Transverse-Traceless Gravitational Waves In A Spatially Flat FLRW Universe: Causal Structure from Dimension Reduction

摘要

This work was mainly driven by the desire to explore, to what extentembedding some given geometry in a higher dimensional flat one is useful forunderstanding the causal structure of classical fields traveling in the former,in terms of that in the latter. We point out, in the 4D spatially flat FLRWuniverse, that the causal structure of transverse-traceless (TT) gravitationalwaves can be elucidated by first reducing the problem to a 2D Minkowski waveequation with a time dependent potential, where the relevant Green's functionis pure tail -- waves produced by a physical source propagate strictly withinthe null cone. By viewing this 2D world as embedded in a 4D one, the 2D Green'sfunction can also be seen to be sourced by a cylindrically symmetric scalarfield in 3D. From both the 2D wave equation as well as the 3D scalarperspective, we recover the exact solution of the 4D graviton tail, for thecase where the scale factor written in conformal time is a power law. There areno TT gravitational wave tails when the universe is radiation dominated becausethe background Ricci scalar is zero. In a matter dominated one, we estimate theamplitude of the tail to be suppressed relative to its null counterpart by boththe ratio of the duration of the source to the age of the universe $\eta_0$,and the ratio of the observer-source spatial distance (at the observer's time)to the same $\eta_0$. In a universe driven primarily by a cosmologicalconstant, the tail contribution to the background FLRW geometry after thesource has ceased, is the conformal factor $a^2$ times a spacetime-constantsymmetric matrix proportional to the spacetime volume integral of the TT partof the source's stress-energy-momentum tensor. In other words, massless spin-2gravitational waves exhibit a tail-induced memory effect in 4D de Sitterspacetime.