Physical Foundations of Cosmology

304 Gravitational instability in General Relativity

η^2

δε

ε 1

9

20

∼

⏐Φ⏐

ηeq ∼k −^1 η

9

10

1

2

Fig. 7.2.

atη∼k−^1 .It follows from (7.47) that for a constant potential,the amplitude of
δε/ε 0 is always equal to− 2 on superhorizon scales.
The change in the amplitude of can also be inferred from a widely used
“conservation law” for the quantity

ζ≡

2

3

(

8 πG

3

) 1 / 2

θ^2

(u

θ

)′

. (7.72)

Substituting the long-wavelength solution (7.67) into (7.72), we see thatζremains
constant (is conserved) even ifw≡p/εis changing. Recalling the definitions of
u,θand using the background equations of motion,ζbecomes

ζ=

2

3

H−^1 ′+

1 +w

+. (7.73)

Let us assume the equation of statewis initially a constantwi,and later on changes
to another constantwf.In this case the initial and final values of are also constants