We consider the steepest rate at which the power spectrum from single field inflation can grow, with the aim of providing a simple explanation for the $k^4$ growth found recently. With this explanation in hand we show that a slightly steeper $k^5(\log k{)}^2$ growth is, in fact, possible. Moreover, we argue that the power spectrum after a steep growth cannot immediately decay, but must remain large for the $k$ modes that exit during a $\sim 2$ $e$-fold period. We also briefly consider how a strong growth can affect the spectral index of longer wavelengths preceding the growth and show that even the conversion of isocurvature modes likely cannot lead to a stronger growth. These results have implications for the formation of primordial black holes and other phenomena that require a large amplitude of power spectrum at short scales.

• Received 2 August 2019

DOI:https://doi.org/10.1103/PhysRevD.100.103529