166 Cosmic Inflation
with퐻given by
퐻= 2√
휋
3
푚휙
푀P
휙푎. (7.48)
At time휏, the Universe has expanded from a linear size푅(푡푎)to
푅(휏)≃푅(푡푎)exp(퐻휏)=푅(푡푎)exp(
4 휋휙^2 푎
푀P^2
)
. (7.49)
For instance, a universe of linear size equal to the Planck length푅(푡푎)≃ 10 −^35 mhas
grown to
푅(휏)≃푅(푡푎)exp(
4 휋푀P^2
푚^2 휙
)
. (7.50)
For a numerical estimate we need a value for the mass푚휙of the inflaton. This is
not known, but we can make use of the condition that the chaotic model must be able
to form galaxies of the observed sizes. Then the scalar mass must be of the order of
magnitude
푚휙≃ 10 −^6 푀P. (7.51)Inserting this estimate into Equation (7.43) we obtain the completely unfathomable
scale
푅(휏)≃ 10 −^35 +exp(^4 휋×^10(^12) )
m≃ 105.^5 ×^10
12
m. (7.52)
It is clear that all the problems of the standard Big Bang model discussed in Sec-
tion 7.1 then disappear. The homogeneity, flatness and isotropy of the Universe turn
out to be consequences of the inflaton field having been large enough in a region of
size푀P−^1 at time푡P. The inflation started in different causally connected regions of
space-time ‘simultaneously’ to within 10−^43 s, and it ended at about 10−^35 s. Our part
of that region was extremely small. Since the curvature term in Friedmann’s equations
decreased exponentially, the end result is exactly as if푘had been zero to start with. A
picture of this scenario is shown in Figure 7.3.
Quantum Fluctuations. At Planck time the universe cannot have been completely
homogeneous and isotropic because of quantum fluctuations. Across the horizon of
size푀P−^1 the field may have varied by an amount
훥휑푎≃푀p. (7.53)
But in quantum mechanics we noted that at Planck time the field휙was indefinite
by푀p, at least, so that there were deviations from a pure de Sitter universe. Even if
this universe was empty, quantum field theory tells us that empty space is filled with
zero-point quantum fluctuations of all kinds of physical fields, here fluctuations from
the classical de Sitter inflaton field.
The vacuum fluctuation spectrum of the slowly rolling scalar field during the
inflationary expansion turns out to be quite unlike the usual spectrum of thermal