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 λ to be the ratio of the change in the potential ΔV to the change in the scalar field (Δψ)⁴; i.e., λ measures the required degree of flatness in the potential. For a quartic polynomial potential, the quartic coupling constant λq is bounded by |λq|≤36λ. 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 λ10^-6-10^-8. The recently proposed "extended" inflationary scenario is even more tightly constrained, with λ10^-15.