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Gravitational waves can be produced by the merger of two black holes or two neutron stars and those have been detected. Now people also thinking about GWs produced in the early universe. By what mechanism(s) could GWs be produced in the early universe at a time when were no massive structures to massively distort the spacetime?

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Inflation ? It doesn't have to be massive structures. As long you have a time dependent energy momentum tensor $T_{\mu \nu}$ that has non zero quadrupole moment, it will generate gravitational waves.

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