## 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 [245]. Two time delays along each arm must be used, say and for clockwise or counter-clockwise propagation as they enter in any of the TDI combinations. Furthermore, since and 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  [13], 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 [34].

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. These 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 [1] or by applying the delay operator introduced in the previous Section 4,

We then impose a second time delay :
A third time delay gives
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 [25], 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

Note that this is not like an advance operator one might expect, since it advances not by but rather .