Gravitational Wave Memory In dS$_{4+2n}$ and 4D Cosmology

TL;DR

This work demonstrates that massless gravitons in all even-dimensional de Sitter spacetimes with d ≥ 4 exhibit a linear gravitational-wave memory effect arising from tail propagation inside the light cone. By solving linearized Einstein equations on de Sitter backgrounds and relating tail contributions to the source's mass and pressure quadrupole moments, the authors show that the TT metric perturbation settles to a spacetime-constant shift after the GW source ends. In 4D cosmologies, a similar tail-induced memory exists in matter-dominated universes (but not radiation-dominated ones), with the memory amplitude linked to the change in TT quadrupole moments and modulated by cosmic expansion. The results extend prior 4D cosmological memory findings and provide a framework for expressing higher-dimensional memory in terms of source dynamics via explicit Green's functions.

Abstract

We argue that massless gravitons in all even dimensional de Sitter (dS) spacetimes higher than two admit a linear memory effect arising from their propagation inside the null cone. Assume that gravitational waves (GWs) are being generated by an isolated source, and over only a finite period of time. Outside of this time interval, suppose the shear-stress of the GW source becomes negligible relative to its energy-momentum and its mass quadrupole moments settle to static values. We then demonstrate, the transverse-traceless (TT) GW contribution to the perturbation of any dS$_{4+2n}$ written in a conformally flat form -- after the source has ceased and the primary GW train has passed -- amounts to a spacetime constant shift in the flat metric proportional to the difference between the TT parts of the source's final and initial mass quadrupole moments. As a byproduct, we present solutions to Einstein's equations linearized about de Sitter backgrounds of all dimensions greater than three. We then point out there is a similar but approximate tail induced linear GW memory effect in 4D matter dominated universes. Our work here serves to improve upon and extend the 4D cosmological results of arXiv:1504.06337, which in turn preceded complementary work by Bieri, Garfinkle and Yau (arXiv:1509.01296) and by Kehagias and Riotto (arXiv:1602.02653).

Figures (1)

• Figure 1: (Figure borrowed from Chu:2015yua.) This is a spacetime diagram depicting the core collapse of a massive star, which then goes supernova (right world line). The dashed-dotted segment of the right world line denotes the full duration during which TT GWs are produced, corresponding to $\eta_\text{i} \leq \eta \leq \eta_\text{f}$ in the main text. For $\eta \leq \eta_\text{i}$, the collapse has not started; afterwards, $\eta \geq \eta_\text{f}$, the system has settled down completely. (We assume equations \ref{['ShearStressZero']}, \ref{['PressureQuadrupoleZero']} and \ref{['MassQuadrupoleDotZero']}.) The TT GWs are heard by a distant detector (left world line). In this paper the background geometry is either dS$_{4+2n}$, a spatially flat 4D FLRW radiation dominated or matter dominated universe. The black dashed lines emanating from the worldline of the GW detector 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 starting point of the stellar collapse; the top pair is that of the ending point. The light gray shaded region of spacetime is filled with TT GWs propagating both on and inside the light cone. The darker-gray region of spacetime is filled with TT GW tails only, whose detailed properties depends in principle on the entire history of the source (i.e., the dashed-dotted segment). A detector that was operational from $A$ through $C$ would sense a permanent change in $D_{ij} \equiv \chi_{ij}^\text{TT}$ if the TT GW tails in this darker-gray region were spacetime constant. This happens in dS$_{4+2n}$ (eq. \ref{['dS_Memory']}) and approximately so in 4D matter dominated universes (eq. \ref{['4D_GW_Tail_MatterDom']}) -- but not in 4D radiation dominated ones, because there are no TT GW tails there.