PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

CHAPTER 1. INTRODUCTION
with the scalar field are given by [6, 7]
ρ φ = ˙φ 2
+V(φ), (1.5a)
2
p φ = ˙φ 2
−V(φ), (1.5b)
2
where the overdots again represent differentiation with respect to the cosmic time, while
V(φ) denotes the potential that describes the inflaton 2 . For the above energy density
and pressure, the condition that leads to an accelerated expansion reduces to ˙φ 2 < V(φ),
i.e. when the kinetic energy of the scalar field is less than the potential energy associated
with the field.
The equation of motion satisfied by the scalar field in the homogeneous and isotropic
Friedmann background is given by [6, 7]
¨φ+3H ˙φ+V φ = 0, (1.6)
whereV φ = dV/dφ. It is clear from this equation that, while the termV φ drives the field in
the potential, the frictional term 3H ˙φ, which arises due to the expansion of the universe,
slows the field down. We saw above that the condition for inflation to occur corresponds
to ˙φ 2 < V(φ). Though this condition ensures accelerated expansion, it is often demanded
that the field rolls slowly in the potential so that one has ˙φ 2 ≪ V(φ), which guarantees
that inflation does indeed take place. Moreover, the field also needs to be slowly rolling
for a sufficient duration in order to achieve the necessary amount of inflation. A suitably
long duration of inflation is accomplished provided the frictional term is strong enough
to overcome the acceleration of the field, i.e. when ¨φ ≪ 3H ˙φ. The above two inequalities,
in fact, constitute the conditions for the so-called slow roll inflationary scenario [33].
Actually, nowadays, it is more common to characterize the evolution of the scalar field in
terms of a hierarchy of slow roll parameters. The first slow roll parameter is defined as
ǫ 1 = − Ḣ
H2, (1.7)
and it is useful to note here that the condition for inflation, viz. ä > 0, corresponds to
ǫ 1 < 1. The second and the higher order slow roll parameters are defined in terms of the
first as follows:
ǫ i+1 = dlnǫ i
dN , (1.8)
2 It is common to refer to the scalar field that drives inflation as the inflaton.
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