Asymptotics with a positive cosmological constant. III. The quadrupole formula

Abstract

Almost a century ago, Einstein used a weak field approximation around Minkowski space-time to calculate the energy carried away by gravitational waves emitted by a time changing mass-quadrupole. However, by now there is strong observational evidence for a positive cosmological constant, Λ. To incorporate this fact, Einstein's celebrated derivation is generalized by replacing Minkowski space-time with de Sitter space-time. The investigation is motivated by the fact that, because of the significant differences between the asymptotic structures of Minkowski and de Sitter space-times, many of the standard techniques, including the standard 1/r expansions, can not be used for Λ > 0. Furthermore since, e.g., the energy carried by gravitational waves is always positive in Minkowski space-time but can be arbitrarily negative in de Sitter space-time irrespective of how small Λ is, the limit Λ → 0 can fail to be continuous. Therefore, a priori it is not clear that a small Λ would introduce only negligible corrections to Einstein's formula. We show that, while even a tiny cosmological constant does introduce qualitatively new features, in the end, corrections to Einstein's formula are negligible for astrophysical sources currently under consideration by gravitational wave observatories.

Gravitational wave memory in dS 4+2 n and 4D cosmology

“…(18) and the ensuing equations below, are really a generalization of the 4D ones in [7], [8] and [9]. At this point, eq.…”

“…Because the GW tail here is a constant in spacetime, notice this linear GW memory effect does not decay with distance from the source, unlike the expected 1/(spatial distance) Remark I Before shifting our attention to 4D cosmology, we mention here that [8] advocates exploiting the Killing vector in de Sitter spacetime…”

“…(76) is consistent with eq. (4.26) in [8]. Note that, their Q (ρ) ab is our Qij and their Q (p) ab is our Pij; according to them, £T Q…”

“…(37), however, we need to first -following [8] and [9] -relate spatial-volume integrals of the shear-stress T ij of the isolated matter source to its mass Q ij and pressure P ij quadrupole moments.…”

“…Remark III If we follow [8] and assume the static condition in eq. (65), it is then possible to justify why both the mass and pressure quadrupole moments stay constant outside the interval of active GW production, η i ≤ η ≤ η f .…”