Introduction to Cosmology

Consensus Inflation 159

In small-field models the field moves over a small (subPlanckian) distance. A gen-
eral parametrization is the Higgs-like potential

푉(휙)=푉 0 [ 1 −(휙∕휇)푝]+... (7.26)

where the dots represent higher-order terms that become important near the end of
inflation.
In large-field models the inflaton field starts at large field values and then evolves to
a minimum at the origin휙=0. The prototypical large-field model ischaotic inflation
where a single monomial term dominates the potential

푉(휙)=휆푝휙푝. (7.27)

If the field is sufficiently homogeneous, we have

(∇휙)^2 ≪푉(휙), (7.28)

and the(∇휙)^2 term in Equation (7.25) then drops out.
The stress-energy tensor for a scalar field is

푇휇휈=(휕휇휙휕휈휙−푔휇휈)휙, (7.29)

and, for a homogeneous field, it takes the form of a perfect fluid with energy density

휌=

1

2

휙̇^2 +푉(휙),

and pressure

푝=

1

2

휙̇^2 −푉(휙).

In the de Sitter limit when푝≃−휌, the potential energy of the field dominates the
kinetic energy,휙̇^2 ≪푉(휙), and the speed of the expansion,퐻=푎̇∕푎is large. The poten-
tial energy then acts almost as a cosmological constant 8휋퐺푉 0 ≡휆.Also퐻is almost
constant and the Universe expands quasi-exponentially

푎(푡)≃exp

(

∫ 퐻d푡

)

≡푒−푁. (7.30)

This limit is referred to asslow-roll,and푁is the number of e-folds that the Universe
expands.
Let us rewrite theRaychauduri Equation(5.6) in the form
푎̈
푎

=퐻^2 ( 1 −휖), (7.31)

where the parameter휖specifies the Equation of State

휖≡^3

2

(

푝

휌

+ 1

)

=^4 휋퐺

푐^2

( ̇

휙

퐻

) 2

=−dln퐻

dln푎

=퐻−^1 d퐻

d푁

. (7.32)

The de Sitter limit푝≃−휌is equivalent to휖→0 and accelerated expansion푎̈푎>0is
equivalent to휖<1. Inflation takes place whenever휖<1.