MODERN COSMOLOGY

Cosmological models and constraints 245

in equations (7.31) is only an approximate equality is the pixelization noise.
Most current experiments oversample the sky with respect to their beam, so
the pixelization noise is negligible. Now assume that the noise is uncorrelated
between pixels and is well represented by a normal distribution. Also, assume that
the map is created with a Gaussian beam with widthθb. Then it is straightforward
to show that the variance of the temperature moments is given by (Knox 1995)

〈dlmTdlT′m∗′〉=(Cle−l

(^2) σ 2
b+w−^1 )δll′δmm′, (7.32)
whereσb= 0 .007 42(θb/ 1 ◦)and
w−^1 =
4 π
Npix
〈(Tinoise)^2 〉
T 02

(7.33)

is the inverse statistical weight per unit solid angle, a measure of experimental
sensitivity independent of the pixel size.
Now the power spectrum can be estimated via equation (7.32) as

ClT=(DTl −w−^1 )el

(^2) σb 2
(7.34)
where
DlT=

1

2 l+ 1

∑l

m=−l

dTlmdlmT∗. (7.35)

The individual coefficientsdlmT are Gaussian random variables. This means that
CTl is a random variable with aχ 22 l+ 1 distribution, and its variance is (Knox 1995)

(ClT)^2 =

2

2 l+ 1

(Cl+w−^1 el

(^2) σb 2
). (7.36)
Note that even forw−^1 =0, corresponding to zero noise, the variance is non-
zero. This is the cosmic variance, arising from the fact that we have only one
sky to observe: the estimator in equation (7.35) is the sum of 2l+1 random
variables, so it has a fundamental fractional variance of( 2 l+ 1 )−^1 /^2 simply due
to Poisson statistics. This variance provides a benchmark for experiments: if the
goal is to determine a power spectrum, it makes no sense to improve resolution
or sensitivity beyond the level at which cosmic variance is the dominant source of
error.
Equation (7.36) is extremely useful: it gives an estimate of how well the
power spectrum can be determined by an experiment with a given beam size
and detector noise. If only a portion of the sky is covered, the variance estimate
should be divided by the fraction of the total sky covered. With these variances in
hand, standard statistical techniques can be employed to estimate how well a given
measurement can recover a given setsof cosmological parameters. Approximate
the dependence ofClTon a given parameter as linear in the parameter; this will