Physical Foundations of Cosmology

370 Cosmic microwave background anisotropies

average over a scalex∼ηr,whereηris roughly the duration of recombina-
tion. Clearly, the contribution of inhomogeneities with scales smaller thanηrto
the temperature fluctuations will be smeared out and therefore strongly suppressed.
Let us calculate the probability that the photon was scattered within the time
intervaltLat physical timetL(corresponding to the conformal timeηL) and then
avoided further scattering until the present timet 0. We can divide the time interval
t 0 >t>tLintoNsmall intervals of durationt, where thejth interval begins at
timetj=tL+jtandN>j> 1 .The probability is then

P=

tL

τ(tL)

(

1 −

t

τ(t 1 )

)

···

(

1 −

t

τ

(

tj

)

)

···

(

1 −

t

τ(tN)

)

, (9.46)

where

τ

(

tj

)

=

1

σTnt

(

tj

)

X

(

tj

)

is the mean free time for Thomson scattering,ntis the number density of all (bound
and free) electrons andX is the ionization fraction. Taking the limitN→∞
(t→0), and converting from physical timetto conformal timeη,we obtain

dP(ηL)=μ′(ηL)exp[−μ(ηL)]dηL, (9.47)

where the prime denotes the derivative with respect to conformal time andμ(ηL)
is the optical depth:

μ(ηL)≡

∫t^0

tL

dt

τ(t)

=

∫η^0

ηL

σTntXea(η)dη. (9.48)

The uncertainty in the last scattering time causes us to modify our expression for
the temperature fluctuation (9.32), replacing the recombination momentηrbyηL,
and integrating overηLwith the probability weight (9.47):

δT

T

=

∫ [

+

δ

4

−

3 δ′

4 k^2

∂

∂η 0

]

ηL

eik·(x^0 +l(ηL−η^0 ))μ′e−μdηL

d^3 k

( 2 π)^3 /^2

. (9.49)

Unlike in (9.32), here we cannot neglectηLcompared toη 0 because fork>
(ηr)−^1 an oscillating exponential factor in (9.49) changes significantly during the
time intervalηrwhen the visibility function

μ′(ηL)exp[−μ(ηL)]

is substantially different from zero. The visibility function vanishes at very small
ηL(becauseμ1) and at largeηL(whereμ′→0), and reaches its maximum at