Constraints on the scalar-field potential in inflationary models

In this paper, we quantify the degree of fine tuning required for successful inflationary scenarios. We define a "fine-tuning" parameter $\ensuremath{\lambda}$ to be the ratio of the change in the potential $\ensuremath{\Delta}V$ to the change in the scalar field ${(\ensuremath{\Delta}\ensuremath{\psi})}^{4}$; i.e., $\ensuremath{\lambda}$ measures the required degree of flatness in the potential. For a quartic polynomial potential, the quartic coupling constant ${\ensuremath{\lambda}}_{q}$ is bounded by $|{\ensuremath{\lambda}}_{q}|\ensuremath{\le}36\ensuremath{\lambda}$. For a general class of inflationary models involving a slowly rolling field, we find that the potential must be very flat, with a fine-tuning parameter $\ensuremath{\lambda}\ensuremath{\lesssim}{10}^{\ensuremath{-}6}\ensuremath{-}{10}^{\ensuremath{-}8}$. The recently proposed "extended" inflationary scenario is even more tightly constrained, with $\ensuremath{\lambda}\ensuremath{\lesssim}{10}^{\ensuremath{-}15}$.