372 Cosmic microwave background anisotropies
ηL=ηr.Then, substituting (9.55) in (9.49) and performing an explicit integration
overηL,one gets
δT
T=
∫(
+
δ
4−
3 δ′
4 k^2∂
∂η 0)
ηre−(σkηr)2
eik·(x^0 +l(ηr−η^0 ))d^3 k
( 2 π)^3 /^2(9.56)
where
σ≡1
√
6 (κHη)r. (9.57)
In deriving (9.56) we replaced(k·l)^2 withk^2 /3, using the fact that the perturbations
field is isotropic. Now it is safe to neglectηrcompared toη 0.
To find out howσdepends on cosmological parameters, we have to calculate
(Hη)r.At recombination, the dark energy contribution can be ignored and the scale
factor is well described by (1.81); hence,
(Hη)r= 2 ×1 +(ηr/η∗)
2 +(ηr/η∗). (9.58)
The ratio(ηr/η∗)can be expressed through the ratio of the redshifts at equality and
recombination using an obvious relation
(
ηr
η∗
) 2
+ 2
(
ηr
η∗)
zeq
zr. (9.59)
Taking this into account and substituting (9.58) in (9.57) we obtain
σ 1. 49 × 10 −^2[
1 +
(
1 +
zeq
zr)− 1 / 2 ]
. (9.60)
The exact value ofzeqdepends on the matter contribution to the total energy density
and the number of ultra-relativistic species present in the early universe. For three
types of light neutrinos we have
zeq
zr
12. 8
(
(^) mh^275
)
. (9.61)
The parameterσis only weakly sensitive to the amount of cold matter and number
of light neutrinos: for (^) mh^275 0 .3,σ 2. 2 × 10 −^2 ,and for (^) mh^275 1 ,σ
- 9 × 10 −^2.
Problem 9.6Find howσdepends on the number of light neutrinos for a given cold
matter density.
Next let us consider how noninstantaneous recombination influences the Silk
dissipation scale. As mentioned above, the ionization fraction atη=ηrisκ≈ 13. 7
times bigger than at decoupling and the mean free path is consequentlyκtimes