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

From Eq. (B.3) we have,
∞
y ν1 = τ − y νp
p=2
(−τ) p−1
(p − 1)!
which, after integrating relation (B.7), implies that,
L = c + τ2
2
∞ (−τ)
+ y νp
p=2
p
p!
(B.8)
= c + τ2
2 + yζ(τ) + y ν1 τ, (B.9)
which leads to (the integration constant c = 0 is such that ϕ(y) ∼ −y 2 at
leading order in y),
ϕ(y) = yζ(τ) − 1
2 yτζ ′ (τ). (B.10)
This equation, with Eq. (B.3), gives the tree generating function expressed as
a function of the vertex generating function ζ.
B.2 For Two Fields
We can extend the previous results to joint tree summations. It corresponds
to either 2 different fields taken at the same position (as the density and the
velocity divergence for instance), or to two fields taken at different locations.
We want to construct the joint generating function, ϕ(y1, y2), of the joint
cumulants,
ϕ(y1, y2) = −
n,m,n+m≥2
(−y1)
Cnm
n (−y2)
n!
m
m!
(B.11)
where Cnm is the value of each cumulant. In this case for each diagram there
are n vertices of type 1, and m of type 2. They take respectively the value νp
and µq if they are connected respectively to p or q neighbors. Obviously if the
two fields are identical the two series identify. Moreover in order to account
for cell separation, a weight ξ is put for each line connecting points of different
nature.
The generating function ϕ is then a function of y1, y2, ξ, ν1, . . .,µ1, . . .. One
can define the two functions τ1 and τ2 by,
τ1 = 1 ∂(−ϕ)
, τ2 =
−y1 ∂ν1
1 ∂(−ϕ)
. (B.12)
−y2 ∂µ1
It is easy to see that the functions τ1 and τ2 are given respectively by,
τ1 =y1
∞
p=1
(−τ1)
νp
p−1
(p − 1)!
∞ (−τ2)
+ ξ y2 µp
p=1
p−1
, (B.13)
(p − 1)!
268