We propose a method of calculation of the power spectrum of cosmological perturbations by means of a direct numerical integration of hydrodynamic equations in the Fourier space for a random ensemble of initial conditions with subsequent averaging pro
cedure. This method can be an alternative to the cosmological N-body simulations. We test realizability of this method in case of one-dimensional motion of gravitating matter pressureless shells. In order to test the numerical simulations, we found an analytical solution which describes one-dimensional collapse of plane shells. The results are used to study a nonlinear interaction of different Fourier modes.
We analyze the quasar two-point correlation function (2pCF) within the redshift interval $0.8<z<2.2$ using a sample of 52303 quasars selected from the recent 7th Data Release of the Sloan Digital Sky Survey. Our approach to 2pCF uses a concept of loc
ally Lorentz (Fermi) frame for determination of the distance between objects and permutation method of the random catalogue generation. Assuming the spatially flat cosmological model with given $Omega_{Lambda}=0.726$, we found that the real-space 2pCF is fitted well with the power-low model within the distance range $1<sigma<35$ $h^{-1}$ Mpc with the correlation length $r_{0}=5.85pm0.33$ $h^{-1}$ Mpc and the slope $gamma=1.87pm0.07$. The redshift-space 2pCF is approximated with $s_{0}=6.43pm0.63$ $h^{-1}$ Mpc and $gamma=1.21pm0.24$ for $1<s<10$ $h^{-1}$ Mpc, and $s_{0}=7.37pm0.81$ $h^{-1}$ Mpc and $gamma=1.90pm0.24$ for $10<s<35$ $h^{-1}$ Mpc. For distances $s>10,h^{-1}$ Mpc the parameter describing the large-scale infall to density inhomogeneities is $beta=0.63pm0.10$ with the linear bias $b=1.44pm0.22$ that marginally (within 2$sigma$) agrees with the linear theory of cosmological perturbations. We discuss possibilities to obtain a statistical estimate of the random component of quasars velocities (different from the large-scale infall). We note rather slight dependence of quasars velocity dispersion upon the 2pCF parameters in the region $r<2$ Mpc.
