On the Production of Scalar and Tensor Perturbations in Inflationary Models

The amplitude and spectrum of the scalar and tensor perturbations depend upon the shape of the inflationary potential in the small interval where the scalar field responsible for inflation was between about 46 and 54 e-folds before the end of inflation. By expanding the inflationary potential in a Taylor series in this interval we show that the amplitude of the perturbations and the power-law slope of their spectra can be expressed in terms of the value of the potential 50 e-folds before the end of inflation, $V_{50}$, the steepness of the potential, $x_{50} \equiv \mpl V_{50}^\prime /V_{50}$, and the rate of change of the steepness, $x_{50}^\prime$ (prime denotes derivative with respect to the scalar field).