The Scalar Field Potential in Inflationary Models: Reconstruction and Further Constraints

In this paper, we present quantitative constraints on the scalar field potential for a general class of inflationary models. (1) We first consider the reconstruction of the inflationary potential for given primordial density fluctuation spectra. Our work differs from previous work on reconstruction in that we find a semi-analytic solution for the potential for the case of density fluctuations with power-law spectra. In addition, for the case of more general spectra, we show how constraints on the density fluctuation spectra imply corresponding constraints on the potential. We present a series of figures 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 $\rave$ of the amplitude of tensor perturbations (gravity wave perturbations) to scalar density perturbations is bounded from above: $\rave \le$ 1.6. We also show that the ratio $\rave$ is proportional to the change $\Delta \phi$ in the field: $\rave \approx 0.42 \Delta \phi/\mp$. Thus, if tensor perturbations are important for the formation of structure, then the width $\Delta \phi$ must be comparable to the Planck mass. (3) We constrain the change $\Delta V$ of the potential and the change $\Delta \phi$ of the inflation field during the portion of inflation when cosmological structure is produced. We find both upper and lower bounds for $\Delta \phi$ and for $\Delta V$. In addition, these constraints are then used to derive a bound on the scale $\Lambda$, which is the scale of the height of the potential