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

in [152]. Similarly to Eq. (472), the weighted factorial moment estimator reads
ˆF C k
= 1
C
C
i=1
(Ni)k ω(ri)
, (478)
[φR(ri)] k
where φR(r) is the average of the selection function over a cell.
To simplify the writing of the cosmic error as a function of the sampling weight,
the variations of the function ω and of the selection function are assumed to
be negligible within the cells, which is equivalent to points (1) and (3) above.
Then the relative cosmic error σFk [ω, φ] = (∆ ˆ Fk/Fk) 2 is
σ 2 Fk [ω, φ] = σ2 F [ω] + σ2 E [ω] + σ2 D [ω, φ], (479)
where the finite volume, edge effect and discreteness contributions read, respectively
σ 2 F[ω]= σ2 F
¯ξ( ˆ L) ˆ V
σ 2 E[ω]= σ2 E
ˆV
σ 2 D[ω, φ]= 1
ˆV
ˆV
ˆV
ˆV
d 3 r1d 3 r2 ω(r1) ω(r2) ξ(r12), (480)
d 3 r ω(r), (481)
d 3 r ω 2 (r) σ 2 D(r). (482)
In these equations, there are terms such as σ2 F = σ2 F [1] or σ2 E = σ2 E [1]. They correspond
to the finite volume and edge effect errors in the case of homogeneous
sampling weight. They do not depend on the number density and are given
by analytical expressions in Appendix F. The term σ2 D (r) is similar, but there
is a supplementary r dependence because the average count ¯ N is proportional
to the selection function φ.
Using Lagrange multipliers, it is easy to write the following integral equation
which determines the optimal weight [152]
σ 2 F
¯ξ( ˆ L) ˆ V
ˆV
d 3 u ω(u) ξ(|r − u|) + [σ 2 E + σ2 D (r)]ω(r) + λ = 0. (483)
The constant λ is determined by appropriate normalization of the weight
function
1
d
ˆV
3 r ω(r) = 1. (484)
ˆV
The solution of this integral equation can be found numerically. However, the
approximation (473) was found to be excellent, i.e. almost perfectly minimizes
the cosmic error [152].
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