MODERN COSMOLOGY

Inflationary cosmology 39

Figure 2.4.This plot shows how fluctuations in the scalar field transform themselves into
density fluctuations at the end of inflation. Different points of the universe inflate from
points on the potential perturbed by a fluctuationδφ, like two balls rolling from different
starting points. Inflation finishes at times separated byδtin time for these two points,
inducing a density fluctuationδ=Hδt.

where the last step uses the crucial input of quantum field theory, which says that
the rmsδφis given byH/ 2 π. This is the classical amplitude that results from the
stretching of sub-horizon flat-space quantum fluctuations. We will not attempt to
prove this key result here (see chapter 12 of Peacock 1999, or Liddle and Lyth
1993, 2000).
Because the de Sitter expansion is invariant under time translation, the
inflationary process produces a universe that is fractal-like in the sense that scale-
invariant fluctuations correspond to a metric that has the same ‘wrinkliness’ per
log length-scale. It then suffices to calculate that amplitude on one scale—i.e.
the perturbations that are just leaving the horizon at the end of inflation, so that
super-horizon evolution is not an issue. It is possible to alter this prediction of
scale invariance only if the expansion is non-exponential; we have seen that such
deviations plausibly do exist towards the end of inflation, so it is clear that exact
scale invariance is not to be expected. This is discussed further later.
In summary, we have the following three key equations for basic inflationary
model building. The fluctuation amplitude can be thought of as supplying the
variance per lnkin potential perturbations, which we show later does not evolve
with time:

δH^2 ≡^2 (k)=

H^4

( 2 πφ) ̇^2

H^2 =

8 π

3

V

m^2 P

3 Hφ ̇=−V′.

We have also written once again the exact relation betweenHandVand the