Conformal Einstein equation and symplectic flux with a positive cosmological constant

TL;DR

The paper addresses gravitational radiation in spacetimes with a positive cosmological constant by deriving fall-off conditions for linearized fields from the conformal Einstein equations and constructing a gauge-invariant presymplectic structure at future infinity $\mathscr{I}^+$. Employing a Geroch–Xanthopoulos style regularization and the Wald–Zoupas framework, it expresses the finite boundary flux in terms of the rescaled electric part of the Weyl tensor $\widetilde{e}_{ab}$ and the holographic boundary stress tensor $T^{ab}$, with the Gibbons–Hawking term and counterterm playing essential roles. The analysis reveals a minimal fall-off with $\alpha=0$, $\beta=0$, $\gamma=1$ for the linearized fields in the presence of $\Lambda>0$, yielding a nontrivial, gauge-invariant presymplectic flux across $\mathscr{I}^+$ and a boundary energy flux formula that remains conformally invariant. The work highlights key differences from the $\Lambda=0$ case, provides a gauge-covariant route to gravitational radiation in de Sitter-like spacetimes, and lays groundwork for further exploration of memory effects and hyperbolic well-posedness in the conformal framework.

Abstract

We analyze the conformal Einstein equation with a positive cosmological constant to extract fall-off conditions of the gravitational fields. The fall-off conditions are consistent with a finite, non-trivial presymplectic current on the future boundary of de Sitter. Hence our result allows a non-zero gravitational flux across the boundary of the de Sitter. We present an explicit gauge-free computation to show that the Gibbons-Hawking boundary term, counterterm in the action, and fall-off condition of gravitational field in conformal Einstein equation are crucial to reproduce the finite symplectic flux.