Physical Foundations of Cosmology

8.3 Quantum cosmological perturbations 335

and in many cases (8.43) can be rearranged to givep=p(ε), the equation of state
for an isentropic fluid. Forp∝Xnwe havep=ε/( 2 n− 1 ), so, for example, the
Lagrangianp∝X^2 describes an “ultra-relativistic fluid” with equation of state
p=ε/3. In the general case,p=p(X,φ), the pressure cannot be expressed only
in terms ofεsinceXandφare independent. However, even in this case, the
hydrodynamical analogy is still useful. For a canonical scalar field we havep=
X−V(φ)and, correspondingly,ε=X+V.

8.3.1 Equations

Here we derive the equations for perturbations and recast them in a simple, conve-
nient form. The reader interested only in the final result can go directly to (8.56)–
(8.58).

Background The state of a flat, homogeneous universe is characterized completely
by the scale factora(η)and the homogeneous fieldφ 0 (η), which satisfy the familiar
equations

H^2 =

8 π

3

a^2 ε, (8.44)

and

ε′=ε,XX′ 0 +ε,φφ 0 ′=− 3 H(ε+p), (8.45)

whereX 0 =φ′ 02 /

(

2 a^2

)

and we have setG=1. Substitutingεfrom (8.44) into the
left hand side of (8.45), we obtain the relation

H′−H^2 =− 4 πa^2 (ε+p), (8.46)

which is useful in what follows.

PerturbationsTo derive the equations for inhomogeneities we must first express
the gauge-invariant perturbations of the energy–momentum tensorδT
α
βin terms of
the scalar field and metric perturbations. The calculation can easily be done in the
longitudinal gauge, where the metric takes the form

ds^2 =a^2 (η)

[

( 1 + 2 )dη^2 −( 1 − 2 )δikdxidxk

]

. (8.47)

To linear order in perturbation we have

δX=

1

2

δg^00 φ′ 02 +g^00 φ′ 0 δφ′= 2 X 0

(

− +

δφ′

φ′ 0

)

, (8.48)