MODERN COSMOLOGY

Quantifying large-scale structure 67

eigenmodes in the new coordinates:

C′′·ψ′′i=λψ′′i.

These eigenmodes must be expressible in terms of some modes in the original
coordinates,ei:
ψ′′i =(W·R)·ei.

In these terms, the eigenproblem is

(W·R)·C·(W·R)†·(W·R)·ei=λ(W·R)·ei.

This can be simplified usingW†·W=N′−^1 andN′−^1 =R·N−^1 R†,togive

C·N−^1 ·ei=λei,

so the required modes are eigenmodes ofC·N−^1. However, care is required when
considering the orthonormality of theei:ψ†i·ψj=e†i·N−^1 ·ej,sotheeiare
not orthonormal. If we writed=

∑

iaiei,then

ai=(N−^1 ·ei)†·d≡ψ†i·d.

Thus, the modes used to extract the compressed data by dot product satisfy
C·ψ=λN·ψ, or finally
S·ψ=λN·ψ,

given a redefinition ofλ. The optimal modes are thus eigenmodes ofN−^1 ·S,
hence the namesignal-to-noise eigenmodes(Bond 1995, Bunn 1995).
It is interesting to appreciate that the set of KL modes just discussed is also
the ‘best’ set of modes to choose from a completely different point of view:
they are the modes that are optimal for estimation of a parameter via maximum
likelihood. Suppose we write the compressed data vector,x, in terms of a non-
square matrixA(whose rows are the basis vectorsψ∗i):

x=A·d.

The transformed covariance matrix is

D≡〈xx†〉=A·C·A†.

For the case where the original data obeyed Gaussian statistics, this is true for the
compressed data also, so the likelihood is

−2lnL=ln detD+x∗·D−^1 ·x+constant.

The normal variance on some parameterp(on which the covariance matrix
depends) is
1
σp^2

=

d^2 [−2lnL]

dq^2

.