More On Cosmological Gravitational Waves And Their Memories

TL;DR

The paper analyzes how linear gravitational and electromagnetic signals propagate and imprint memories in spatially flat FLRW spacetimes across dimensions $d\ge 4$ with a constant equation-of-state $w$. It develops a gauge-invariant perturbation framework, derives analytic Green's functions via conformal rescaling, dimension reduction, and Nariai's ansatz, and examines both tensor and scalar perturbations as well as electric memory from Maxwell theory, including a discussion of a GR $=$ YM$^2$ double-copy-like relation. Key findings show that tensor tails can yield spacetime-constant memory in de Sitter and matter-dominated regimes, while scalar modes propagate at speed $\sqrt{w}$ and can contribute to memory for $0<w\le1$, with scalar tails possible in some backgrounds; in 4D radiation, tensor tails vanish but acoustic-cone scalar memory may persist. The work provides analytic tools for understanding memory effects in realistic cosmologies, with implications for GW and EM observations and for exploring nonlinear memory and gravity–gauge dualities in the future.

Abstract

We extend recent theoretical results on the propagation of linear gravitational waves (GWs), including their associated memories, in spatially flat Friedmann--Lemaître--Robertson--Walker (FLRW) universes, for all spacetime dimensions higher than 3. By specializing to a cosmology driven by a perfect fluid with a constant equation-of-state $w$ -- conformal re-scaling, dimension-reduction and Nariai's ansatz may then be exploited to obtain analytic expressions for the graviton and photon Green's functions, allowing their causal structure to be elucidated. When $0 < w \leq 1$, the gauge-invariant scalar mode admits wave solutions, and like its tensor counterpart, likely contributes to the tidal squeezing and stretching of the space around a GW detector. In addition, scalar GWs in 4D radiation dominated universes -- like tensor GWs in 4D matter dominated ones -- appear to yield a tail signal that does not decay with increasing spatial distance from the source. We then solve electromagnetism in the same cosmologies, and point out a tail-induced electric memory effect. Finally, in even dimensional Minkowski backgrounds higher than 2, we make a brief but explicit comparison between the linear GW memory generated by point masses scattering off each other on unbound trajectories and the linear Yang-Mills memory generated by color point charges doing the same -- and point out how there is a "double copy" relation between the two.

Figures (1)

• Figure 1: (Figure borrowed from Chu:2015yua and modified slightly.) This is a spacetime diagram depicting a hypothetical GW event involving an isolated astrophysical system (right world line). The dashed-dotted segment of the right world line denotes the full duration during which GWs are produced, corresponding to $\eta_\text{i} \leq \eta \leq \eta_\text{f}$ in the main text. The GWs are heard by a distant detector (left world line). The black dashed lines emanating from the worldline of the GW detector are either the past acoustic or null cones of events $A$, $B$ and $C$. The bottom pair of light gray dashed lines emanating from the right world line is either the forward acoustic or null cone of the starting point of the GW-generation process; the top pair is that of the ending point. Acoustic/Null cone GWs: We have highlighted, in equations \ref{['GWMemory_NullCone']} and \ref{['ScalarMemory_NullCone_Delta']}, that the portion of GW memory transmitted on the acoustic/null cone arises because the GW source configuration does not return to its original state after the main GW event. Specifically, if the $\sigma_{ij}$ part of the astrophysical system at $C'$ is not the same as that it were at $A'$, the GW detector operational from $A$ through $C$ would experience a non-trivial tensor null-cone memory effect. Similarly, over human GW experiment timescales, if $(\dot{\Sigma},\rho,\dot{\Upsilon})$ have changed significantly from $A'$ to $C'$, there will be a non-zero scalar acoustic-cone memory effect.