Second-Generation Time-Delay Interferometry

Time-Delay Interferometry (TDI) is the data processing technique that cancels the large laser phase fluctuations affecting the heterodyne Doppler measurements made by unequal-arm space-based gravitational wave interferometers. The space of all TDI combinations was first derived under the simplifying assumption of a stationary array, for which the three time-delay operators commute. In this model, any element of the TDI space can be written as a linear combination of four TDI variables, the generators of the "first-generation" TDI space. To adequately suppress the laser phase fluctuations in a realistic array configuration, the rotation of the array and the time-dependence of the six inter-spacecraft light-travel-times have to be accounted for. In the case of the Laser Interferometer Space Antenna (LISA), a joint ESA-NASA mission characterized by slowly time varying arm-lengths, it has been possible to identify data combinations that, to first order in the inter-spacecraft velocities, either exactly cancel or suppress the laser phase fluctuations below the level identified by the noise sources intrinsic to the heterodyne measurements (the so called "secondary" noises). Here we reanalyze the problem of exactly canceling the residual laser noise terms linear in the inter-spacecraft velocities. We find that the procedure for obtaining elements of the $2^{\rm nd}$-generation TDI space can be generalized in an iterative way. This allows us to "lift-up" the generators of the $1^{\rm st}$-generation TDI space and construct elements of the higher order TDI space.