Physical Foundations of Cosmology

7.1 Perturbations and gauge-invariant variables 293

Problem 7.1Consider a 4-scalarq(xρ)=(^0 )q(xρ)+δq,where(^0 )qis its back-
ground value, and verify that the perturbationδqtransforms under (7.8) as

δq→δq ̃=δq−(^0 )q,αξα. (7.13)

Similarly, show that for acovariant4-vector,

δuα→δu ̃α=δuα−(^0 )uα,γξγ−(^0 )uγξ,αγ. (7.14)

Of course the value of a 4-scalarqat a given point of the manifold does not change
as a result of the coordinate transformation, but the way we split it into a background
value and a perturbation depends on the coordinates used.

Let us write the spatial components of the infinitesimal vectorξα≡(ξ^0 ,ξi)as

ξi=ξ⊥i+ζ,i, (7.15)

whereξ⊥i is a 3-vector with zero divergence

(

ξ⊥i,i= 0

)

andζis a scalar function.
Since in the Friedmann universe(^0 )g 00 =a^2 (η)and(^0 )gij=−a^2 (η)δij,we obtain
from (7.12)

δg ̃ 00 =δg 00 − 2 a

(

aξ^0

)′

,

δg ̃ 0 i=δg 0 i+a^2

[

ξ⊥′i+

(

ζ′−ξ^0

)

,i

]

,

δg ̃ij=δgij+a^2

[

2

a′

a

δijξ^0 + 2 ζ,ij+

(

ξ⊥i,j+ξ⊥j,i

)

]

,

(7.16)

whereξ⊥i≡ξ⊥iand a prime denotes the derivative with respect to conformal timeη.
Combining these results with (7.4)–(7.6), we immediately derive the transformation
laws for the different types of perturbations.

Scalar perturbationsFor scalar perturbations the metric takes the form

ds^2 =a^2

[

( 1 + 2 φ)dη^2 + 2 B,idxidη−

(

( 1 − 2 ψ)δij− 2 E,ij

)

dxidxj

]

. (7.17)

Under the change of coordinates we have

φ→φ ̃=φ−

1

a

(

aξ^0

)′

, B→B ̃=B+ζ′−ξ^0 ,

ψ→ψ ̃=ψ+

a′

a

ξ^0 , E→E ̃=E+ζ.

(7.18)

Thus, onlyξ^0 andζcontribute to the transformations of scalar perturbations and by
choosing them appropriately we can make any two of the four functionsφ,ψ,B,E
vanish. The simplestgauge-invariantlinear combinations of these functions, which