6.4. Testing the idea of inflation

The moral of the previous section was that different inflation models lead to very different models of structure formation, spanning a wide range of possibilities. That means, for example, that a definite measure of say the spectral index n would rule out most inflation models. But it would always be possible to find models which did give that value of n. Is there any way to try and test the idea of inflation, independently of the model chosen?

The answer, in principle, is yes. In the previous section we introduced three observables (in addition to the overall normalization), namely n, R and nG. However, they depend only on two fundamental parameters, namely and . [3] We can therefore eliminate and to obtain a relation between observables, the consistency equation

(65)

This relation has been much discussed in the literature. [26, 21] It is independent of the choice of inflationary model (though it does rely on the slow-roll and power-law approximations).

The idea of a consistency equation is in fact very general. The point is that we have obtained two continuous functions, H(k) and AG(k), from a single continuous function V(). This can only be possible if the functions H(k) and AG(k) are related, and the equation quoted above is the simplest manifestation of such a relation.

Vindication of the consistency equation would be a remarkably convincing test of the inflationary paradigm, as it would be highly unlikely that any other production mechanism could entangle the two spectra in the way inflation does. Unfortunately though, measuring nG is a much more challenging observational task than measuring n or R and is likely to be beyond even next generation observations. Indeed, this is a good point to remind the reader that even if inflation is right, only one model can be right and it is perfectly possible (and maybe even probable, see Ref. [27]) that that model has a very low amplitude of gravitational waves and that they will never be detected.