V.M. Sliusar , V.I. Zhdanov Astronomical Observatory,

1. One-dimensional model of cosmological perturbations: direct integration in Fourier space V.M. Sliusar, V.I. Zhdanov Astronomical Observatory, TarasShevchenko National University of Kyiv Observatornastr., 3, Kiev 04053 Ukraine vitaliy.slyusar@gmail.com, valeryzhdanov@gmail.com

2. Introduction We test a method of investigation of the power spectrum of cosmological perturbations with a special choice of initial conditions corresponding to the one-dimensional evolution of cold dark matter. The method deals with a direct numerical integration of hydrodynamical equations for the perturbations in the Fourier space for a random ensemble of initial conditions with subsequent averaging procedure. This method is an alternative the cosmological N-body simulations. We have obtained the power spectrum of the density contrast in non-linear regime up to scales ~0.3 Mpc. The results are used to study a nonlinear interaction of different Fourier modes and for a comparison with the available approximation schemes. We point out a significant growth of dispersion of the power spectrum in the non-linear region.

3. Method description Here the alternative approach to calculate correlation function and power spectrum is proposed. The idea is not to study the dynamic of particles (N-Body simulations), but to integrate directly the hydrodynamical equations in the Fourier space. This approach is effective mostly for the weakly nonlinear regime, but it is also possible to consider the strongly nonlinear regime. For the initial moment of time we set values of δin(k)θin(k) according to the Gaussian distribution with power-law power spectrum (n = - 2) or an and bn. At the moment we consider only the onedimensional case to simplify the equation solution process and to reduce the total machine time required to perform this calculations. We used the GPGPU instead of the classic CPU because the problem can be easily processed in parallel environment. The results of trial onedimentional run show that it is possible to work in three dimensions as well.

4. Hydrodynamical equations in the Fourier space • Onedimentional case: • Threedimentional case Scoccimarro& Frieman 1996; Pietroni 2008; Scoccimarro 1997

5. Hydrodynamical equations’ solution in the Lagrangian description: initial conditions in the form of the Fourier expansion • Symmetric onedimentional case , where L is the size of a periodic “box”

6. Comparison with N-Body method Aquarius simulation;Springel et al. (2008)

7. Inter wave number interaction One of the most basic tests that we performed to test the method was where we set the initial conditions in such a way that the density contrast δ had non-zero value for only one value of k. This allows us to see how the perturbations appear on other k-scales.

8. “Uniform” initial conditions It is very complicated or even impossible to solve the previously shown equations analytically, so it was interesting to test the solutions in different cases. On this slide you can see the density contrast rations in case where all the initial conditions of δwere set according to the Gaussian distribution with constant dispersion .

9. Results We integrate the equations while . Initial conditions are specified for z = 99. Totally we calculated 100 random realizations of initial conditions that are ensemble averaged to get the resulting curves of density contrast ratios in nonlinear and linear regime. Time spent for evolution of one realization is 5 seconds.

10. Conclusions • The method was tested in one dimentional case and it works. • At the moment with the help of GPGPU (OpenCL software) we can reach the 0.1 Mpc/h scale and at the moment there are no limits to reach 0.01 Mpc/h scale and even smaller.

11. Thank you for attention