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 fluctuations that seed structure formation. In general, these perturbations are nearly scale invariant (Harrison-Zel’dovich spectrum), though 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 present 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 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, V50, the steepness of the potential, x50 ≡ mPlV ′ 50/V50, and the rate of change of the steepness, x′50 (prime denotes derivative with respect to the scalar field). In addition, the power-law index of the cosmic-scale factor at this time, q50 ≡ [d lnR/d ln t]50 ≃ 16π/x50. (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 x50, or potentials with rapidly changing steepness, large x′50. In the latter case, only the spectrum of scalar perturbations is significantly tilted. Steep potentials are characterized by large tensor-mode contribution to the quadrupole CBR temperature anisotropy, similar tilt in both scalar and tensor perturbations, and slower expansion rate, i.e., smaller q50. Measurements of the amplitude and tilt of the scalar and tensor perturbations over determine V50, x50, and x ′ 50, and can in principle be used to infer these quantities as well as 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.