Non-Gaussianity consistency relations and their consequences for the peaks

Strong deviations from scale invariance and the appearance of high peaks in the primordial power spectrum have been extensively studied for generating primordial black holes (PBHs) or gravitational waves (GWs). It is also well-known that the effect of non-linearities can be significant in both phenomena. In this paper, we advocate the existence of a general single-field consistency relation that relates the amplitude of non-Gaussianity in the squeezed limit $f_{\text{NL}}$ to the power spectrum and remains valid when almost all other consistency relations are violated. In particular, it is suitable for studying scenarios where scale invariance is strongly violated. We discuss the general and model-independent consequences of the consistency relation on the behavior of $f_{\text{NL}}$ at different scales. Specifically, we study the size, sign and slope of $f_{\text{NL}}$ at the scales where the power spectrum peaks and argue that generally the peaks of $f_{\text{NL}}$ and the power spectrum occur at different scales. As an implication of our results, we argue that non-linearities can shift or extend the range of scales responsible for the production of PBHs or GWs, relative to the window as determined by the largest peak of the power spectrum, and may also open up new windows for both phenomena.