and Cosmology

4. Cosmology I: Homogeneous Isotropic World Models
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Fig. 4.17. During an inflationary phase, indicated here by the
gray bar, the Universe expands exponentially; see (4.77). This
phase comes to an end when a phase transition transforms
the vacuum energy into matter and radiation, after which the
Universe follows the normal Friedmann expansion
the exponential expansion. For illustration we consider
a very small region in space of size L < ct i at a time
t i ∼ 10 −34 s prior to inflation which is in causal contact.
Through inflation, it expands tremendously, e.g.,
by a factor ∼ 10 40 ; the original L ∼ 10 −24 cm inflate to
about 10 16 cm by the end of the inflationary phase, at
t f ∼ 10 −32 s. By today, this spatial region will have expanded
by another factor of ∼ 10 25 by following (for
t > t f ) the normal cosmic expansion, to ∼ 10 41 cm. This
scale is considerably larger than the size of the currently
visible Universe, c/H 0 . According to this scenario, the
whole Universe visible today was in causal contact prior
to inflation, so that the homogeneity of the physical conditions
at recombination, and with it the nearly perfect
isotropy of the CMB, is provided by causal processes.
Inflation Solves the Flatness Problem as well. Due
to the tremendous expansion, any initial curvature is
straightened out (see Fig. 4.18). Formally this can be
seen as follows: during the inflationary phase we have
Ω Λ =
Λ
3H 2 = 1 ,
and since it is assumed that the inflationary phase lasts
long enough for the vacuum energy to be completely
Fig. 4.18. Due to tremendous expansion during inflation, even
a Universe with initial curvature will appear to be a flat
Universe by the end of the inflationary phase
dominant, when it ends we then have Ω 0 = 1. Hence the
Universe is flat to an extremely good approximation.
The inflationary model of the very early Universe
predicts that today Ω 0 = 1 is valid to very high precision;
any other value of Ω 0 would require another
fine-tuning. Thus the Universe is flat.
The physical details of the inflationary scenario are
not very well known. In particular it is not yet understood
how the phase transition at the end of the
inflationary phase took place and why it did not occur
earlier. But the two achievements presented above (and
some others) make an inflationary phase appear a very
plausible scenario. As we will see below (Chap. 8), the
prediction of a flat Universe was recently accurately
tested and it was indeed confirmed. Furthermore, the
inflationary model provides a natural explanation for
the origin of density fluctuations in the Universe which
must have been present at very early epochs as the seeds
of structure formation. We will discuss these aspects
further in Chap. 7.