192	5 Cosmology and General Relativity

5.9.2.2 Evaluation of the Effective Mass Term in the Slow Roll Approximation

Let us now evaluate the effective η-dependent mass term

2	(η) =	θ	(5.9.35)
meff		θ (η)

in the slow roll approximation.

Starting from its definition in (5.9.27), the function θ (η) can be rewritten as:

H

dt

1

θ =

=

a˙ da a

=

H

(5.9.36)

aϕ

dt

ϕ˙

aϕ

a da

˙

Calculating the first derivative of θ we obtain:

θ = a dt

aϕ

= a aϕ − a2ϕ2 a˙ ϕ˙ − a2ϕ2 ϕ¨

d

H

H

H

H a

˙

˙

˙

˙

˙

H 2

H

H 2

= ϕ˙ −

−

ϕ¨ ≈ ϕ˙ −

(5.9.37)

ϕ

ϕ2

ϕ

˙

˙

˙

where we used the exact result (5.8.23) and where the last approximate equality follows from the slow-rolling condition ϕ¨ ≈ 0. Using this approximate result in the calculation of the second derivative we obtain:

d

H 2

d H 2

2H H

H 2

θ = a

ϕ˙ −

−a

= −a

ϕ

˙

−

ϕ¨

dt

ϕ

dt

ϕ

ϕ2

˙

˙

˙

˙

˙

≈ −2 H H ϕ˙

Consequently we obtain:

	m2	(η)	≡	θ	(η) slow roll	2a2H
				θ
	eff				−	˙

During an almost exponential expansion the scale factor behaves as:

a − 1 H η

Hence we conclude that in the slow roll approximation we have

mslow2 -roll	≈ η2 × μ;		μ ≡	− H˙2
	2			H

(5.9.38)

(5.9.39)

(5.9.40)

(5.9.41)

In other words the effective scalar degree of freedom propagates as if it were in de Sitter space yet with an effective η-dependent mass term depressed by the almost

		H	'	H 2	in the slow roll phase. Note
constant factor μ which is very small since ˙
also that μ is positive since	˙	is negative. Indeed in the slow rolling phase the
	H

expansion is slowly decelerating.

5.9 Primordial Perturbations of the Cosmological Metric and of the Inflaton

193

5.9.2.3 Derivation of the Propagation Equation

Having discussed its meaning, let us derive the propagation equation (5.9.26) of the scalar perturbation.

Combining the result (5.9.18)–(5.9.20) for the Einstein tensor with the result (5.9.22)–(5.9.24) for the stress-energy tensor we obtain a set of equations which immediately yield the following constraints. Since the perturbation δTij of the stressenergy tensor is diagonal and proportional to the Kronecker δij , the same must be true of the corresponding Einstein tensor. This occurs if and only if:

Φ = Ψ

(5.9.42)

which therefore has to be imposed. Next let us consider the implication of the equation δG0i = 4π GδT 0i . This latter can be rewritten as follows:

1	Ψ	+ H Φ − 2π Gϕ δϕ = 0	(5.9.43)
a2 ∂i

Using (5.9.42) and fixing the boundary condition at some reference time, (5.9.43) implies:

Ψ + H Ψ − 2π Gϕ δϕ = 0

(5.9.44)

By means of identities following from the equations satisfied by the background fields, the constraint (5.9.44) can be rewritten in the following way:

d

a2

Ψ

aϕ

2

δϕ

= 2π G

H

+ Ψ

(5.9.45)

∂η

H

H

ϕ

To prove such a result we just compare the following elaborations of the l.h.s. and r.h.s. of (5.9.45).

d

a2Ψ

l.h.s. =

∂η

H

2aa

a2

a2

=

Ψ +

Ψ

−

H Ψ

H

H

H

= 2a2Ψ +

a2

−

a2

H

Ψ

(5.9.46)

Ψ

H

H

2

H ϕ

+ Ψ

r.h.s. = 2π G aϕ

δϕ

= a2

H

+

H 2

)

Ψ

2πH 2 )

2

H ϕ

2

G(ϕ

δϕ

2π G(ϕ

=

a2

2π Gϕ δϕ

+

−H + H

2

Ψ

H

H 2

Ψ

H

= a2

−

+ 2Ψ

(5.9.47)

H

H 2

194	5 Cosmology and General Relativity

showing that they are indeed equal. In going from the second to the third line of (5.9.47) we used the background field equation (5.9.8), while in going from the third to the fourth we used the constraint equation (5.9.44).

Let us next consider the equation δG00 = 4π GδT 00. In view of (5.9.18) and (5.9.22) we can write:

2Ψ − 3H Ψ + H Φ = 2π G ϕ δϕ − ϕ 2Φ + a2 dϕ δϕ
				dV
= 2π G ϕ δϕ − ϕ 2Φ − ϕ + 2H ϕ δϕ
= 2π G ϕ 2 ϕ		− 2H	ϕ − Φ		(5.9.48)
	δϕ		δϕ

where, in stepping from the first to the second line we have used the back-ground equation (5.9.6). Using the constraint equations (5.9.42) and (5.9.44) we can further rewrite (5.9.48) as follows:

2Ψ = 2π G ϕ 2 δϕ + H δϕ − Ψ

ϕϕ

	2π G(ϕ	2		d		δϕ
=		)			H		+ Ψ	(5.9.49)
	H			dη		ϕ

In stepping from the first to the second line of the above equation one makes once again use of (5.9.8) and (5.9.44).

Consider now the following redefined fields:

u =		a				(5.9.50)
			Ψ
		ϕ
				ϕ
v =	2π Ga δϕ +				Ψ	(5.9.51)
				H

in terms of them and of the function z(η) of the conformal time η defined in (5.9.27), (5.9.49) and (5.9.45) can be rewritten as:

2u = z			d				v		(5.9.52)
		dη					z
v =	1 d					(zu)			(5.9.53)
	z			dη

By taking the Laplacian 2 of the latter equation and the derivative dηd of the former one obtains a system from which we can eliminate v obtaining the second order equation (5.9.26) satisfied by the field u(η, xi ). Alternatively we can eliminate u obtaining the following field equation for the v(η, xi ) field:

v − 2v − z v = 0

(5.9.54)

z