The Chaotic Model 165The metric perturbations can be decomposed into scalar, vector, and tensor pertur-
bations, of which the vectors can be ignored because they are not created by inflation
and they decay with the expansion. The scalar perturbations are observed and the ten-
sor ones will be observed as gravitational waves in the future. The ratio of tensor to
scalar perturbations, denoted푟, is an observable signalling gravitational waves. By the
Lyth bound푟correlates with the inflaton field moving over superPlanckian distance
during inflation, provided the scalar field space is large,>푀푃, and if also the flatness
of the inflaton potential is controlled dynamically over a superPlanckian field range.
During inflation the inflationary energy is the dominant contribution to the
stress-energy tensor of the universe so that the inflaton perturbations back-react on
the spacetime geometry.
Primordial perturbations can give rise to nonGaussianities, but in single-field
slow-roll inflation they are expected to be small. Large nonGaussianities can only arise
if inflaton interactions are significant during inflation.
Initially all space-time regions of size퐻−^1 would contain inhomogeneities inside
their respective event horizons. At every instant during the inflationary de Sitter stage
an observer would see himself surrounded by a black hole with event horizon퐻−^1
(but remember that ‘black hole’ really refers to a static metric). There is an analogy
between the Hawking radiation of black holes and the temperature in an expanding
de Sitter space. Black holes radiate at the Hawking temperature푇H[Equation (5.83)],
while an observer in de Sitter space will feel as if he is in a thermal bath of temperature
푇dS=퐻∕^2 휋.
Within a time of the order of퐻−^1 all inhomogeneities would have traversed the
Hubble radius. Thus they would not affect the physics inside the de Sitter universe
which would be getting increasingly homogeneous and flat. On the other hand, the
Hubble radius is also receding exponentially, so if we want to achieve homogeneity it
must not run away faster than the inhomogeneities.
The theory of perturbations during inflation is, however, beyond the scope of the
present monograph. If nonGaussianities were to be observed they would be most use-
ful to select between different inflationary scenarios.
7.3 The Chaotic Model
Consider the case of chaotic inflation with the potential
푉(휙)=^1
2푚^2 휙휙^2. (7.45)
The time dependence of the field is then
휙(푡)=휙푎−푚휑푀P
2
√
3 휋
푡≡휙푎
(
1 −푡
휏
)
, (7.46)
where휏(휙푎)is the characteristic timescale of the expansion. At early times when푡≪휏
the scalar field remains almost constant, changing only slowly from a value휙푎≫푀푃
to its ultimate value휙 0. The scale factor then grows quasi-exponentially as
푅(푡)=푅(푡푎)exp(
Ht−^1
6