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

is positive and compact in Fourier space and 〈ˆg〉 ≃ f, so that to keep the
interpretation of the power in this new representation as giving the power
centered about some well-defined scale [294,296].
The above line of thoughts can in fact be pushed even further by applying the
so called “pre-whitening” technique to ˆ f: if ˆ f is decomposed in terms of signal
plus noise, pre-whitening basically consists in multiplying ˆ f by a function h
such that the noise becomes white or constant. If the noise is uncorrelated, this
method allows one to diagonalize simultaneously the covariance matrix of the
signal and the noise. When non-Gaussian contributions to the power spectrum
covariance matrix are included, however, such a diagonalization is not possible
anymore. However, in the FKP approximation, as described in the previous
section, it was shown that an approximate diagonalization (where two of the
contributions coming from two- and four-point functions are exactly diagonal,
whereas the third coming from the three-point function is not) works extremely
well, at least when non-Gaussianity is modeled by the hierarchical ansatz [296].
The quantity whose covariance matrix has these properties corresponds to
the so-called prewhitened power spectrum, which is easiest written in real
space [296]
ˆξ(r) →
2 ˆ ξ(r)
1 + [1 + ξ(r)] 1/2.
(512)
Note that in the linear regime, ˆ ξ(k) reduces to the linear power spectrum; however,
unlike the non-linear power spectrum, ˆ ξ(k) has almost diagonal cosmic
covariance matrix even for nonlinear modes. More details on the theory and
applications to observations can be found in e.g. [296,297] and [298,487,299]
respectively.
6.11.4 Data Compression and the Karhunen-Loève Transform
A problem to face is with modern surveys such as the 2dFGRS and SDSS,
is that the data set ˆx becomes quite large for “brute force” application of
estimation techniques. Before statistical treatment of the data as discussed in
the previous sections, it might be necessary to find a way to reduce their size,
but keeping as much information as possible. The (discrete) Karhunen-Loève
transform (KL) provides a fairly simple method to do that (see e.g. [680,646]
and references therein for more technical details and e.g. [487,443] for practical
applications to observations). Basically, the idea is to work in the space of
eigenvectors Ψj of the cross-correlation matrix M ≡ 〈δˆx · δˆx t 〉, i.e. to diagonalize
the cosmic covariance matrix of the data,
M · Ψj = λj Ψj, (513)
where the matrix Ψ is unitary, Ψ −1 = Ψ t . A new set of data, ˆy, can be defined
ˆy ≡ Ψ t · ˆx, (514)
177