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

TL;DR

This work extends Einstein's gravitational-wave quadrupole formula to a universe with a positive cosmological constant by formulating linearized gravity on de Sitter space and carefully treating the altered asymptotic structure. It introduces a late-time, post-Newtonian framework that expresses the metric perturbation in terms of mass and pressure quadrupole moments, including a curvature tail term that reflects back-scattering by de Sitter curvature. The radiated energy and angular momentum are computed via a covariant phase-space Hamiltonian approach, yielding a quadrupole-type energy flux that remains positive for retarded sources and reduces to the standard Minkowski result as $\Lambda\to 0$. The analysis shows that, for astrophysical sources relevant to current detectors, the leading corrections due to $\Lambda$ are negligible, though tail effects and memory could be significant for cosmological or long-wavelength sources. Overall, the paper provides a controlled bridge between flat-space intuition and de Sitter physics, clarifying the role of the cosmological constant in gravitational-wave emission and energy positivity.

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 \emph{irrespective of how small $Λ$ is}, the limit $Λ\to 0$ 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.

Figures (1)

• Figure 1: Left Panel: A time-changing quadrupole emitting gravitational waves whose spatial size is uniformly bounded in time. The causal future of such a source covers only the future Poincaré patch $M^{+}_{\rm P}$ (the upper triangle of the figure). There is no incoming radiation across the past boundary $E^{+}(i^{-})$ of $M^{+}_{\rm P}$ because we use retarded solutions. The shaded region represents a convenient neighborhood of $\mathcal{I}^{+}$ in which perturbations satisfy a homogenous equation and the approximation (\ref{['approxsoln']}), discussed below, holds everywhere. The dashed (red) lines with arrows show the integral curves of the 'time translation' Killing field $T^{a}$ (adapted to the rest frame of the source). Right Panel: The rate of change of the quadrupole moment at the point $(-|\vec{x}|, \vec{0})$ on the source creates the retarded field at the point $(0, \vec{x})$ on $\mathcal{I}^{+}$. The figure also shows the cosmological foliation $\eta= {\rm const}$ and the time-like surfaces $r:= |\vec{x}|={\rm const}$. As $r$ goes to infinity, the $r={\rm const}$ surfaces approach $E^{+}(i^{-})$. Therefore, in contrast with the situation in Minkowski space-time, for sufficiently large values of $r$, there is no flux of energy across the $r={\rm const}$ surfaces.