Physical Foundations of Cosmology

6.3 Instability in an expanding universe 275

Problem 6.4If we use the redshiftzas a time parameter, then the integral in (6.55)
can be calculated explicitly for a matter-dominated universe with arbitrary
0.
Find the corresponding solutionδ(z)and show that for
0 1 the perturbation
amplitude freezes out atz∼ 1 /
0.

6.3.2 Vector perturbations

Withδ=0, (6.43)–(6.45) reduce to

∇δv= 0 ,

∂δv

∂t

+Hδv= 0. (6.57)

From the first equation it follows that for a plane wave perturbation,δv∝
δvk(t)exp(ikq), the peculiar velocityδvis perpendicular to the wavenumberk.
The second equation becomes

δv ̇k+

a ̇

a

δvk= 0 , (6.58)

and has the obvious solutionδvk∝ 1 /a.Thus, the vector perturbations decay as the
universe expands. These perturbations can have significant amplitudes at present
only if their initial amplitudes were so large that they completely spoiled the isotropy
of the very early universe. In an inflationary universe there is no room for such large
primordial vector perturbations and they do not play any role in the formation of
the large-scale structure of the universe. Vector perturbations, however, can be
generated at late times, after nonlinear structure has been formed, and can explain
the rotation of galaxies.

6.3.3 Self-similar solution

For large-scale perturbations we can neglect the pressure, and the spatial derivatives
drop out of (6.46). In this case the solution for perturbations can be written directly
in coordinate space

δ(q,t)=A(q)δi(t)+B(q)δd(t), (6.59)

whereδi andδdare growing and decaying modes respectively. Without losing
generality we can setδi(t 0 )=δd(t 0 )=1 at some initial moment of timet 0 .If the
density distribution at this time is described by the functionδ(q,t 0 )and the matter
is at rest with respect to the Hubble flow (δv∝δ ̇(q,t 0 )=0), then, expressingA(q)
andB(q)in terms ofδ(q,t 0 ), we obtain

δ(q,t)=δ(q,t 0 )

(

δi(t)

1 −

(

̇δi/δ ̇d

)

t 0

+

δd(t)

1 −

(

δ ̇d/δ ̇i

)

t 0

)

. (6.60)