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 and we defined earlier. [3]

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

(59)

where the spectral indices n and n_G are given by

(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 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

(61)

(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

(64)

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

From these expressions we immediately see

• If and only if << 1 and || << 1 do we get n 1 and R 0.
• Because the coefficient in Eq. (64) is so large, gravitational waves can have a significant effect even if is quite a bit smaller than one.

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.

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 + B2	1	0
Hybrid inflation (extreme)	1 + B2	1 < n < 1.15	~ 0

⁸ Namely, Back.