The aLIGO signal is a discrete event. In addition to such events that can be resolved into individual chirps, I would think that there would be a background of gravitational waves that would look like noise, but that could be distinguished from terrestrial noise (trucks passing by, etc.) because it's correlated between the Hanford and Livingston detectors. To some extent this would be analogous to the CMB, but for gravity rather than electromagnetism. I don't know if it would be primordial in origin, or whether it would be thermalized (I would guess not). But it seems like it would be of interest to measure its power spectrum, angular distribution, and polarization, just as we do with the CMB. Is this actually possible with aLIGO? It seems like it should be possible in principle, although conceivably when one attempted to measure it, it would be statistically consistent with zero. But I don't see how its strength could be estimated a priori, since we know so little about possible sources, including primordial ones. Since matter is transparent to gravitational waves, it would seem like this noise would probe an earlier era than the CMB does.