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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 [192120]. 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 V~ -- 2 in the low-part of the frequency band, and slightly more than V~ -- 3 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.


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