Production of scalar and tensor perturbations in inflationary models
Abstract
Scalar (density) and tensor (gravitywave) 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 (HarrisonZel'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 efolds 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 powerlaw slopes of their spectra can be expressed in terms of the value of the potential 50 efolds before the end of inflation, V_{50}, its steepness x_{50}≡m_{Pl}V^{'}_{50}/V_{50}, and the rate of change of its steepness, x^{'}_{50} (a prime denotes a derivative with respect to the scalar field). In addition, the powerlaw index of the cosmicscale factor at this time is q_{50}≡[dlnR/dlnt]_{50}~=16π/x^{2}_{50}. (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 x_{50}, 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 a large tensormode contribution to the quadrupole CBR temperature anisotropy, a similar tilt in both scalar and tensor perturbations, and a slower expansion rate, i.e., smaller q_{50}. Measurements of the amplitude and tilt of the scalar and tensor perturbations overdetermine V_{50}, x_{50}, and x^{'}_{50}, 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.
 Publication:

Physical Review D
 Pub Date:
 October 1993
 DOI:
 10.1103/PhysRevD.48.3502
 arXiv:
 arXiv:astroph/9302013
 Bibcode:
 1993PhRvD..48.3502T
 Keywords:

 98.80.Cq;
 98.80.Es;
 98.80.Hw;
 Particletheory and fieldtheory models of the early Universe;
 Observational cosmology;
 Astrophysics;
 High Energy Physics  Phenomenology
 EPrint:
 32 pages FERMILABPub93/026A