hep-ph 9401006 THE SCALAR FIELD POTENTIAL IN INFLATIONARY MODELS: RECONSTRUCTION AND FURTHER CONSTRAINTS

In this paper, we present quantitative constraints on the scalar eld potential for a general class of in ationary models. (1) We rst consider the reconstruction of the in ationary potential for given primordial density uctuation spectra. Our work di ers from previous work on reconstruction in that we nd a semi-analytic solution for the potential for the case of density uctuations with power-law spectra. In addition, for the case of more general spectra, we show how constraints on the density uctuation spectra imply corresponding constraints on the potential. We present a series of gures which show how the shape of the potential depends on the shape of the perturbation spectrum and on the relative contribution of tensor modes. (2) We show that the average ratio hRi of the amplitude of tensor perturbations (gravity wave perturbations) to scalar density perturbations is bounded from above: hRi 1.6. We also show that the ratio hRi is proportional to the change in the eld: hRi 0:42 =Mpl. Thus, if tensor perturbations are important for the formation of structure, then the width must be comparable to the Planck mass. (3) We constrain the change V of the potential and the change of the in ation eld during the portion of in ation when cosmological structure is produced. We nd both upper and lower bounds for and for V . In addition, these constraints are then used to derive a bound on the scale , which is the scale of the height of the potential during the portion of in ation when cosmological perturbations are produced; we nd 10 Mpl. Thus, the last 60 e-foldings of in ation must take place after the GUT epoch. This bound on , although comparable to those found previously, is found here using di erent methods. (4) In an earlier paper, we de ned a ne-tuning parameter FT V=( ) and found an upper bound for FT . In this paper, we nd a lower bound on FT . The ne-tuning parameter is thus constrained to lie in the range 6 10 11 ( =10GeV) FT 10 . (5) Finally, we consider the e ects of requiring a non-scale-invariant spectrum of perturbations (i.e., with spectral index n 6=1) on the ne-tuning parameter FT . For spectral indices n less than unity, the upper bound on the ne-tuning parameter FT becomes more restrictive than the n = 0 case by a factor F 2 { 5. PAC Numbers: 98.80.{k, 98.80.Cq, 12.10.Dm