426 Numerical simulations in cosmology
In order to set the initial conditions, we choose the size of the computational
boxLand the number of particlesN^3. The phase space is divided into small equal
cubes of size 2π/L. Each cube is centred on a harmonick= 2 π/L×{i,j,k},
where{i,j,k}are integer numbers with limits from zero toN/2. We realize the
spectrum of perturbationsakandbk, and find the displacement and the momenta
of particles withq=L/N×{i,j,k}using equation (15.9). Herei,j,k= 1 ,N.
15.2.3.2 Power spectrum
There are approximations of the power spectrumP(k)for a wide range of
cosmological models. The publicly available COSMICS code (Bertschinger
1996) gives accurate approximations for the power spectrum. Here we follow
Klypin and Holtzman (1997) who give the following fitting formula:
P(k)=kn
( 1 +P 2 k^1 /^2 +P 3 k+P 4 k^3 /^2 +P 5 k^2 )^2 P^6. (15.12)
The coefficientsPiare presented by Klypin and Holtzman (1997) for a variety
of models. A comparison of some of the power spectra with the results from
COSMICS (Bertschinger 1996) indicate that the errors of the fits are smaller than
5%. Table 15.1 gives the parameters of the fits for some popular models. The
power spectrum of cosmological models is often approximated using a fitting
formula given by Bardeenet al(1986, BBKS):
P(k)=knT^2 (k),T(k)=ln( 1 + 2. 34 q)
2. 34 q[ 1 + 3. 89 q+( 16. 1 q)^2 +( 5. 4 q)^3 +( 6. 71 q)^4 ]−^1 /^4 ,(15.13)whereq=k/( 0 h^2 Mpc−^1 ). Unfortunately, the accuracy of this approximation
is not great and it should not be used for accurate simulations. We find that the
following approximation, which is a combination of a slightly modified BBKS fit
and the Hu and Sugiyama (1996) scaling with the amount of baryons, provides
errors in the power spectrum which are less than 5% for the range of wavenumbers
k=( 10 −^4 –40)hMpc−^1 and forb/ 0 < 0 .1:
P(k)=knT^2 (k),T(k)=ln( 1 + 2. 34 q)
2. 34 q[ 1 + 13 q+( 10. 5 q)^2 +( 10. 4 q)^3 +( 6. 51 q)^4 ]−^1 /^4 ,q=k(TCMB/ 2 .7K)^2
0 h^2 α^1 /^2 ( 1 −b/ 0 )^0.^60,α=a− 1 b/^0 a−(b/^0 )3
2 ,a 1 =( 46. 9 0 h^2 )^0.^670 [ 1 +( 32. 1 0 h^2 )−^0.^532 ],
a 2 =( 12 0 h^2 )^0.^424 [ 1 +( 45 0 h^2 )−^0.^582 ]. (15.14)