˆ•c2), which is much smaller than 1 part in 108, which is the level at which the laser frequency noise must be canceled. Thus, we keep terms only to the first degree in Λ™ L and also neglect higher time derivative terms in L.

The result for the Sagnac combination can be generalized. In order to simplify notation we write xk or ym for the time-delay operators, where k,m = 1,2, ...,n and n ≥ 2, that is, xk or ym are any of the operators ′ ′ ′ D1, D2, D3, D1 ,D2 ,D3. Then a commutator is:

[x1x2...xn, y1y2...yn] = x1x2 ...xny1y2...yn − y1y2 ...ynx1x2 ...xn. (71 )
Up to the order of approximation we are working in we compute the effect of the commutator on the phase Ο• (t):
[x1x2 ...xn,y1y2...yn ] Ο•(t) ( ∑n ∑n ∑n ∑n ) ( ∑n ∑n ) = Lx Λ™Lym − Lym Λ™Lx Ο•Λ™ t − Lx − Lym . (72 ) k=1 k m=1 m=1 k=1 k k=1 k m=1

Note that the LHS acts on Ο•(t), while the right-hand side multiplies Λ™Ο• 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 k, we have, x = π’Ÿ ′ k 2 say, then L = L ′ xk 2 and so on; the same holds for ym for a given m. From this equation it immediately follows that if the operators y1,y2,...,yn are a permutation of the operators x1,x2,...,xn, then the commutator,

[x1x2 ...xn,y1y2...yn ] = 0 , (73 )
up to the order we are working in. We can understand this by the following argument. If y1...yn is a permutation of x1...xn then both polynomials trace the 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 L¨ terms and those of higher orders, we have effectively assumed that Λ™ Ls 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 𝒰 generated by the commutators such as those given by Eq. (73*). Then we quotient the ring 𝒦 by 𝒰, thereby constructing a smaller ring 𝒦 βˆ•π’° ≡ ¯π’¦. 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.

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 π’Ÿ2, π’Ÿ2′,π’Ÿ3, π’Ÿ3′ and we can set the polynomials ′ q2 = q3 = 0. This simplifies the Eqs. (68*) considerably because the last two equations reduce to ′ q2 = q1π’Ÿ3 and ′ q3 = q1π’Ÿ2′. Substituting these in the first equation gives just one equation:

q1(1 − π’Ÿ3 π’Ÿ3′) + q′1(1 − π’Ÿ2 ′π’Ÿ2) = 0 , (74 )

If we can solve this equation for ′ q1,q1 then the full polynomial vector can be obtained because q′2 = q1π’Ÿ3 and q3 = q′1π’Ÿ2′. It is clear that solutions are of the Michelson type. Also notice that the coefficients of this equation has the operators a = π’Ÿ3 π’Ÿ3 ′ and b = π’Ÿ2′π’Ÿ2 occurring in them. So the solutions q,q ′ 1 1 too will be in terms of a and b only. Physically, the operators a and b correspond to round trips.

One solution has already been given in the literature [1, 61*]. This solution in terms of a,b is:

2 q1 = 1 − b − ba + ab , q′1 = − (1 − a − ab + ba2) . (75 )
Δ = q1(1 − a ) + q′(1 − b), (76 ) 1
we get for Eq. (75*), Δ = [ba,ab], which is a commutator that vanishes since ab is a permutation of ba. Thus, it is an element of 𝒰 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 q1,q′ 1 of Eq. (75*). Note that these are of degree 3 in a and b. The commutator corresponding to this solution is 2 2 Δ = [ba,ab] = ba b − ab a 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 q1,q′1 are of degree 7. There are three such commutators at degree 8. We list the solutions and the commutators below. One solution is:

q = 1 − b − ba + ab2 − ba2b + ab2ab + ab2aba − ba2bab2, 1 q′1 = − (1 − a − ab + ba2 − ab2a + ba2ba + ba2bab − ab2aba2), (77 )
whose commutator is:
Δ = [ab2a, ba2b]. (78 )

The second solution is:

q1 = 1 + a − b2 − b2a − b2a2 − b2a3 + a2b4 + a2b4a 2 2 2 2 4 = (1 − b − b a + a b )(1 + a ), q′1 = − (1 + b − a2 − a2b − a2b2 − a2b3 + b2a4 + b2a4b) = − (1 − a2 − a2b2 + b2a4)(1 + b) , (79 )
whose commutator is Δ = [b2a2,a2b2].

The third solution corresponds to the commutator [baba,abab]. This solution is given by:

q1 = 1 − b + ab − bab − baba + abab2 − baba2b + abab2ab , ′ 2 2 2 q1 = − (1 − a + ba − aba − abab + baba − abab a + baba ba ). (80 )

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 4n where n = 1, 2,..., then the solutions ′ q1,q1 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−1Cn −1. So at n = 3 we have 10 commutators and so as many solutions q1,q′1 of degree 11.

These are the degrees of polynomials of ′ q1,q1 in the operators a,b. But for the full polynomial vector, which has ′ q2 and q3, we need to go over to the operators expressed in terms of π’Ÿi,π’Ÿi ′. Then the degree of each of the q1,q′1 is doubled to 8n − 2, while q′2 and q3 are each of degree 8n − 1. Thus, for a general value of n, the solution contains polynomials of maximum degree 8n − 1 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 ¨L and Λ™L2 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 L¨. We now investigate this limit and make a very rough estimate of it. As mentioned earlier, the LISA model in [12] gives L¨ ∼ 10− 6 m βˆ•s2. From ¨ L we compute the error in L, namely, 1 2¨ ΔL ∼ 2Δt L. If we allow the error to be no more than say 10 meters, then we find Δt ∼ 4500 s. 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 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.

Some remarks are in order:

  • 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 second-generation TDI observables can be found in [30].

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