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4 Algebraic Approach to Cancelling Laser and Optical Bench Noises

In ground based detectors the arms are chosen to be of equal length so that the laser light experiences identical delay in each arm of the interferometer. This arrangement precisely cancels the laser frequency/phase noise at the photodetector. The required sensitivity of the instrument can thus only be achieved by near exact cancellation of the laser frequency noise. However, in LISA it is impossible to achieve equal distances between spacecraft, and the laser noise cannot be cancelled in this way. It is possible to combine the recorded data linearly with suitable time-delays corresponding to the three arm lengths of the giant triangular interferometer so that the laser phase noise is cancelled. Here we present a systematic method based on modules over polynomial rings which guarantees all the data combinations that cancel both the laser phase and the optical bench motion noises.

We first consider the simpler case, where we ignore the optical-bench motion noise and consider only the laser phase noise. We do this because the algebra is somewhat simpler and the method is easy to apply. The simplification amounts to physically considering each spacecraft rigidly carrying the assembly of lasers, beam-splitters, and photodetectors. The two lasers on each spacecraft could be considered to be locked, so effectively there would be only one laser on each spacecraft. This mathematically amounts to setting Di = D'i = 0 and pi = p'i. The scheme we describe here for laser phase noise can be extended in a straight-forward way to include optical bench motion noise, which we address in the last part of this section.

The data combinations, when only the laser phase noise is considered, consist of the six suitably delayed data streams (inter-spacecraft), the delays being integer multiples of the light travel times between spacecraft, which can be conveniently expressed in terms of polynomials in the three delay operators D1, D2, D3. The laser noise cancellation condition puts three constraints on the six polynomials of the delay operators corresponding to the six data streams. The problem therefore consists of finding six-tuples of polynomials which satisfy the laser noise cancellation constraints. These polynomial tuples form a module1 called the module of syzygies. There exist standard methods for obtaining the module, by which we mean methods for obtaining the generators of the module so that the linear combinations of the generators generate the entire module. The procedure first consists of obtaining a Gröbner basis for the ideal generated by the coefficients appearing in the constraints. This ideal is in the polynomial ring in the variables D1, D2, D3 over the domain of rational numbers (or integers if one gets rid of the denominators). To obtain the Gröbner basis for the ideal, one may use the Buchberger algorithm or use an application such as Mathematica [35]. From the Gröbner basis there is a standard way to obtain a generating set for the required module. This procedure has been described in the literature [2Jump To The Next Citation Point16Jump To The Next Citation Point]. We thus obtain seven generators for the module. However, the method does not guarantee a minimal set and we find that a generating set of 4 polynomial six-tuples suffice to generate the required module. Alternatively, we can obtain generating sets by using the software Macaulay 2. The importance of obtaining more data combinations is evident: They provide the necessary redundancy - different data combinations produce different transfer functions for GWs and the system noises so specific data combinations could be optimal for given astrophysical source parameters in the context of maximizing SNR, detection probability, improving parameter estimates, etc.

 4.1 Cancellation of laser phase noise
 4.2 Cancellation of laser phase noise in the unequal-arm interferometer
 4.3 The module of syzygies
 4.4 Gröbner basis
 4.5 Generating set for the module of syzygies
 4.6 Canceling optical bench motion noise
 4.7 Physical interpretation of the TDI combinations

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