Physical Foundations of Cosmology

7.3 Hydrodynamical perturbations 309

by the time of equality. Forη>ηeq,the entropy perturbations evolve like the non-
decaying mode of the adiabatic perturbations. There is a key difference, however:
it follows from (7.83) that for adiabatic perturbations(δS= 0 )we always have

δεγ

εγ

=

4

3

δεb

εb

, (7.89)

whereas for entropy perturbations

δεγ

εγ

− 2

δεb

εb

(7.90)

after equality.
We can also define the isocurvature mode of perturbations by imposing the

conditions (^) i=0 and ′i=0 at some initial moment of timeηi= 0 .One can
easily verify that this mode soon approaches the entropy mode.

7.3.2 Vector and tensor perturbations

For a perfect fluid, the only nonvanishing vector components ofδTβαareδTi^0 =
a−^1 (ε 0 +p 0 )δu⊥i.Equations (7.45) become

Vi= 16 πGa(ε 0 +p 0 )δu⊥i, (7.91)

(

Vi,j+Vj,i

)′

+ 2 H

(

Vi,j+Vj,i

)

= 0. (7.92)

The solution of the second equation is

Vi=

C⊥i

a^2

, (7.93)

whereC⊥iis a constant of integration. Taking into account that thephysicalvelocity
isδvi=a

(

dxi/ds

)

=−a−^1 δu⊥i,we obtain

δvi∝

1

a^4 (ε 0 +p 0 )

. (7.94)

Thus, in a matter-dominated universe, the rotational velocities decay in inverse
proportion to the scale factor, in agreement with the result of Newtonian theory. In
a radiation-dominated universeδv=const. In both cases the metric perturbations
given by (7.93) decay very quickly and the primordial vector fluctuations have
significant amplitudes at present only if they were originally very large. There is
no reason to expect such large primordial vector perturbations and from now on we
will completely ignore them.
Tensor perturbationsare more interesting, and as we will see in the following
chapter, they can be generated during an inflationary stage. In a hydrodynamical