236 Inflation I: homogeneous limit
from the Klein–Gordon equation (1.57) or by substituting (5.22) and (5.23) into the
conservation law (1.65). The result is
φ ̈+ 3 Hφ ̇+V,φ= 0 , (5.24)whereV,φ≡∂V/∂φ. This equation has to be supplemented by the Friedmann
equation:
H^2 =
8 π
3(
1
2
φ ̇^2 +V(φ))
, (5.25)
where we have setG=1 andk=0. We first find the solutions of (5.24) and (5.25)
for a free massive scalar field and then study the behavior of the scalar field in the
case of a general potentialV(φ).
5.4.1 Simple example: V=^12 m^2 φ^2.SubstitutingHfrom (5.25) into (5.24), we obtain the closed form equation forφ,
φ ̈+√
12 π(
φ ̇^2 +m^2 φ^2) 1 / 2
φ ̇+m^2 φ= 0. (5.26)This is a nonlinear second order differential equation with no explicit time depen-
dence. Therefore it can be reduced to a first order differential equation for ̇φ(φ).
Taking into account that
φ ̈=φ ̇dφ ̇
dφ,
(5.26) becomes
dφ ̇
dφ=−
√
12 π(
φ ̇^2 +m^2 φ^2) 1 / 2
φ ̇+m^2 φ
φ ̇, (5.27)
which can be studied using the phase diagram method. The behavior of the solutions
in theφ– ̇φplane is shown in Figure 5.3. The important feature of this diagram is the
existence of an attractor solution to which all other solutions converge in time. One
can distinguish different regions corresponding to different effective equations of
state. Let us consider them in more detail. We restrict ourselves to the lower right
quadrant (φ>0, ̇φ<0); solutions in the other quadrants can easily be derived
simply by taking into account the symmetry of the diagram.
Ultra-hard equation of stateFirst we study the region where|φ ̇|mφ. It describes
the situation when the potential energy is small compared to the kinetic energy, so
that ̇φ^2 V. It follows from (5.22) and (5.23) that in this case the equation of state