Sensitivity functions for geometric time-delay interferometry combinations

Time delay interferometry (TDI) is primarily proposed for space-borne heterodyne interferometers to suppress the intrinsic laser phase noise. According to the TDI algorithm, the targeted sensitivity is achieved by constructing a virtual equal-arm interferometer through a linear combination of appropriately time-shifted independent Doppler measurements. To maximize the potential capability of the experimental apparatus, it is desirable to obtain the sensitivity limit of a specific detector among different TDI combinations. In this paper, we evaluate the detector's sensitivity functions in terms of generators of the kernel of a homomorphism map determined by the underlying TDI equation. The analysis is based on decomposing an arbitrary second-generation TDI combination as a linear combination of independent first and zeroth-generation generators. As one ignores the difference in the armlengths, it is found that the sensitivity function can be classified in terms of a smaller set of independent generators. In particular, the equal armlength approximation simplifies the underlying TDI equation as well as the resultant analytic expressions of sensitivity functions. By enumerating all 45 distinct TDI combinations up to sixteen links obtained using the geometric TDI approach, eleven distinct elementary sensitivity curves are derived in terms of the relevant generators. The observed degeneracies in sensitivity curves are explained using algebraic arguments. The obtained analytic results are understood to be helpful in characterizing the quality and improving the implementation of specific TDI solutions.