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

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).