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It is shown that a gravitationally bound system with a one-dimensionalvelocity dispersion sigma can at most dissipate a fraction ~36(sigma/c)^3 ofthe gravitational wave energy propagating through it, even if their dynamicaltime is shorter than the wave period. The limit is saturated for particles witha mean-free-path equal to the size of the system, such as hot protons in galaxyclusters, strongly-interacting dark matter particles in halos, or massive blackholes in clusters. For such systems with random motions and no resonances, thedissipated fraction, <10^{-6}, does not degrade the use of gravitational wavesas cosmological probes. At high wave frequencies, the dissipated fraction isadditionally suppressed by the square of the ratio between the collisionfrequency and the wave frequency. The electromagnetic counterparts that resultfrom the dissipation are too faint to be detectable at cosmological distances.

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