We present quantitative constraints on the scalar field potential for a general class of inflationary models. (1) We first study aspects of the reconstruction of the inflationary potential from primordial fluctuation spectra. Specifically, we consider the case of a pure power law spectrum for the total perturbations (density modes as well as tensor gravitational modes); for this case the reconstruction of the potential can be done semianalytically. We find the solutions and present a series of figures. The figures show how the shape of the potential depends on the shape of the perturbation spectrum and on the relative contribution of tensor modes and scalar density perturbations. When tensor modes provide a significant fraction of the total, the potentials V(φ) are concave upward; when tensor modes provide a negligible contribution, the potentials are concave downward. (2) We show that the ratio scrR of the amplitude of tensor perturbations (gravity wave perturbations) to scalar density perturbations is bounded from above: scrR≤6.1. We also show that the average ratio 〈scrR〉 is proportional to the change Δφ in the field: 〈scrR〉≊1.6Δφ/${\mathit{M}}_{\mathrm{Pl}}$. Thus, if tensor perturbations are important for the formation of structure, then the width Δφ must be comparable to the Planck mass. (3) We constrain the change ΔV of the potential and the change Δφ of the inflation field during the portion of inflation when cosmological structure is produced.

These constraints are then used to derive a bound on the scale Λ of the height of the potential during the portion of inflation when cosmological perturbations are produced; we find Λ≤${10}^{\mathrm{-}2}$${\mathit{M}}_{\mathrm{Pl}}$. (4) In an earlier paper, we defined a fine-tuning parameter ${\lambda }_{\mathrm{FT}}$≡ΔV/(Δφ${)}^4$ and found an upper bound for ${\lambda }_{\mathrm{FT}}$. In this paper, we find a lower bound on ${\lambda }_{\mathrm{FT}}$. The fine-tuning parameter is thus constrained to lie in the range 4×${10}^{\mathrm{-}10}$(Λ/${10}^{17}$ GeV${)}^8$≤${\lambda }_{\mathrm{FT}}$≤${10}^{\mathrm{-}7}$. (5) We consider the effects of requiring a non-scale-invariant spectrum of perturbations (i.e., with a spectral index n≠1) on the fine-tuning parameter ${\lambda }_{\mathrm{FT}}$. (6) We also present a very rough argument which indicates that inflation at very low energy scales will encounter some difficulty: the fractional change in the height of the potential during the N=8 e-foldings of structure formation is very small when the energy scale Λ is small. It is then difficult for the potenital to drop to (roughly) zero in the remaining e-foldings for a normally shaped potential.

• Received 12 January 1994

DOI:https://doi.org/10.1103/PhysRevD.51.6722