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

and the angular power spectrum [27] in the APM survey used an iterative deconvolution
procedure suggested by Lucy [413] to solve integral equations.
However, although Lucy’s method can provide a stable inversion, it does not
provide a covariance matrix of the recovered 3D power spectrum. Error bars
on the reconstructed 3D power spectrum have been estimated by computing
the scatter in the spectra recovered from four different zones of the APM survey
[26,27]; this can only be considered as a crude estimate and cannot be used
to constrain cosmological parameters in terms of rigorous confidence intervals.
A number of methods have emerged in the last couple of years to overcome
these limitations. These techniques involve some way of constraining
the smoothness of the 3D power spectrum to suppress features in it that lead
to minimal effects on the angular clustering and thus make the inversion process
unstable. A method using a Bayesian prior on the smoothness of the 3D
power spectrum was proposed in [188]. An improved method, based on SVD
decomposition [204], identifies and discards those modes that lead to instability.
Both methods give the covariance matrix for the estimates of the 3D power
spectrum given a covariance matrix of the angular correlations, which can be
done beyond the Gaussian approximation. The resulting 3D covariance matrix
shows significant anti-correlations between neighboring bins [188,204]; this is
expected since oscillatory features in the power spectrum are washed out by
projection and thus are not well constrained from angular clustering data.
Another technique based on maximum likelihood methods for performing the
inversion is presented in [203] (see e.g. discussion in Sect. 6.11). This has the
advantage of being optimal for Gaussian fluctuations, on the other hand, the
assumption of Gaussianity means that errors and their covariances are underestimated
at scales affected by non-linear evolution where non-Gaussianity
becomes important. Including the covariance matrix of angular correlations
showed that constraints on the recovered large-scale 3D power spectrum of
APM galaxies become less stringent by a factor of two [204,203] compared to
some of the previous analyses that assumed a diagonal covariance matrix.
Figure 51 displays the APM 3D power spectrum P(k) reconstructed from the
angular two-point correlation function [26,253] inverting Limber’s Eq. (570)
using Lucy’s method. The errorbars are obtained from the dispersion on w2(θ)
over four zones as shown in Figure 50 and should thus be considered as a
crude estimate, especially at large scales (see [203] for comparison of errors in
different inversion methods). The solid curve corresponds to a reconstruction
of the linear part of the spectrum, which can be fitted by:
P APM
linear
(k) ≃ 7 × 105
k
[1 + (k/0.05) 2 ] 1.6
(627)
for k < 0.6h/Mpc, and Ωm = 1 [30]. This linearization has been obtained
assuming no bias between APM galaxies and dark matter 114 , following the
114 Unfortunately, as shown in [30], this assumption is inconsistent at small scales:
230