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

TL;DR

This work analyzes the causal structure of transverse-traceless gravitational waves in a spatially flat FLRW universe by reducing the 4D problem to a 2D Minkowski wave equation with a time-dependent potential $U[\eta] = -\ddot{a}/a$, enabling a clean separation of the direct (null cone) part from a tail. For power-law cosmologies, the tail is exactly solvable via a homogeneous 2D equation for the tail function $J$, yielding no tail during radiation domination, a decaying tail in matter domination, and a constant-tail memory effect in de Sitter spacetime, the latter producing a permanent shift in the metric proportional to $(H/M_{pl})^2$. The paper also provides a 3D scalar perspective by embedding the 2D problem in 4D Minkowski, showing that the 2D tail can be sourced by a cylindrically symmetric 3D field $J$, and demonstrates consistency between the 2D reduction and this embedding. Overall, the study highlights the utility of dimension reduction and embedding in understanding GW tails and memory, and it discusses open questions about higher-dimensional embeddings.

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 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 $η_0$, and the ratio of the observer-source spatial distance (at the observer's time) to the same $η_0$. 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.

Figures (2)

• Figure 1: Causal structure of TT GWs. This spacetime diagram depicts a hypothetical astrophysical process, where a massive star undergoes core collapse and goes supernova (right world line). The dashed-dotted segment of the right world line represents the full duration during which GWs are produced: before that, the collapse has not started; after that, the system has settled down completely. These GWs are detected by a distant detector (left world line). We assume that the background geometry is that of a 4D spatially flat FLRW universe. The black dashed lines emanating from the detector's world line are the past light cones of events $A$, $B$ and $C$. The bottom pair of light gray dashed lines emanating from the right world line is the forward light cone of the beginning of the collapse; the top pair is that of the end of the process. The light gray shaded region of spacetime is filled with GWs propagating both on and inside the null cone. The darker-gray region of spacetime is filled with GW tails only. The detector at $A$ sees no signal. The detector at $B$ sees a "direct" signal from $B'$ but also a tail from the dashed-dotted segment of the massive star before $B'$. At $C$ the signal received is the accumulation of the GW tails from the entire dashed-dotted segment. As we show in the main text, in a radiation dominated universe, there are in fact no tails, so the detector sees nothing at $C$. In a matter dominated universe, the signal at $C$ is independent of the spatial location but decays with time in an expanding universe. In a de Sitter background, the Green's function tail is a constant; the detector that was operational from $A$ through $C$ would sense a permanent change in $D_{ij}$ that is proportional to $(H/M_{\rm pl})^2$ and to the total $\Pi_{ij}^\text{(T)}$ contained in the dashed-dotted segment, i.e., $(H/M_{\rm pl})^2 \int \text{d}^4 x' \sqrt{|\bar{g}[\eta']|} \Pi_{ij}^\text{(T)}[\eta',\vec{x}']$.
• Figure 2: 1D space embedded in 3D space. This figure illustrates the causal structure encapsulated in the integral representation of $\widehat{G}_2$ in eq. \ref{['2DFrom4DSource']}. The large shaded oval is to be viewed as a portion of the infinite 2D plane source $J[\eta,\eta';r_\perp]$, which comes into existence for only an instant at $\eta'$; whereas the observer's time is $\eta$, i.e., the elapsed time between observation and emission is $\eta-\eta'$. The 1D space of the 2D world whose scalar waves are described by $\widehat{G}_2$, pierces the 2D plane source $J$ orthogonally at its origin (denoted by $O$). The 2D observer is located $R$ away from $O$ along the 1D line. (In 2D $R\equiv|x-x'|$, where $x$ and $x'$ are, respectively, the spatial locations of the observer and the source of the Green's function. In 4D $R\equiv|\vec{x}-\vec{x}'|$, with similar meanings for $\vec{x}$ and $\vec{x}'$.) Because the ambient 4D spacetime is Minkowskian (the $\overline{G}_4$ in eq. \ref{['2DFrom4DSource']}), for a fixed $R$ and elapsed time $|\eta-\eta'|$, no signal from any part of the 2D plane source $J$ can reach the observer whenever $|\eta-\eta'| < R$. Suppose instead $|\eta-\eta'| \geq R$, then because massless scalars in 4D Minkowski travel strictly on the light cone, the observed signal -- from this 4D perspective -- receives contributions only from the (dotted) circle on $J$ of infinitesimal thickness $\text{d} r_\perp$ and radius defined by $|\eta-\eta'|^2 = R^2 + r_\perp^2$. But from the 2D point of view $|\eta-\eta'| > R$ is simply the statement that the observed signal is the tail of $\widehat{G}_2$; from eq. \ref{['2DFrom4DSource_J']} we also see the 2D signal has no $\delta$-function impulse at $|\eta-\eta'|=R$. Finally, since it is the integrated signal that is being observed, there is no need to allow for the source $J$ to have an azimuthal dependence.