Next Contents Previous

6.3. Observational consequences

Observations have moved on beyond us wanting to know the overall normalization of the potential. The interesting things are

  1. The scale-dependence of the spectra.
  2. The relative influence of the two spectra.

These can be neatly summarized using the slow-roll parameters epsilon and eta we defined earlier. [3]

The standard approximation used to describe the spectra is the power-law approximation, where we take

Equation 59 (59)

where the spectral indices n and nG are given by

Equation 60 (60)

The power-law approximation is usually valid because only a limited range of scales are observable, with the range 1 Mpc to 104 Mpc corresponding to Deltaln k appeq 9.

The crucial equation we need is that relating phi values to when a scale k crosses the Hubble radius, which from Eq. (58) is

Equation 61 (61)

(since within the slow-roll approximation k appeq exp N). Direct differentiation then yields [3]

Equation 62 (62)
Equation 63 (63)

where now epsilon and eta are to be evaluated on the appropriate part of the potential.

Finally, we need a measure of the relevant importance of density perturbations and gravitational waves. The natural place to look is the microwave background; a detailed calculation which I cannot reproduce here (see e.g. Ref. [2]) gives

Equation 64 (64)

Here the Cl are the contributions to the microwave multipoles, in the usual notation. (8)

From these expressions we immediately see

Table 1 shows the predictions for a range of inflation models. The information I've given you so far should be sufficient to allow you to reproduce them. Even the simplest inflation models can affect the large-scale structure modelling at a level comparable to the present observational accuracy. The predictions of the different models will be wildly different as far as future high-accuracy observations are concerned.

Table 1: The spectral index and gravitational wave contribution for a range of inflation models.
Polynomial phi2 0.97 0.1
chaotic inflation phi4 0.95 0.2
Power-law inflation exp(- lambda phi) any n < 1 2pi(1 - n)
`Natural' inflation 1 + cos(phi / f ) any n < 1 0
Hybrid inflation (standard) 1 + Bphi2 1 0
Hybrid inflation (extreme) 1 + Bphi2 1 < n < 1.15 ~ 0

Observations have some way to go before the power-law approximation becomes inadequate. Consequently ...

8 Namely, Equation 64a Back.

Next Contents Previous