Physical Foundations of Cosmology

5.4 How to realize the equation of state p≈−ε 241

5.4.2 General potential: slow-roll approximation

Equation (5.24) for a massive scalar field in an expanding universe coincides with
the equation for a harmonic oscillator with a friction term proportional to the Hubble
parameterH. It is well known that a large friction damps the initial velocities and
enforces a slow-roll regime in which the acceleration can be neglected compared to
the friction term. Because for a general potentialH∝

√

ε∼

√

V, we expect that
for large values ofVthe friction term can also lead to a slow-roll inflationary stage,
where ̈φis negligible compared to 3Hφ ̇. Omitting the ̈φterm and assuming that
φ ̇^2 V, (5.24) and (5.25) simplify to

3 Hφ ̇+V,φ 0 , H≡

(

dlna

dt

)



√

8 π

3

V(φ). (5.48)

Taking into account that

dlna

dt

=φ ̇

dlna

dφ

−

V,φ

3 H

dlna

dφ

,

equations (5.48) give

−V,φ

dlna

dφ

 8 πV (5.49)

and hence

a(φ)aiexp

(

8 π

∫φi

φ

V

V,φ

dφ

)

. (5.50)

This approximate solution is valid only if the slow-roll conditions
∣
∣φ ̇^2

∣

∣|V|, |φ ̈| 3 Hφ ̇∼

∣

∣V,φ

∣

∣, (5.51)

used to simplify (5.24) and (5.25), are satisfied. With the help of (5.48), they can
easily be recast in terms of requirements on the derivatives of the potential itself:
(
V,φ
V

) 2

 1 ,

∣

∣

∣∣V,φφ

V

∣

∣

∣∣ 1. (5.52)

For a power-law potential,V=( 1 /n)λφn, both conditions are satisfied for|φ|1.
In this case the scale factor changes as

a(φ(t))aiexp

(

4 π

n

(

φi^2 −φ^2 (t)

)

)

. (5.53)

It is obvious that the bulk of the inflationary expansion takes place when the scalar
field decreases by a factor of a few from its initial value. However, we are interested
mainly in the last 50–70 e-folds of inflation because they determine the structure
of the universe on present observable scales. The detailed picture of the expansion