Time Delay Interferometry with Moving Spacecraft Arrays

TL;DR

This paper develops a generalized time-delay interferometry (TDI) framework for space-based gravitational wave detectors with moving spacecraft, accounting for time-varying light delays to improve noise cancellation.

Contribution

It introduces a method using non-commuting operators to derive TDI combinations that handle time-dependent delays in moving spacecraft arrays.

Findings

Derived TDI expressions for constant delays.

Developed a general procedure for time-dependent delays.

Re-derived and generalized key TDI combinations.

Abstract

Space-borne interferometric gravitational wave detectors, sensitive in the low-frequency (millihertz) band, will fly in the next decade. In these detectors the spacecraft-to-spacecraft light-travel-times will necessarily be unequal, time-varying, and (due to aberration) have different time delays on up- and down-links. Reduction of data from moving interferometric laser arrays in solar orbit will in fact encounter non-symmetric up- and downlink light time differences that are about 100 times larger than has previously been recognized. The time-delay interferometry (TDI) technique uses knowledge of these delays to cancel the otherwise dominant laser phase noise and yields a variety of data combinations sensitive to gravitational waves. Under the assumption that the (different) up- and downlink time delays are constant, we derive the TDI expressions for those combinations that rely only on four inter-spacecraft phase measurements. We then turn to the general problem that encompasses time-dependence of the light-travel times along the laser links. By introducing a set of non-commuting time-delay operators, we show that there exists a quite general procedure for deriving generalized TDI combinations that account for the effects of time-dependence of the arms. By applying our approach we are able to re-derive the ``flex-free'' expression for the unequal-arm Michelson combinations $X_1$, first presented in \cite{STEA}, and obtain the generalized expressions for the TDI combinations called Relay, Beacon, Monitor, and Symmetric Sagnac.