Revisitation of algebraic approach for time delay interferometry

TL;DR

This paper revisits the algebraic approach to Time Delay Interferometry (TDI) in space-based gravitational wave detectors, revealing that under realistic, time-varying conditions, the operator equation has only the trivial solution, challenging previous assumptions.

Contribution

It demonstrates that the operator equation for TDI has only the zero solution in non-static scenarios, and proposes a new strategy to find higher-generation TDI combinations.

Findings

The operator equation has only the zero solution under time-varying armlengths.

No nonzero TDI combination can fully suppress laser noise in realistic conditions.

Introduces a method to connect TDI combinations across different generations.

Abstract

Time Delay Interferometry (TDI) is often utilized in the data pre-processing of space-based gravitational wave detectors, primarily for suppressing laser frequency noise. About twenty years ago, assuming armlengths remain constant over time, researchers presented comprehensive mathematical descriptions for the first-generation and modified first-generation TDI. However, maintaining a steady distance between satellites is pragmatically challenging. Hence, the operator equation that neutralizes laser frequency noise, though provided, was deemed difficult to resolve. In this paper, we solve this equation in the context of a non-static scenario where distances between spacecrafts vary over time. Surprisingly, contrary to what previous researchers thought, the study reveals that the equation has only the zero solution, which suggests that no nonzero TDI combination can entirely suppress laser frequency noise under time-varying armlengths. This necessitates the persistent search for second-generation TDI combinations through alternative methods besides directly solving the operator equation. We establish the connections between TDI combinations of different generations and propose a search strategy for finding higher-generation TDI combinations by using generators of lower-generation TDI. The findings contribute to the ongoing discussion on gravitational waves and provide a novel insight into the hurdles faced in space-based gravitational wave detection.