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To my poor knowledge on the topic, the gravitational waves that are most likely to be detected by LIGO or other experiments do not have thermal spectrum. But I'm not certain.

I know that Hawking's radiation is thermal and it is partially make up of gravitational radiation. However, as far as I know (correct me), its detection is not possible in practice.

I think that the gravitational wave background is thermal, but I have no idea about its direct or indirect detection. Will (Advanced) LIGO be able to say anything about this? Maybe the B modes of CMB can tell us something?

Any information about possible sources of gravitational radiation with thermal spectrum will be welcome.

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The majority of the sources that LIGO (and other gravitational wave detectors) are aiming for are astrophysical (e.g. neutron stars, black holes, supernovae, pulsars).

The expected cosmological gravitational radiation from standard inflationary models (see this recent article from P. Steinhardt) would be very weak in the LIGO band (10-1000 Hz). There are a few space interferometer mission concepts (DECIGO and Big Bang Observer) which would be focused on the 0.1-1 Hz frequency band where the foreground of galactic white dwarf binaries is small enough to make this detection, in principle.

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