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

Almost a century ago, Einstein used a weak field approximation around Minkowski spacetime 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, Lambda. To incorporate this fact, Einstein's celebrated derivation is generalized by replacing Minkowski spacetime with de Sitter spacetime. The investigation is motivated by the fact that, because of the significant differences between the asymptotic structures of Minkowski and de Sitter spacetimes, many of the standard techniques, including the usual 1/r expansions, cannot be used for Lambda > 0. Furthermore, since, e.g., the energy carried by gravitational waves is always positive in Minkowski spacetime but can be arbitrarily negative in de Sitter spacetime irrespective of how small Lambda is, the limit Lambda -> 0 can fail to be continuous. Therefore, a priori it is not clear that a small. 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.