Physical Foundations of Cosmology

5.4 How to realize the equation of state p≈−ε 239

time. It follows from (5.33) that inflation lasts for

ttf−ti

√

12 π(φi/m). (5.36)

During this time interval the scale factor increases

af

ai

exp

(

2 πφi^2

)

(5.37)

times. The results obtained are in good agreement with the previous rough estimates
(5.18) and (5.19). Inflation lasts more than 75 e-folds if the initial value of the scalar
field,φi, is four times larger than the Planckian value. To obtain an estimate for the
largest possible increase of the scale factor during inflation, let us consider a scalar
field of mass 10^13 GeV. The maximal possible value of the scalar field for which
we still remain in the sub-Planckian domain isφi∼ 106 , and hence
(
af
ai

)

max

∼exp

(

1012

)

. (5.38)

Thus, the actual duration of the inflationary stage can massively exceed the 75
e-folds needed. In this case our universe would constitute only a very tiny piece of
an incredibly large homogeneous domain which originated from one causal region.
The other important feature of inflation is that the Hubble constant decreases only
by a factor 10−^6 , while the scale factor grows by the tremendous amount given in
(5.38), that is,

Hi

Hf



af

ai

.

Graceful exit and afterwardsAfter the field drops below the Planckian value it
begins to oscillate. To determine the attractor behavior in this regime we note that

φ ̇^2 +m^2 φ^2 =

3

4 π

H^2 (5.39)

and use the Hubble parameterHand the angular variableθ, defined via

φ ̇=

√

3

4 π

Hsinθ, mφ=

√

3

4 π

Hcosθ, (5.40)

as the new independent variables. It is convenient to replace (5.27) by a system of
two first order differential equations forHandθ:

H ̇=− 3 H^2 sin^2 θ, (5.41)

θ ̇=−m−^3

2

Hsin 2θ, (5.42)