MODERN COSMOLOGY

Methods 425

φorφ′. At early stages of evolution perturbations still grow almost linearly. In
this case we expect thatδdm∝ a,φ≈ constant andφ′ ≈ a. Thus,φcan
be a better choice because it does not change. This is especially helpful, if
the code uses the gravitational potential from a previous moment of time as an
initial ‘guess’ for the current moment, as happens in the case of the ART code.
In any case, it is better to have a variable which does not change much. For
equations of motion we can choose, for example, either the first equations in
(15.5)–(15.6) or the second equations. If we choose the ‘momentum’p=a^2 x ̇
as the effective velocity and take the expansion parameteraas the time variable,
then for linear growth we expect the change of coordinates per each step to be
constant:x∝a. Numerical integration schemes should not have a problem
with this type of growth. For thetandvvariables, the rate of change is more
complicated:x∝a−^1 /^2 t, which may produce some errors at small expansion
parameters. The choice of variables may affect the accuracy of the solution even
at a very nonlinear stage of the evolution as was argued by Quinnet al(1997).

15.2.3 Initial conditions

15.2.3.1 The Zeldovich approximation

The Zeldovich approximation is commonly used to set initial conditions. The
approximation is valid in mildly nonlinear regimes and is much superior
to the linear approximation. We slightly rewrite the original version of
the approximation to incorporate cases (like CHDM) when the growth rates
g(t)depends on the wavelength of the perturbation|k|. In the Zeldovich
approximation the comoving and Lagrangian coordinates are related in the
following way:

x=q−α

∑

k

g|k|(t)S|k|(q), p=−αa^2

∑

k

g|k|(t)

(

g ̇|k|

g|k|

)

S|k|(q),(15.9)

where the displacement vectorSis related to the velocity potentialand the
power spectrum of fluctuationsP(|k|):

S|k|(q)=∇q|k|(q), |k|=

∑

k

akcos(kq)+bksin(kq), (15.10)

whereaandbare Gaussian random numbers with mean zero and dispersion
σ^2 =P(k)/k^4 :

ak=

√

P(|k|)

Gauss( 0 , 1 )

|k|^2

, bk=

√

P(|k|)

Gauss( 0 , 1 )

|k|^2

. (15.11)

The parameterα, together with the power spectrum P(k),definethe
normalization of the fluctuations.