About the propagation of the Gravitational Waves in an asymptotically de-Sitter space: Comparing two points of view

We analyze the propagation of gravitational waves (GWs) in an asymptotically de-Sitter space by expanding the perturbation around Minkowski and introducing the effects of the Cosmological Constant ($\Lambda$), first as an additional source (de-Donder gauge) and after as a gauge effect ($\Lambda$-gauge). In both cases the inclusion of the Cosmological Constant $\Lambda$ impedes the detection of a gravitational wave at a distance larger than $L_{crit}=(6\sqrt{2}\pi f \hat{h}/\sqrt{5})r_\Lambda^2$, where $r_\Lambda=\frac{1}{\sqrt{\Lambda}}$ and f and $\hat{h}$ are the frequency and strain of the wave respectively. We demonstrate that $L_{crit}$ is just a confirmation of the Cosmic No hair Conjecture (CNC) already explained in the literature.