MODERN COSMOLOGY

228 The cosmic microwave background

is damping of baryon fluctuations on scales belowLD, known as Silk damping or
diffusion damping. This effect can be modelled by the replacement

' 0 (k,η∗)→' 0 (k,η∗)e−(kLD)

2

(7.5)

although detailed calculations are needed to defineLDprecisely. As a result of
this damping, microwave background fluctuations are exponentially suppressed
on angular scales significantly smaller than a degree.

7.2.6 The resulting power spectrum

The fluctuations in the universe are assumed to arise from some random statistical
process. We are not interested in the exact pattern of fluctuations we see from our
vantage point, since this is only a single realization of the process. Rather, a
theory of cosmology predicts an underlying distribution, of which our visible sky
is a single statistical realization. The most basic statistic describing fluctuations
is their power spectrum. A temperature map on the skyT(nˆ)is conventionally
expanded in spherical harmonics,

T(nˆ)

T 0

= 1 +

∑∞

l= 1

∑l

m=−l

a(Tlm)Y(lm)(nˆ) (7.6)

where

a(Tlm)=

1

T 0

∫

dnˆT(nˆ)Y(∗lm)(nˆ) (7.7)

are the temperature multipole coefficients andT 0 is the mean CMB temperature.
Thel=1 term in equation (7.6) is indistinguishable from the kinematic dipole
and is normally ignored. The temperature angular power spectrumClis then
given by
〈a(Tlm∗)a(Tl′m′)〉=CTlδll′δmm′, (7.8)

where the angled brackets represent an average over statistical realizations of the
underlying distribution. Since we have only a single sky to observe, an unbiased
estimator ofClis constructed as

CˆTl =^1

2 l+ 1

∑l

m=−l

almT∗aTlm. (7.9)

The statistical uncertainty in estimatingClTby a sum of 2l+1 terms is known as
‘cosmic variance’. The constraintsl=l′andm=m′follow from the assumption
of statistical isotropy:ClTmust be independent of the orientation of the coordinate
system used for the harmonic expansion. These conditions can be verified via an
explicit rotation of the coordinate system.
A given cosmological theory will predictClTas a function ofl, which can be
obtained from evolving the temperature distribution function as described earlier.