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 nG 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 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
(since within the slow-roll approximation
k exp N). Direct
differentiation then yields
[3]
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 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.
Observations have some way to go before the power-law approximation becomes
inadequate. Consequently ...
(62)
(63)
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