258 Inflation I: homogeneous limit
coincide with (5.109) if we setF=∂f/∂Rand take the following potential for the
scalar field:
V(φ)=1
16 πf−R∂f/∂R
(∂f/∂R)^2. (5.114)
Problem 5.16Study the inflationary solutions inR^2 gravity:
S=−1
16 π∫(
R−
1
6 M^2
R^2
)
√
−gd^4 x. (5.115)What is the physical meaning of the constantM?
Thus, the higher derivative gravity theory is conformally equivalent to Einstein
gravity with an extra scalar field. If the scalar field potential satisfies the slow-roll
conditions, then we have an inflationary solution in the conformal frame for the
metricg ̃μν. However, one should not confuse the conformal metric with the original
physical metric. They generally describe manifolds with different geometries and
the final results must be interpreted in terms of the original metric. In our case the
use of the conformal transformation is a mathematical tool which simply allows
us to reduce the problem to one we have studied before. The conformal metric is
related to the physical metricgμνby a factorF, which depends on the curvature
invariants; it does not change significantly during inflation. Therefore, we also have
an inflationary solution in the original physical frame.
So far we have been considering inflationary solutions due to the potential of
the scalar field. However, inflation can be realized even without a potential term.
It can occur in Born–Infeld-type theories, where the action depends nonlinearly on
the kinetic energy of the scalar field. These theories do not have higher-derivative
terms, but they have some other peculiar properties.
Problem 5.17Consider a scalar field with action
S=∫
p(X,φ)√
−gd^4 x, (5.116)wherepis an arbitrary function ofφandX≡^12
(
∂μφ∂μφ)
. Verify that the energy–
momentum tensor for this field can be written in the form
Tνμ=(ε+p)uμuν−pδμν, (5.117)where the Lagrangianpplays the role of the effective pressure and
ε= 2 X
∂p
∂X−p, uν=
∂νφ
√
2 X. (5.118)
If the Lagrangianpsatisfies the conditionX∂p/∂Xpfor some range ofXand
φ, then the equation of state isp≈−εand we have an inflationary solution. Why