Physical Foundations of Cosmology

9.7 Anisotropies on small angular scales 375

the source function at recombination is
(

(^) k+
δk
4

)

r



⎡

⎣Tp

(

1 −

1

3 c^2 s

)

+To

√

cscos

⎛

⎝k

∫ηr

0

csdη

⎞

⎠e−(k/kD)^2

⎤

⎦ (^0) k (9.63)
and
(
δ′k

)

r−^4 Tokc

3 / 2

s sin

⎛

⎝k

∫ηr

0

csdη

⎞

⎠e−(k/kD)^20 k, (9.64)

where the transfer functionsTpandTocorrespond to the constants of integration
in the WKB solution and depend on whether the perturbation entered the horizon
before or after equality. They were calculated in Section 7.4.2 in two limiting
cases, namely, for the perturbations withkηeq 1 ,which entered the horizonlong
enoughafter equality, see (7.134),

Tp→

9

10

; To→

9

10

× 3 −^3 /^4  0. 4 , (9.65)

and for the perturbations withkηeq 1 ,which entered the horizonwell before
equality, see (7.143),

Tp→

ln

(

0. 15 kηeq

)

(

0. 27 kηeq

) 2 →0; To→

35 /^4

2

 1. 97. (9.66)

Note that the transfer functions change very significantly. In particular,Tpis
negligible for the perturbations which entered the horizon during the radiation-
dominated stage and close to unity for the perturbations which entered the horizon
long after that. The physical reason for such behavior is obvious. As we found in
Section 6.4.3, for the subhorizon modes the gravitational instability in the cold dark
matter component is suppressed during radiation-dominated stage and the gravi-
tational potential decays; for the perturbation which enters the horizon when cold
matter already dominates the amplitude of inhomogeneity in the cold component
grows and the potential does not change.To,which defines the amplitude of the
sound wave, is about 5 times greater for modes that enter the horizon well before
equality than for those that enter the horizon long afterwards. This effect is due to the
gravitational field of the radiation; it is significant when the modes with largekηeq
enter the horizon and boosts the amplitude of the resulting sound wave compared to
the case when the contribution of radiation to the gravitational potential is negligible.
It is clear that for those perturbations which entered the horizon near equality,
the appropriate values of the transfer functions should lie somewhere between their
asymptotic values. As we mentioned above these perturbations withkηeq∼O( 1 )
determine the amplitude of the temperature fluctuations in the region of the first