Can lightcone fluctuations be probed with cosmological backgrounds?

Finding signatures of quantum gravity in cosmological observations is now actively pursued both from the theoretical and the experimental side. Recent work has concentrated on finding signatures of light-cone fluctuations in the CMB. Because in inflationary scenarios a Gravitational Wave Background (GWB) is always emitted much before the CMB, we can ask, in the hypothesis where this GWB could be observed, what is the imprint of light cone fluctuations on this GWB. We show that due to the flat nature of the GWB spectrum, the effect of lightcone fluctuations are negligible.