Physical Foundations of Cosmology

9.7 Anisotropies on small angular scales 381
This estimate is not very reliable because the imperfect fluid approximation
breaks down when the visibility function is near maximum. Nevertheless, compar-
ison with exact numerical calculations shows that the discrepancy is less than 10%
with numericallSbeing slightly smaller than in (9.90). In contrast tolf, the scale
lSdoesdepend on the baryon density, characterized byξ. However, this depen-
dence is very strong only forξ1, when the second term inside the parenthesis

in (9.90) dominates. Forξ= 0 .6, we findlS1100 if (^) mh^275  0 .3 andlS 980
for (^) mh^275  1.
The parameterρ, which determines the location of the acoustic peaks, can be
calculated by substituting the expression forcs(η)
cs(η)=

1

√

3

[

1 +ξ

(

a(η)

a(ηr)

)]− 1 / 2

, (9.91)

witha(η) given by (1.81) into (9.77) and then integrating.

Problem 9.11Verify that

!

I

√

3 zrξ

ln

⎡

⎣

√(

1 +zr/zeq

)

ξ+

√

( 1 +ξ)

1 +

√

ξ

(

zr/zeq

)

⎤

⎦. (9.92)

Although it is clear that!depends on both baryon and matter densities, it is not

apparent from this expression how!varies when we change their values. For this

reason, it is useful to find a fitting formula for!. Verify that the numerical fit

! 0. 014 ( 1 + 0. 13 ξ)−^1

(

(^) mh^275

) 1 / 4

I (9.93)

reproduces the exact result (9.92) to within 7% everywhere in the region 0<ξ< 5 ,

0. 1 <

mh 75 <1, whereas the function!itself varies by roughly a factor of 3.

Combining (9.93) with the numerical fit forIin (9.85), we obtain

! 0. 014 ( 1 + 0. 13 ξ)−^1

(

(^) mh^375.^1

) 0. 16

. (9.94)

Note the unusual combination (^) mh^375.^1. We will see later that because of this we
can hope to determine (^) mandh 75 separately by combining the measurements of
the location of the peaks with the measurements of other features of the microwave
spectrum which depend only on (^) mh^275.
The parameter!characterizes the angular size of the sound horizon at recom-
bination on today’s sky. The size of the sound horizon drops as the baryon density
increases. For a given physical size of the sound horizon, its angular scale on
today’s sky should of course also depend on the evolution of the universe after