Upper Limit on the Dissipation of Gravitational Waves in Gravitationally Bound Systems

It is shown that a gravitationally bound system with a one-dimensional velocity dispersion σ can at most dissipate a fraction $\sim 36{\left(\sigma /c\right)}^{3}$ of the gravitational wave (GW) energy propagating through it, even if their dynamical time is shorter than the wave period. The limit is saturated for low-frequency waves propagating through a system of particles with a mean-free-path equal to the size of the system, such as hot protons in galaxy clusters, strongly interacting dark matter particles in halos, or massive black holes in clusters. For such systems with random motions and no resonances, the dissipated fraction, $\lesssim {10}^{-6}$, does not degrade the use of GWs as cosmological probes. At high-wave frequencies, the dissipated fraction is additionally suppressed by the square of the ratio between the collision frequency and the wave frequency. The electromagnetic counterparts that result from the dissipation are too faint to be detectable at cosmological distances.