Initial conditions: CDM and CHDM models

We use the Zeldovich approximation to set initial conditions. The approximation is valid in mildly nonlinear regime 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 b(t) depend on the wavelength of the perturbation |k| . In the Zeldovich approximation the comoving and the lagrangian coordinates are related in the following way:

 

$$\alpha \sum_{\mathbf{k}}^{} \alpha \sum_{\mathbf{k}}^{} \left( {\dot b_{\vert k\vert}\over b_{\vert k\vert}}\right)\mathbf$$ (17)

where the displacement vector S is related to the velocity potential $\Phi$ and the power spectrum of fluctuations P(|k|) :

 

$$\nabla_{q}^{} \Phi_{\vert k\vert}^{} \Phi_{\vert k\vert}^{} \sum_{\mathbf{k}}^{}$$ (18)

where a and b are gaussian random numbers with the mean zero and dispersion $\sigma^{2}_{}$ = P(k)/k ⁴:

 

$$\sqrt{P(\vert k\vert)} \cdot {Gauss(0,1)\over \vert k\vert^2} \sqrt{P(\vert k\vert)} \cdot {Gauss(0,1)\over \vert k\vert^2}$$ (19)

The parameter $\alpha$ , together with the power spectrum P(k) , define the normalization of the fluctuations.

We estimate the power spectrum P(k) for a wide range of cosmological models using a Boltzman code (Holtzman 1989). As compared with the original version of the code, the current version allows for more accurate estimates at high wavenumbers. For each cosmological model the numerical data points were fitted using the following fitting formula:

 

$${k^n\exp(P_1) \over (1 + P_2k^{1/2} +P_3k +P_4k^{3/2}+P_5k^{2})^{2P_6}}$$ (20)

The coefficients P_i are presented in the file cdm.fit for a variety of models. The errors of the fits are smaller than 5% in the power spectrum. The top panel in Figure 1 shows the errors of the fits for CDM models ( $\Omega_{0}^{}$ = 1 ) with a Hubble constant H = 50 km/s/Mpc. Errors at a level of $\sim$ 2% level at k $\sim$ 3h Mpc ^{- 1} and at k $\sim$ 30h Mpc ^{- 1} are due to small mismatch in approximations used at high wavenumbers. The fits smooth out the jumps and, thus, provide better approximations to the real power spectra at those large wavenumbers. The waves around k $\sim$ 0.1h Mpc ^{- 1} are due to acoustic oscillations in baryons. They are larger for larger $\Omega_{b}^{}$/$\Omega_{0}^{}$ ratios. For very small $\Omega_{b}^{}$/$\Omega_{0}^{}$ the errors introduced by using the fits are extremely small. Thus, if one can neglect (or smooth out) the acoustic oscillations, the maximum errors of our fits are expected to be smaller than 1-2% in the power. The comparison of some of our power spectra with the results from COSMICS (Bertschinger 1996) support our conclusion. We recommend the use of the fits whenever it is possible.

The power spectrum of cosmological models is often approximated using a fitting formula given by Bardeen et al.(1986, BBKS):

 

$${\ln(1+2.34q)\over 2.34q}$$ (21)

where q = k/($\Omega_{0}^{}$h ² Mpc ^{- 1}) . Unfortunately, the accuracy of this approximation is not great. Peacock & Dodds (1994) modified the fit using another relation between q and k :

 

$$\Omega_{0}^{} \Omega_{b}^{}$$ (22)

This approximation was criticized by Sugiyama (1995), who introduced a better scaling for low- $\Omega_{0}^{}$ cases:

 

$${k(T_{\rm CMB}/2.7K)^2 \over \Omega_0h^2\exp(-\Omega_b-\sqrt{h/0.5}\Omega_b/\Omega_0)~{\rm Mpc}^{-1}}$$ (23)

These approximations have been frequently used in a number of publications (e.g. Liddle et al.1996). The bottom panel of Figure 1 shows the ratio of the power spectrum given by this approximation to the power spectrum obtained from our fits for several choices of baryon fraction. For comparison, we also present the error of the eqs.(21-23) relative to the power spectrum obtained by COSMICS for $\Omega_{b}^{}$ = 0.05 (triangles), showing the good agreement of our results with those of COSMICS. In all cases, there is a large decline (around 20% in power) between a peak at k = 0.2h Mpc ^{- 1} and small scales k $\sim$ (10 - 30)h Mpc ^{- 1}. This decline was noticed by Hu & Sugiyama (1996), who studied the small-scale perturbations. Note that if we take T_CMB = 2.70 K instead of 2.726K, than the peak of the error at k = 0.2 increases up to 15%. The error in the power is rather small for small k < 0.1h Mpc ^{- 1} and for a realistic amount of baryons $\Omega_{b}^{}$ $\sim$ 0.07 . One can easily miss it if instead of an error of the power spectrum, one plots the transfer function in a double logarithmic scale. But the error is very significant for galactic-scale events. It can result in serious errors in the epoch of galaxy formation or in the amount of gas in damped Ly- $\alpha$ clouds at high redshifts.

Hu & Sugiyama (1996) recommend changing the last parameter in the BBKS fit from 6.71 to 6.07. We do not find that this correction gives an accurate fit to our spectrum. We find that the following approximation, which is a combination of a slightly modified BBKS fit and the Hu & Sugiyama (1996) scaling with the amount of baryons, provides errors in the power spectrum smaller than 5% for the range of wavenumbers k = (10^{- 4} - 40)h Mpc ^{- 1} and for $\Omega_{b}^{}$/$\Omega_{<}^{}$0.1 :

 

P(k) = k ⁿT ²(k),

$${\ln(1+2.34q)\over 2.34q}$$ T(k) =

$${k(T_{\rm CMB}/2.7K)^2\over \Omega_0h^2\alpha^{1/2}(1-\Omega_b/\Omega_0)^{0.60}} \qquad \alpha \Omega_{b} \Omega_{0} \Omega_{b} \Omega_{0}$$ q =

$$\Omega_{0}^{} \Omega_{0}^{} \Omega_{0}^{} \Omega_{0}^{}$$ a₁ = (24)

Figures 2 and 3 show errors of the fits for the CDM and for the $\Lambda$ CDM models.