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 $\lambda$ to be the ratio of the change in the potential $\Delta V$ to the change in the scalar field ${\left(\Delta \psi \right)}^4$; i.e., $\lambda$ measures the required degree of flatness in the potential. For a quartic polynomial potential, the quartic coupling constant ${\lambda }_q$ is bounded by $\left|{\lambda }_q\right|\le 36\lambda$. 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 $\lambda \lesssim {10}^{-6\mathrm{}}-{10}^{-8\mathrm{}}$. The recently proposed "extended" inflationary scenario is even more tightly constrained, with $\lambda \lesssim {10}^{-15\mathrm{}}$.