Relic gravitons and non-stationary processes

Stationary processes do not accurately describe the diffuse backgrounds of relic gravitons whose correlations are homogeneous in space (i.e. only dependent upon the distance between the two spatial locations) but not in time. The symmetries of the autocorrelations ultimately reflect the quantum mechanical origin of the diffuse backgrounds and lead to non-stationary observables at late time. In particular, large oscillations are believed to arise in the spectral energy density that is customarily (but approximately) related to the tensor power spectrum. When the full expression of the spectral energy density is employed the amplitudes of oscillation are instead suppressed in the large-scale limit and the non-stationary features of the late-time signal practically disappear. For similar reasons the relations between the spectral energy density and the spectral amplitude are ambiguous in the presence of non-stationary features. While it is debatable if the non-stationary features are (or will be) directly detectable, we argue that the spectral amplitude following from the Wiener-Khintchine theorem is generally inappropriate for a consistent description of the relic signal. Nevertheless the strong oscillatory behaviour of the late-time observables is naturally smeared out provided the spectral energy density is selected as pivotal variable.