Consistent Truncation of Linearized gravitational waves in de Sitter space-time

TL;DR

The paper addresses how to relate near-source multipole dynamics to far-field gravitational waves in de Sitter space by implementing a consistent truncation of the linearized solution. It develops a procedure that integrates the gauge conditions and source conservation in conformal time and then truncates the spatial TT part $[\chi_{ij}]^{TT}$, with the remaining components deduced from the gauge constraints; this avoids inconsistencies that previously led to $\log(r)$ terms. It shows that inconsistent truncations generate log terms after transforming to Bondi-Sachs form, and that the log-free behavior can be achieved with the proposed approach, including an alternative truncation based on moments and conservation equations. The two-step Bondi-Sachs transformation then yields a log-free radiative solution, enabling reliable interpretation of source information in a de Sitter background.

Abstract

An important step in using observations of gravitational waves from bounded sources is to relate the observed waveforms far away from a source to the local dynamics and environment of the source. To isolate and identify different features of the source, multipole expansion of radiation field as well as that of the source distribution is crucial. Unlike the waves on Minkowski background, those on de Sitter background do not have a convenient, close form relation among these multipoles of arbitrary order to draw general conclusions. This requires truncation to some multipole order to be employed and a consistency issue arises. This was flagged in \cite{CHK} as emergence of disallowed $log(r)$ terms in certain asymptotic forms and a particular quadrupolar truncation was proposed which was consistent with the source conservation. In \cite{NKD80}, it was noticed that a truncated solution while satisfying the wave equation, may fail to satisfy the gauge condition. Truncation in a fully gauge fixed form should not generate the unwanted $log(r)$ terms. In this work, we clarify the consistency issue and also correct a couple of erroneous statements in \cite{NKD80}. The consistency is analyzed by explicitly integrating the gauge conditions and the source conservation equations over the conformal time. This gives a general procedure for a consistent truncation. The procedure is applied to truncations of completely gauge fixed solution illustrating its consistency.