Large-Scale Structure of the Universe and Cosmological ...

The cosmic error is most useful when the function Υ( Â) is Gaussian. If this is
not the case, full knowledge of the shape of the cosmic distribution function,
including its skewness, is necessary to interpret correctly the measurements 59 .
6.2.2 The Covariance Matrix
As for correlation functions, a simple generalization of the concept of variance
is that of covariance between two different quantities; this can be for example
between two estimators  and ˆ B
Cov( Â, ˆ B) = 〈δÂ δ ˆ
B〉 = δÂ δ ˆ B Υ( Â, ˆ B) dÂd ˆ B, (366)
or simply between estimates of the same quantity at different scales; say, for
the power spectrum, the covariance matrix between estimates of the power at
ki and kj reads,
C P ij ≡ 〈 ˆ P(ki) ˆ P(kj)〉 − 〈 ˆ P(ki)〉〈 ˆ P(kj)〉, (367)
where ˆ P(ki) is the estimator of the power spectrum at a band power centered
about ki.
In general, testing theoretical predictions against observations requires knowledge
of the joint covariance matrix for all the estimators (e.g. power spectrum,
bispectrum) at all scales considered. We will consider some examples below in
Sects. 6.4.4, 6.5.4 and 6.10.2.
The cosmic error and the cosmic bias can be roughly separated in three contributions
[621] if the scale R (or separation) considered is small enough compared
to the typical survey size L, or equivalently, if the volume v ≡ vR ≡
(4/3)πR 3 is small compared to the survey volume, V :
(i) Finite volume effects: they are due to the fact that we can have access to
only a finite number of structures of a given size in surveys (whether they
are 2-D or 3-D surveys), in particular the mean density itself is not always
well determined. These effects are roughly proportional to the average of
the two point correlation function over the survey, ¯ ξ(L). They are usually
designated by “cosmic variance”.
(ii) Edge effects: they are related to the geometry of the catalog. In general,
estimators give less weight to galaxies near the edge than those far away
from the boundaries. As we shall see later, edge effects can be partly corrected
for, at least for N-point correlation functions. At leading order in
v/V , they are proportional to roughly ξv/V . Note that even 2-D surveys
cannot avoid edge effects because of the need to mask out portions of
the sky due to galaxy obscuration, bright stars, etc... Edge effects vanish
only for N-body simulations with periodic boundary conditions.
59 For example, it could be very desirable to impose in this case that a good estimator
should have minimum skewness [610].
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