Physical Foundations of Cosmology

9.11 Reionization 407

such a way that purelyEmode polarization appears to have aBmode component.
Nevertheless, present projections suggest that, during the next decade, the most
likely range of gravitational wave amplitudes predicted by inflationary cosmology
can be fully explored.

9.11 Reionization

At late stages, when nonlinear structure begins to form in the universe, neutral
hydrogen can be reionized. In fact, analyzing the spectra of the most distant quasars,
one can conclude that most of the intergalactic hydrogen is ionized atz 5 .If it
were not, the spectra would be significantly corrupted by the absorptions lines of
the intergalactic neutral hydrogen. After reionization, the CMB photons can scatter
on the free electrons and therefore late reionization affects the resulting CMB
fluctuations.
Let us first establish how reionization influences the spectrum of the temperature
fluctuations. The probability that the photon has avoided scatterings and propagated
freely from timet until the present timet 0 is equal to

P(t)= lim

t→ 0

(

1 −

t

τ(t)

)

···

(

1 −

t

τ(t 0 )

)

=exp

(

−

∫t 0

t

dt

τ(t)

)

≡exp(−μ(t)), (9.166)

whereτ=(σTXnt)−^1 is the mean free time for Thomson scattering,ntis the total
number of all electrons andXis the ionization fraction. The optical depthμ(t)=
μ(z)entering (9.166) can be rewritten as an integral over the redshift parameter

μ(z)=

∫t 0

t

dt

τ(t)

=σT

∫z

0

Xnt(z)

H(z)( 1 +z)

dz. (9.167)

Let us assume instantaneous reionization at redshiftzrzion1 and calculate
the optical depthμ(zion)in a flat universe. The total number of all (free and bound)
electrons is equal to

nt(z) 0. 88 ×

εb(z)

mb

= 0. 88 ×

3 H 02

8 πmb

(^) b(1+z)^3 , (9.168)
where the factor 0.88 accounts for the fact that about 12% of all baryons are
neutrons. At highz,which give the main contribution to the integral (9.167), we
can neglect the cosmological constant in comparison with the cold matter and
use the following expression for the Hubble parameter:H(z)H 0
1 m/^2 ( 1 +z)^3 /^2.
Substituting it with (9.168) into (9.167), and assuming thatX1atz<zion,we