4.3. 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 [6].

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

(44)

where the spectral indices n and n_G are given by

(45)

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. (43) is

(46)

(since within the slow-roll approximation k exp N). Direct differentiation then yields [6]

(47)

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. [18]) gives

(48)

Here the C_l are the contributions to the microwave multipoles, in the usual notation. ⁽⁵⁾

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. (48) is so large, gravitational waves can have a significant effect even if is quite a bit smaller than one.

At present, a large number of inflationary models exist covering a large part of the n-r parameter space. Observations are just beginning to narrow down the allowed region, and in the future satellite microwave anisotropy experiments such as MAP and Planck [20] should determine n sufficiently accurately to exclude almost all models of inflation on that basis, and may be able to measure r as well.

The principal observational challenge is to untangle the effects of the inflationary parameters (_H(k₀), n and r) from all the other parameters required to specify a complete cosmological model, such as the Hubble constant, the density of each component of matter, and so on. The two sets of parameters cannot be studied separately; an attempt to match the observations must fit for both simultaneously. A typical set of parameters likely to be important in determining predictions for observations such as microwave anisotropies contains about ten different parameters, with some authors suggesting this list extends up to fifteen or more. It is a testament to the predicted accuracy of upcoming observations that considerable progress is expected in this direction over the next decade.

⁵ Namely, Back.