Physical Foundations of Cosmology

242 Inflation I: homogeneous limit

during these last 70 e-folds depends on the shape of the potential only within a
rather narrow interval of scalar field values.

Problem 5.7Find the time dependence of the scale factor for the power-law po-
tential and estimate the duration of inflation.

Problem 5.8Verify that for a general potentialVthe system of equations (5.24),
(5.25) can be reduced to the following first order differential equation:

dy

dx

=− 3

(

1 −y^2

)(

1 +

V,x

6 yV

)

, (5.54)

where

x≡

√

4 π

3

φ; y≡

√

4 π

3

dφ

dlna

.

Assuming thatV,φ/V→0as|φ|→∞, draw the phase diagram and analyze the
behavior of the solutions in different asymptotic regions. Consider separately the
case of the exponential potential. What is the physical meaning of the solutions in
the regions corresponding toy>1?

After the end of inflation the scalar field begins to oscillate and the universe enters
the stage of deceleration. Assuming that the period of oscillation is smaller than the
cosmological time, let us determine the effective equation of state. Neglecting the
expansion and multiplying (5.24) byφ, we obtain

(φφ ̇)·−φ ̇^2 +φV,φ 0. (5.55)

As a result of averaging over a period, the first term vanishes and hence

〈

φ ̇^2

〉

〈 

φV,φ

〉

. Thus, the averaged effective equation of state for an oscillating scalar
field is

w≡

p

ε



〈

φV,φ

〉

−〈 2 V〉

〈

φV,φ

〉

+〈 2 V〉

. (5.56)

It follows that forV∝φnwe havew(n− 2 )/(n+ 2 ). For an oscillating massive
field(n= 2 )we obtainw0 in agreement with our previous result. In the case
of a quartic potential(n= 4 ), the oscillating field mimics an ultra-relativistic fluid
withw 1 /3.
In fact, inflation can continue even after the end of slow-roll. Considering the
potential which behaves as

V∼ln(|φ|/φc)