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

Extension to minimum variance cubic estimators for the angular bispectrum
in the Gaussian limit is considered in [307,245].
Note that, the full minimum variance estimator involves inverting a rank 4
matrix, a very demanding computational task, which however simplifies significantly
in the Gaussian limit where ˜ C factorizes. Another case in which the
result becomes simpler is the so-called FKP limit [212], where the selection
function ng(r) can be taken as locally constant, compared to the scale under
consideration. This becomes a good approximation at scales much smaller
than the characteristic size of the survey, which for present surveys is where
non-Gaussian contributions become important, so it is a useful approximation.
In this case the minimum variance pair weighting for a pair ij is only a function
of the separation α of the pair, not on their position or orientation, since
ni and nj are assumed to be constants locally. As a result, the power spectrum
covariance matrix can be written in terms of a two by two reduced covariance
matrix, which although not diagonal due to non-Gaussian contributions, becomes
so in the Gaussian limit, leading to the standard result Eq. (476). We
refer the reader to [296] for more details.
6.11.3 Uncorrelated Error Bars
Clearly, minimum variance estimates can be deceptive if correlations between
them are substantial. Ideally one would like to obtain not only an optimal estimator
(with minimum error bars), but also estimates which are uncorrelated
(with diagonal covariance matrix), like in the case of the power spectrum of a
Gaussian field in the infinite volume limit. Once the optimal (or best possible)
estimator ˆ f is found, it is possible to work in a representation where the cosmic
covariance matrix C becomes diagonal,
C · Ψj = λjΨj, (509)
where the eigenvectors Ψj form an orthonormal basis. A new set of estimators
can be defined
ˆg ≡ Ψ −1 · ˆ f, (510)
which are statistically orthogonal
〈δˆgiδˆgj〉 = λiδij = Ψ t i · C · Ψi δij. (511)
These new estimators can in principle be completely different from the original
set, but if by chance the diagonal terms of C are dominant, then we have ˆg ≃ ˆ f.
In fact, if one takes the example of the two-point correlation function (or higher
order) in case the galaxy number density is known, using the new estimator
ˆg is equivalent to changing the binning function Θ defined previously to a
more complicated form. Among those estimators which are uncorrelated, it is
however important to find the set ˆg such that the equivalent binning function
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