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

the size of the catalog and optimizing its geometry.
In this chapter, we concentrate mainly on the point (iii). Dynamical biases
mentioned in point (ii) will be addressed in the next chapter. These effects
can also be taken into account in the formalism, by simply replacing the values
of the statistics intervening in the equations giving cosmic errors and crosscorrelations
with the “distorted” ones, as we shall implicitly assume in the rest
of this chapter 57 . Segregation effects and incompleteness due to instrument
biases, obscuration or to selection in magnitude will be partly discussed here
through weighted estimators, and in Chapter 8 when relevant.
This chapter is organized as follows. In Sect. 6.2, we discuss the basic concepts
of cosmic bias, cosmic error and the covariance matrix. Before entering
in technical details, it is important to discuss the fundamental assumptions
implicit in any measurement in a galaxy catalog, namely the fair sample hypothesis
[500] and the local Poisson approximation. This is done in Sect. 6.3,
where basic concepts on count-in-cell statistics and discreteness effects corrections
are introduced to illustrate the ideas. In Sect. 6.4, we study the most
widely used statistic, the two-point correlation function, with particular attention
to the Landy and Szalay estimator [393] introduced in Sect. 6.4.1.
The corresponding cosmic errors and biases are given and discussed in several
regimes. Section 6.5 is similar to Sect. 6.4, but treats the Fourier counterpart
of ξ, the power spectrum. Generalization to higher-order statistics is discussed
in Sect. 6.6.
Section 6.7 focuses on the count-in-cell distribution function, which probes
the density field smoothed with a top-hat window. In that case a full analytic
theory for estimators and corresponding cosmic errors and biases is available.
Section 6.8 discusses multivariate counts-in-cells statistics. In Sect. 6.9 we introduce
the notion of optimal weighting: each galaxy or fraction of space can
be given a specific statistical weight chosen to minimize the cosmic error. Section
6.10 deals with cross-correlations and the shape of the cosmic distribution
function and discusses the validity of the Gaussian approximation, useful for
maximum likelihood analysis. Section 6.11 reinvestigates the search for optimal
estimators in a general framework in order to give account of recent
developments. In particular, error decorrelation and the discrete Karhunen-
57 Of course, this step can be non trivial. Measurements in galaxy catalogs (Sect. 8)
and in N-body simulations suggest that in the nonlinear regime the hierarchical
model is generally a good approximation (e.g. [87,234,147,150,472]), but it can
fail to describe fine statistical properties (e.g. for the power spectrum covariance
matrix [564,296]). In the weakly nonlinear regime, PT results including redshift
distortions (Sect. 7.4), projection along the line of sight (Sect. 7.2) and biasing
(Sect. 7.1) can help to compute the quantities determining cosmic errors, biases
and cross-correlations. In addition to the hierarchical model, extensions of PT to
the nonlinear regime, such as EPT, E 2 PT (Sect. 5.13) and HEPT (Sect. 4.5.6),
coupled with a realistic description of galaxy biasing can be used to estimate the
errors.
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