408 Cosmic microwave background anisotropies
obtain
μ(zion) 0. 03(^) bh^275
√
(^) mh^275
z^3 ion/^2. (9.169)
In the concordance model, where (^) bh^275 0 .04 and (^) mh^275 0. 3 ,the reionization
atzion20 implies an optical depthμ 0. 2 .If reionization happens atzion 5 ,
thenμ 0. 02 ,and in this case the overall effect of reionization on the CMB
fluctuations does not exceed 2% or so of the total fluctuations.
As a result of reionization, the fraction
1 −exp(−μ(zion))
of all photons will be rescattered on the electrons. The remaining fraction
exp(−μ(zion))will not be influenced and will give the usual contribution to the
fluctuations. For instance, in the model withzion 20 ,about 80% of all photons
will not be influenced by reionization. The contribution of the 20% of the photons
that are rescattered to the angular power spectrum depends on the multipolel. After
rescattering, the photon changes its direction of propagation and then the position
from which it appears to emanate can be any point that is remote from the original
scattering point at a distance not exceeding the horizon scale at this time. As a
result, the contribution of the rescattered photons is smeared out within the angular
scales corresponding to the reionization horizon scale and does not give rise to
temperature fluctuations. For thoselwhich correspond to superhorizon scales the
fluctuations will not be influenced because of causality. Of course, atz<zion,the
horizon scale continues to grow but the optical depth decreases and the fraction
of the photons which are rescattered drops; therefore we neglect this effect. The
angular size of the horizon atzion,corresponding to the conformal timeηion,can
easily be found from (2.69) if we take into account that the physical size of the
reionization horizon is equal toa(ηion)ηionandχem=η 0 −ηionη 0 .Then, in
a flat universe, we getθionηion/η 0 and the appropriate multipole moment is
equal to
lionπ/θion=πη 0 /ηionπz^1 ion/^2
0 m.^09 , (9.170)
where for the ratioη 0 /ηionwe have used the results of Problem 9.9.Thus, in the
model with reionization the observed temperature fluctuations are
Clobs=
{
Cl, llion,
exp(−μ)Cl, llion,(9.171)
where the intrinsic temperature fluctuationsClhave been calculated in the previous
sections. If (^) m 0 .3 andzion 20 ,we havelion 12 .In this case the amplitudes