  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 4D spatially flat FLRW
universe, that the causal structure of transverse-traceless (TT) gravitational
waves can be elucidated by first reducing the problem to a 2D Minkowski wave
equation with a time dependent potential, where the relevant Green's function
is 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 3D. From both the 2D wave equation as well as 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 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 cosmological
constant, the tail contribution to the background FLRW geometry 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.
