Physical Foundations of Cosmology

9.6 Delayed recombination and the finite thickness effect 369

forlup to 20 or so. Forl> 20 ,the neglected effects become essential, leading
first to the rise in amplitude of the temperature fluctuations and then to the acoustic
peaks.

Problem 9.4Find the correction to (9.44) if the initial spectrum is not scale-
invariant,|

(

(^0) k

) 2

k^3 |=BknS−^1 ,assuming that|nS− 1 | 1.

Problem 9.5Determine how (9.42) is modified for the entropy perturbations con-
sidered in Section 7.3.

Unfortunately, the information about statistical properties of the primordial spec-
trum gathered from asinglevantage point is limited by cosmic variance. Since there
are only 2l+1 independentalm, the variance is

Cl

Cl

( 2 l+ 1 )−^1 /^2. (9.45)

The typical fluctuation is about 50% for the quadrupole (l=2) and 15% forl∼20.
Therefore, we are forced to go to smaller angular scales to obtain precise constraints
on the spectrum of primordial inhomogeneities. The bad news is that, for these
scales, we can no longer ignore evolution. On the other hand, if we can deconvolve
the effects of evolution, we gain information about both the primordial spectrum
and the parameters that control cosmic evolution.

9.6 Delayed recombination and the finite thickness effect

On small angular scales, recombination can no longer be approximated as instanta-
neous. The finite duration of recombination introduces uncertainty as to the precise
moment and position when a given photon last scatters. As a result, photons arriving
from a given direction yield only “smeared out” information. In turn this leads to a
suppression of the temperature fluctuations on small angular scales known as the
finite thickness effect. The spread in the time of last scattering also increases the
Silk damping scale, changing the conditions in the region from which the photon
last scatters.
We first consider the finite thickness effect. A photon arriving from directionl
might last scatter at any value of the redshift in the interval 1200>z>900. If the
last scattering occurs at conformal timeηL, the photon carries information about
conditions at position

x(ηL)=x 0 +l(ηL−η 0 ).

Since the total flux of radiation arriving from directionlconsists of photons that
last scattered over a range of times, the information it carries represents a weighted