Physical Foundations of Cosmology

284 Gravitational instability in Newtonian theory

If we ignore vector perturbations thenfi=∂ψ/∂qi, whereψis the potential for
the peculiar velocities.For a one-dimensional perturbation,ψdepends only on one
of the coordinates, sayq^1 .Then the strain tensor takes the form

J=a(t)

⎛

⎝

1 −λ

(

q^1 ,t

)

00

010

001

⎞

⎠, (6.104)

and hence

J=a^3 ( 1 −λ), tr

[(

J ̇·J−^1

) 2 ]

=

(

H−

λ ̇

1 −λ

) 2

+ 2 H^2 , (6.105)

whereλ

(

q^1 ,t

)

≡∂f^1 /∂q^1 .Substituting (6.105) into (6.89), we find that for! 0 (q)=
const this equation reduces to two independent equations:

H ̇+H^2 =−^4 πG

3

ε 0 , (6.106)

λ ̈+ 2 Hλ ̇− 4 πGε 0 λ= 0 , (6.107)

whereε 0 (t)≡! 0 /a^3. The first equation is the familiar Friedmann equation for the
homogeneous background. The second equationcoincideswith (6.46) for linear
perturbations in pressureless matter. However, it must be stressed that in deriving
(6.107) we did not assume that the perturbations were small, and hence its solutions
are valid in both the linear and the nonlinear regime.
According to (6.88), the energy density is equal to

ε(q,t)=

ε 0 (t)

(

1 −λ

(

q^1 ,t

)), (6.108)

andλ

(

q^1 ,t

)

can be written as (see (6.59))

λ

(

q^1 ,t

)

=α

(

q^1

)

δi(t)+κ

(

q^1

)

δd(t). (6.109)

Hereδi(t)andδd(t)are the growing and decaying modes from the linearized the-
ory. For example, in a flat matter-dominated universeδi∝t^2 /^3 andδd∝t−^1 .For
λ

(

q^1 ,t

)

1, the exact solution in (6.108) obviously reproduces the results of the
linearized theory.

Problem 6.11How must (6.89) be modified in the presence of a homogeneous
relativistic component? Find and analyze the corresponding Zel’dovich solutions
in a flat universe with a cosmological constant.