The Chaotic Model 167fluctuations. This can be seen if one transforms the de Sitter metric (5.60) into the
metric of a Euclidean four sphere [Equation (2.28)]. Bose fields (like the inflaton)
obeying a massless Klein–Gordon equation turn out to oscillate harmonically on
this sphere with period 2휋∕퐻, which is equivalent to considering quantum statis-
tics at a temperature푇dS=퐻∕ 2 휋. However, the temperature in de Sitter space is
highly unusual in that the fluctuations on the four sphere are periodic in all four
dimensions [4, 5].
The fate of a bubble of space-time clearly depends on the starting value of휙.Only
when it is large enough will inflationary expansion commence. If휙is very much
larger than푀P, Equation (7.41) shows that the rate of expansion is faster than the
timescale휏,
퐻≫ 2√
휋
3
푚휙≃^2
휏
. (7.54)
Although the wavelengths of all quantum fields then grow exponentially, the change
훥휙in the value of the inflaton field itself may be small. In fact, when the physical
wavelengths have reached the size of the Hubble radius퐻−^1 , all changes in휑are
impeded by the friction 3퐻̇휑in Equation (7.37), and fluctuations of size훿휙freeze to
an average nonvanishing amplitude of
|훿휙(푥)|≃퐻
2 휋
. (7.55)
Consequently, the vacuum no longer appears empty and devoid of properties.
Fluctuations of a length scale exceeding퐻−^1 are outside the present causal horizon
so they no longer communicate, crests and troughs in each oscillation mode remain
frozen. But at the end of inflation, the expansion during radiation and matter dom-
ination starts to return these frozen fluctuations inside the horizon. With time they
become the seeds of perturbations we now should observe in the CMB and in the
density distribution of matter.
The quantum fluctuations remaining in the inflaton field will cause the energy to be
dumped into entropy at slightly fluctuating times. Thus, the Universe will also contain
entropy fluctuations as seeds of later density perturbations.
Note that the quantum fluctuations amplified during inflation also lead to selfre-
production and therefore do not allow inflation to end once it has started. Inflation
continues forever leading to a metaphysical (nonverifiable) concept of eternal uni-
verse and multiverse. This damages the predictive power of the theory because in this
case ‘anything can happen and will happen an infinite number of times’.
Linde’s Bubble Universe. Since our part of the pre-inflationary universe was so
small, it may be considered as just one bubble in a foam of bubbles having differ-
ent fates. In Linde’s chaotic model each original bubble has grown in one e-folding
time휏=퐻−^1 to a size comprising푒^3 mini-universes, each of diameter퐻−^1 .Inhalf
of these mini-universes, on average, the value of휙may be large enough for inflation
to continue, and in one-half it may be too small. In the next e-folding time the same
pattern is repeated. Linde has shown that in those parts of space-time where휙grows