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

the initial conditions.
In Sect. 2.4.6 we presented the solutions involving the full time dependence
from arbitrary initial conditions [561]. Again, we assume Ωm = 1 for simplicity.
The recursion relations for PT kernels including transients results from using
the following ansatz in Eq. (86),
Ψ (n)
a (k, z) =
d 3
k1 . . .
d 3 kn [δD]n F (n)
a (k1, . . .,kn; z)δ1(k1) · · ·δ1(kn),
(298)
where a = 1, 2, z ≡ ln a(τ) with a(τ) the scale factor, and the nth order
solutions for density and velocity fields are components of the vector Ψb, i.e.
Ψ (n)
1 ≡ δn, Ψ (n)
2 ≡ θn. In Eq. (298), [δD]n ≡ δD(k − k1 − . . . − kn).
The kernels F (n)
a now depend on time and reduce to the standard ones when
transients die out, that is F (n)
1 → Fn, F (n)
2 → Gn when z → ∞. Also, Eq. (298)
incorporates in a convenient way initial conditions, i.e. at z = 0, F (n)
a = I(n) a ,
where the kernels I (n)
a describe the initial correlations imposed at the start of
the simulation. For the ZA we have
I (n)
1 = F ZA
n
, I(n)
2 = G ZA
. (299)
n
Although most existing initial conditions codes use the ZA prescription to set
up their initial conditions, there is another prescription to set initial velocities
suggested in [199], which avoids the high initial velocities that result from
the use of ZA because of small-scale density fluctuations approaching unity
when starting a simulation at low redshifts. This procedure corresponds to
recalculate the velocities from the gravitational potential due to the perturbed
particle positions, obtained by solving again Poisson equation after particles
have been displaced according to the ZA. Linear PT is then applied to the
density field to obtain the velocities, which implies instead that the initial
velocity field is such that the divergence field Θ(x) ≡ θ(x)/(−f H) has the
same higher-order correlations as the ZA density perturbations. In this case,
I (n)
1 = F ZA
n
, I(n)
2 = F ZA
. (300)
n
The recursion relations for F (n)
a , which solve the non-linear dynamics at arbitrary
order in PT, can be obtained by replacing Eq. (298) into Eq. (86), which
yields [561]
F (n)
a (k1, . . .,kn; z) =e −nz gab(z) I (n)
n−1 z
+
m=1 0
b (k1, . . .,kn)
ds e n(s−z) gab(z − s)γbcd(k (m) ,k (n−m) )
×F (m)
c (k1, . . .,km; s) F (n−m)
d (km+1, . . .,kn; s), (301)
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