Transverse traceless gravitational waves in a spatially flat FLRW universe: Causal structure from dimensional reduction

Abstract

This work was mainly driven by the desire to explore to what extent embedding some given geometry in a higher dimensional flat one is useful for understanding the causal structure of classical fields traveling in the former, in terms of that in the latter. We point out, in the four-dimensional (4D) spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker universe, that the causal structure of transverse-traceless (TT) gravitational waves can be elucidated by first reducing the problem to a two-dimensional (2D) Minkowski wave equation with a time-dependent potential, where the relevant Green's function is a pure tail---waves produced by a physical source propagate strictly within the null cone. By viewing this 2D world as embedded in a 4D one, the 2D Green's function can also be seen to be sourced by a cylindrically symmetric scalar field in three dimensions (3D). From both the 2D wave equation and the 3D scalar perspective, we recover the exact solution of the 4D graviton tail for the case where the scale factor written in conformal time is a power law. There are no TT gravitational-wave tails when the universe is radiation dominated because the background Ricci scalar is zero. In a matter-dominated one, we estimate the amplitude of the tail to be suppressed relative to its null counterpart by both the ratio of the duration of the (isolated) source to the age of the universe ${\ensuremath{\eta}}_{0}$ and the ratio of the observer-source spatial distance (at the observer's time) to the same ${\ensuremath{\eta}}_{0}$. In a universe driven primarily by a cosmological constant, the tail contribution to the background geometry $a[\ensuremath{\eta}{]}^{2}{\ensuremath{\eta}}_{\ensuremath{\mu}\ensuremath{\nu}}$ after the source has ceased is the conformal factor ${a}^{2}$ times a spacetime-constant symmetric matrix proportional to the spacetime volume integral of the TT part of the source's stress-energy-momentum tensor. In other words, massless spin-2 gravitational waves exhibit a tail-induced memory effect in 4D de Sitter spacetime.