Reply to the Bayle {\it et al.} gr-qc document dated June 7, 2021}

TL;DR

This paper defends a matrix formulation of Time-Delay Interferometry against recent critiques, demonstrating its dependence on boundary conditions and its mathematical consistency for constant and fractional delays in space-based gravitational wave detectors.

Contribution

It clarifies the dependence of TDI solutions on boundary conditions and proves the general validity of the matrix representation for fractional delays.

Findings

TDI-$ abla$ solutions depend on boundary conditions.

The matrix representation of delay operators is mathematically consistent.

Fractional delays can be combined via sequential filtering.

Abstract

We address the two issues raised by Bayle, Vallisneri, Babak, and Petiteau (in their gr-qc document arXiv:2106.03976) about our matrix formulation of Time-Delay Interferometry (TDI) (arXiv:2105.02054) \cite{TDJ21}. In so doing we explain and quantify our concerns about the results derived by Vallisneri, Bayle, Babak and Petiteau \cite{Vallisneri2020} by applying their data processing technique (named TDI-$\infty$) to the two heterodyne measurements made by a two-arm space-based GW interferometer. First we show that the solutions identified by the TDI-$\infty$ algorithm derived by Vallisneri, Bayle, Babak and Petiteau \cite{Vallisneri2020} {\underbar {do}} depend on the boundary-conditions selected for the two-way Doppler data. We prove this by adopting the (non-physical) boundary conditions used by Vallisneri {\it et al.} and deriving the corresponding analytic expression for a laser-noise-canceling combination. We show it to be characterized by a number of Doppler measurement terms that grows with the observation time and works for any time-dependent time delays. We then prove that, for a constant-arm-length interferometer whose two-way light times are equal to twice and three-times the sampling time, the solutions identified by TDI-$\infty$ are linear combinations of the TDI variable $X$. In the second part of this document we address the concern expressed by Bayle {\it et al.} regarding our matrix formulation of TDI when the two-way light-times are constant but not equal to integer multiples of the sampling time. We mathematically prove the homomorphism between the delay operators and their matrix representation \cite{TDJ21} holds in general. By sequentially applying two order-$m$ Fractional-Delay (FD) Lagrange filters of delays $l_1$, $l_2$ we find its result to be equal to applying an order-$m$ FD Lagrange filter of delay $l_1 + l_2$.