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Gravitational waves originate from significant distances. Presumably they are red-shifted the same amount as the galaxies they originated in. In the absence of gravitational “spectral lines”, can this red shift be measured? And if it can be, is this information useful in calculating distance or inferring the event that created the gravitational waves?

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Gravitational waves should be redshifted.

The gravitational wave signal of an inspiralling binary system, which drifts across a range of frequencies, does not however yield the redshift of the binary system, since changing the redshift is indistinguishable from changing the mass of the system.

In practice, what is usually done is to estimate the redshift from the luminosity-distance, which can be determined purely from the gravitational wave signal (the so-called "standard siren" technique), combined with a cosmological model.

Of course, where there is an optical counterpart - such as a kilonova - then the redshift of the gravitational wave source can then be estimated from its host galaxy. In which case this offers an independent route to determine cosmological parameters.

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