6.3. Observational consequences
Observations have moved on beyond us wanting to know the overall normalization of the potential. The interesting things are
These can be neatly summarized using the slow-roll parameters and we defined earlier. [3]
The standard approximation used to describe the spectra is the power-law approximation, where we take
where the spectral indices n and n_{G} are given by
The power-law approximation is usually valid because only a limited range of scales are observable, with the range 1 Mpc to 10^{4} Mpc corresponding to ln k 9.
The crucial equation we need is that relating values to when a scale k crosses the Hubble radius, which from Eq. (58) is
(since within the slow-roll approximation k exp N). Direct differentiation then yields [3]
(62)
(63)
where now and 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
Here the C_{l} 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.
MODEL | POTENTIAL | n | R |
Polynomial | ^{2} | 0.97 | 0.1 |
chaotic inflation | ^{4} | 0.95 | 0.2 |
Power-law inflation | exp(- ) | any n < 1 | 2(1 - n) |
`Natural' inflation | 1 + cos( / f ) | any n < 1 | 0 |
Hybrid inflation (standard) | 1 + B^{2} | 1 | 0 |
Hybrid inflation (extreme) | 1 + B^{2} | 1 < n < 1.15 | ~ 0 |
Observations have some way to go before the power-law approximation becomes inadequate. Consequently ...