Production of scalar and tensor perturbations in inflationary models

Scalar (density) and tensor (gravity-wave) perturbations provide the basis for the fundamental observable consequences of inflation, including CBR anisotropy and the formation of structure in the Universe. These perturbations are nearly scale invariant (Harrison-Zel'dovich spectrum), though a slight deviation from scale invariance (“tilt”) can have significant consequences for both CBR anisotropy and structure formation. In particular, a slightly tilted spectrum of scalar perturbations may improve the agreement of the cold dark matter scenario with the observational data. 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 over this interval we show that the amplitudes of the perturbations and the power-law slopes of their spectra can be expressed in terms of the value of the potential 50 e-folds before the end of inflation, ${\mathrm{V}}_{50}$, its steepness ${\mathrm{x}}_{50}$≡${\mathrm{m}}_{\mathrm{Pl}}$ ${\mathrm{V}}_{50}^{\prime }$/${\mathrm{V}}_{50}$, and the rate of change of its steepness, ${\mathrm{x}}_{50}^{\prime }$ (a prime denotes a derivative with respect to the scalar field). In addition, the power-law index of the cosmic-scale factor at this time is ${\mathrm{q}}_{50}$≡[dlnR/dlnt${]}_{50}$≃16π/${\mathrm{x}}_{50}^2$. (Formally, our results for the perturbation amplitudes and spectral indices are accurate to lowest order in the deviation from scale invariance.) In general, the deviation from scale invariance is such to enhance fluctuations on large scales, and is only significant for steep potentials, large ${\mathrm{x}}_{50}$, or potentials with rapidly changing steepness, large ${\mathrm{x}}_{50}^{\prime }$. In the latter case, only the spectrum of scalar perturbations is significantly tilted. Steep potentials are characterized by a large tensor-mode contribution to the quadrupole CBR temperature anisotropy, a similar tilt in both scalar and tensor perturbations, and a slower expansion rate, i.e., smaller ${\mathrm{q}}_{50}$. Measurements of the amplitude and tilt of the scalar and tensor perturbations overdetermine ${\mathrm{V}}_{50}$, ${\mathrm{x}}_{50}$, and ${\mathrm{x}}_{50}^{\prime }$, and can in principle be used to infer these quantities as well as for testing the inflationary hypothesis. Our formalism has its limitations; it is not applicable to potentials with unusual features in the region that affects astrophysical scales.