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Are gravitational waves streched by the expansion of the universe in the same way as EM radiation is?

In that case how does one differentiate between a gravitational wave from a given event (say neutron star merger) from one that originated from a more violent event (say black hole merger) but that has been cosmologically redshifted?

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Yes, gravitational waves are stretched by cosmological expansion the same way light waves are. See also: Are the gravitational wave mass estimates redshift dependent?.

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Since current black hole observation extend to about 1B light years, the red shift is not a factor. Moreover, a neutron star merger doesn't look (quantitatively) like a time dilated (medium sized) black hole merger. The former signal last on the order of 100 seconds, while that later is on the order of 200 ms. So it would take $z\approx 500$, while the most distant GRB, 090423, was at "only:" $z=8.2$.

Furthermore, the new field of multi-messenger astronomy allows gamma ray observations of neutron star mergers.

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    $\begingroup$ Since current black hole observation extend to about 1B light years, the red shift is not a factor. Why would cosmological redshifts be negligible at a a billion light years? That's a significant distance on the cosmological scale. And it would be more to the point of the OP's question to note that black hole mergers are standard sirens, so we don't just get their distances from their intensity. $\endgroup$
    – user4552
    Commented Aug 4, 2018 at 0:13

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