Physical Foundations of Cosmology

9.7 Anisotropies on small angular scales 379

is proportional to the baryon density and vanishes in the absence of baryons where
c^2 s→ 1 /3. The other two integrals are

N 2 =

cs

2

∫∞

1

To^2 e−(l/lS)

(^2) x 2
x^2

√

x^2 − 1

dx, (9.81)

and

N 3 =

9 cs^3

2

∫∞

1

To^2

√

x^2 − 1

x^4

e−(l/lS)

(^2) x 2
dx. (9.82)
The microwave background anisotropy is a powerful cosmological probe because
the parameters which determine the spectruml(l+ 1 )Cl, namely,cs,lf,lS,!and
the transfer functionsToandTp, can all be directly related to the basic cosmological
parameters (^) b, (^) m, (^) , the dark energy equation of statew,and the Hubble
constanth 75. Before we proceed with the calculation of the integrals determining
the anisotropies, we explore these relations, making the simplifying assumption
that the dark energy is the vacuum energy density, so thatw=−1.

9.7.3 Parameters

The speed of sound csat recombination depends only on the baryon density, which
determines how muchcsdiffers from its value for a purely relativistic gas of photons,
cs= 1 /

√

3. If we define the baryon density parameter

ξ≡

1

3 c^2 s

− 1 =

3

4

(

εb

εγ

)

r

 17

(

(^) bh^275

)

, (9.83)

then the speed of sound is

cs^2 =

1

3 ( 1 +ξ)

.

For baryon density (^) bh^275  0 .035, one hasξ 0. 6 .The physical reason for the
dependence ofcs on the baryon density is clear. The baryons interacting with
radiation make the sound waves more “heavy” and therefore reduce their speed.
The damping scales lfandlS, given in (9.78), each depend on the ratioηr/η 0 .To
calculate this ratio we introduce a supplementary moment of timeη 0 ηxηr,
so that atηxthe radiation energy density isalreadynegligible and the cosmological
term isstillsmall compared with the cold matter energy density. Then we can
separately determineηx/η 0 andηr/ηxusing the exact solutions (1.108) and (1.81)
respectively.