Physical Foundations of Cosmology

314 Gravitational instability in General Relativity

whereis the Laplacian operator andcsis the speed of sound in the baryon–
radiation plasma given in (7.85). In deriving (7.113), we used the relations

ε+p=εb+

4

3

εγ=

4

9 c^2 s

εγ (7.114)

andεγa^4 =const. Neglecting polarization effects, the shear viscosity coefficient
entering (7.113) is given by

η=

4

15

εγτγ, (7.115)

whereτγis the mean free time for photon scattering.
We have, thus far, two perturbation equations, (7.108) and (7.113), for three un-
known variables,δd,δγand. To these equations, we add the 0−0 component of
the Einstein equations (see (7.38)), which in the case under consideration becomes

− 3 H ′− 3 H^2 = 4 πGa^2

(

δεd+δεb+δεγ

)

= 4 πGa^2

(

εdδd+

1

3 c^2 s

εγδγ

)

, (7.116)

where (7.112) has been used to expressδbin terms ofδγ.
Using (7.110), we obtain a useful relation forthe radiation contributionto the
divergence of the 0−icomponents of the energy–momentum tensor:

T 0 i,i=^43 εγu 0 ui,i=

(

4 −δγ

)′

εγ, (7.117)

which will be used in Section 9.3.

7.4.2 Evolution of perturbations and transfer functions

If the density fluctuations are decomposed into modes with comoving wavenumber
k, then their behavior for a givenkdepends on whetherkη<1orkη> 1 .The cross-
over fromkη<1tokη>1 corresponds to the transition in which a mode changes
from having a wavelength exceeding the curvature scale to having a wavelength
less than the curvature scale. In a decelerating universe, as time evolves andη
grows, the curvature scaleH−^1 =a/a ̇increases faster than the physical scale of
the perturbation,λpha/k,and encompasses modes with smaller and smallerk.
We shall refer to modes for whichkη<1 as supercurvature modes and those with
kη>1 as subcurvature ones.
The initial perturbation spectrum produced in inflation can be characterized by

the “frozen” amplitudes of the metric fluctuations (^0) kon supercurvature scales dur-
ing the radiation stage (see the following chapter for details). After the perturbation
enters the curvature scale it evolves in a nontrivial way. Our goal here is to determine