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

6 From Theory to Observations: Estimators and Errors
6.1 Introduction
This chapter focuses on issues regarding accurate estimation of clustering
statistics in large-scale galaxy surveys and their uncertainties, in order to properly
constraint theories against observations. We also consider applications to
measurements in N-body simulations, as briefly described in Sect. 6.12.
In many respects, the theory of estimators of large scale structure statistics
was triggered in the seventies and the early eighties by Peebles and his collaborators.
In a series of seminal works, starting with a fundamental paper [500],
these authors developed the statistical theory of the two-point correlation
function in real and Fourier space, in two- and three-dimensional catalogs,
including estimates of the cosmic errors and the cosmic bias (formulated as
an integral constraint problem), followed soon by investigations on higherorder
statistics. They used several estimators, including count-in-cell statistics.
These results are summarized in [508].
Since then, and particularly in the nineties, a number of techniques were put
forward to allow a more precise testing of cosmological theories against observations.
These include:
- Detailed studies of two-point and higher-order correlation functions estimators.
- Accurate estimation of errors going beyond the simple (and often severe
underestimate) Poisson error bars, to include finite-volume effects, survey
geometry and non-Gaussian contributions due to non-linear evolution.
- The treatment of covariance between measurements at different scales. In
order to properly test theoretical predictions, this is equally important
to an accurate treatment of errors, which are just the diagonal elements
of the covariance matrix. Neglecting off-diagonal elements can lead to a
substantial overestimate of the constraining power of observations (see
e.g. Chapter 8).
- Implementation of techniques for data compression, error decorrelation,
and likelihood analysis for cosmological parameters estimation.
It is clear that the upcoming large-scale galaxy surveys such as 2dFGRS and
SDSS will certainly have to rely heavily on these new developments to extract
all the information encoded by galaxy clustering to constrain cosmological
parameters, primordial non-Gaussianity and galaxy formation models. In addition
to standard second-order statistics such as the power-spectrum or the
two-point correlation function, our review focuses on higher-order statistics
for several reasons:
- As detailed in previous chapters, non-linear evolution leads to deviations
from Gaussianity, so two-point statistics are not enough to characterize
large-scale structure. They do not contain all the information available
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