Physical Foundations of Cosmology

7.2 Equations for cosmological perturbations 297

7.2 Equations for cosmological perturbations

To derive the equations for the perturbations we have to linearize the Einstein
equations

Gαβ≡Rαβ−^12 δαβR= 8 πGTβα,

for small inhomogeneities about a Friedmann universe. The calculation of the
Einstein tensor for the background metric (7.3) is very simple and the result is

( 0 )G 0

0 =

3 H^2

a^2

, (^0 )G^0 i= 0 , (^0 )Gij=

1

a^2

(

2 H′+H^2

)

δij, (7.32)

whereH≡a′/a.It is clear that in order to satisfy the background Einstein equa-
tions, the energy–momentum tensor for the matter,(^0 )Tβα,must have the following
symmetry properties:

( 0 )T 0

i =^0 ,

( 0 )Ti

j∝δij. (7.33)

For a metric with small perturbations, the Einstein tensor can be written as
Gαβ=(^0 )Gαβ+δGαβ+···, whereδGαβdenote terms linear in metric fluctuations.
The energy–momentum tensor can be split in a similar way and the linearized
equations for perturbations are

δGαβ= 8 πGδTβα. (7.34)

NeitherδGαβnorδTβαare gauge-invariant. Combining them with the metric per-
turbations, however, we can construct corresponding gauge-invariant quantities.

Problem 7.4Derive the transformation laws forδTβαand verify that

δT

0

0 =δT

0

0 −

(( 0 )

T 00

)′(

B−E′

)

,

δT

0

i=δT

0

i −

(( 0 )

T 00 −(^0 )Tkk/ 3

)(

B−E′

)

,i,

δT

i

j= δT

i

j−

(( 0 )

Tji

)′(

B−E′

)

,

(7.35)

whereTkkis the trace of the spatial components, are gauge-invariant.

In a similar manner to (7.35), we can construct

δG

0

0 =δG

0

0 −

(( 0 )

G^00

)′(

B−E′

)

,etc. (7.36)

and rewrite (7.34) in the form

δG

α

β=^8 πGδT

α

β. (7.37)