## 7 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 unequal-arm interferometer in order to give a physical insight of TDI.
Here, these methods have been rigorously applied to the idealized case of a stationary LISA for
deriving the generators of the module from which the entire TDI data set can be obtained; they
can be extended in a straight-forward way to more than three spacecraft for possible LISA
follow-on missions. 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
free-falling spacecraft orbiting around the Sun. The TDI data combinations canceling laser phase
noise in this general case are referred to as second generation TDI, and they contain twice
as many terms as their corresponding first generation 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 [19, 21, 20]. The
resulting improvement in sensitivity over that of an unequal-arm Michelson interferometer, in the
case of monochromatic signals randomly distributed over the celestial sphere and of random
polarization, is non-negligible. We have found this to be equal to a factor of in the low-part
of the frequency band, and slightly more than in the high-part 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.

As a final remark we would like to emphasize that this field of research, TDI, is still very young and
evolving. Possible physical phenomena, yet unrecognized, might turn out to be important to account for
within the TDI framework. The purpose of this review was to provide the basic mathematical tools needed
for working on future TDI projects. We hope to have accomplished this goal, and that others will be
stimulated to work in this new and fascinating field of research.