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

TL;DR

This paper addresses the challenge of consistent multipole truncation of gravitational wave solutions in de Sitter space, ensuring gauge conditions and source conservation are maintained, which is crucial for accurate waveform analysis.

Contribution

It provides a general procedure for consistent multipole truncation of gravitational waves in de Sitter space, correcting previous misconceptions and ensuring gauge and conservation conditions are satisfied.

Findings

Established a method for consistent multipole truncation.

Corrected previous errors regarding gauge conditions.

Demonstrated the procedure with explicit examples.

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.