Entropy of gravitons produced in the early Universe

Gravitons produced from quantum vacuum fluctuations during an inflationary stage in the early Universe have zero entropy as far as they reflect the time evolution (squeezing) of a pure state, their large occupation number notwithstanding. A non-zero entropy of the gravitons (classical gravitational waves (GW) after decoherence) can be obtained through coarse graining. The latter has to be physically justified {\it and} should not contradict observational constraints. We propose two ways of coarse graining for which the fixed temporal phase of each Fourier mode of the GW background still remains observable: one based on quantum entanglement, and another one following from the presence of a secondary GW background. The proposals are shown to be mutually consistent. They lead to the result that the entropy of the primordial GW background is significantly smaller than it was thought earlier. The difference can be ascribed to the information about the regular (inflationary) initial state of the Universe which is stored in this background and which reveals itself, in particular, in the appearance of primordial peaks (acoustic peaks in the case of scalar perturbations) in the multipole spectra of the CMB temperature anisotropy and polarization.