Physical Foundations of Cosmology

5.6 “Menu” of scenarios 257

can be achieved by a fundamental scalar field or by a fermionic condensate described
in terms of an effective scalar field. This, however, does not exhaust all possibilities.
The scalar condensate can also be imitated entirely within the theory of gravity it-
self. Einstein gravity is only a low curvature limit of some more complicated theory
whose action contains higher powers of the curvature invariants, for example,

S=−

1

16 π

∫

(

R+αR^2 +βRμνRμν+γR^3 +···

)√

−gd^4 x. (5.107)

The quadratic and higher-order terms can be either of fundamental origin or they can
arise as a result of vacuum polarization. The corresponding dimensional coefficients
in front of these terms are likely of Planckian size. The theory with (5.107) can
provide us with inflation. This can easily be understood. Einstein gravity is the only
metric theory in four dimensions where the equations of motion are second order.
Any modification of the Einstein action introduces higher-derivative terms. This
means that, in addition to the gravitational waves, the gravitational field has extra
degrees of freedom including, generically, a spin 0 field.

Problem 5.15Consider a gravity theory with metricgμνand action

S=

1

16 π

∫

f(R)

√

−gd^4 x, (5.108)

wheref(R)is an arbitrary function of the scalar curvatureR. Derive the following
equations of motion:

∂f

∂R

Rμν−

1

2

δμνf+

(

∂f

∂R

);α

;α

δνμ−

(

∂f

∂R

);μ

;ν

= 0. (5.109)

Verify that under the conformal transformationgμν→g ̃μν=Fgμν, the Ricci ten-
sor and the scalar curvature transform as

Rμν→R ̃μν=F−^1 Rμν−F−^2 F;;νμ−

1

2

F−^2 F;;ααδνμ+

3

2

F−^3 F;νF;μ, (5.110)

R→R ̃=F−^1 R− 3 F−^2 F;;αα+

3

2

F−^3 F;αF;α. (5.111)

Introduce the “scalar field”

φ≡

√

3

16 π

lnF(R), (5.112)

and show that the equations

R ̃νμ−^1

2

R ̃δνμ= 8 πT ̃νμ(φ), (5.113)