Wave-mechanics and the adhesion approximation Chris Short School of Physics and Astronomy The University of Nottingham UK Basics of structure formation • Observations: ? • Simplifications: – – – Newtonian gravity Assume the universe is spatially flat Assume the universe is dominated by collisionless CDM The linearised fluid approach • Equations of motion for a fluid of CDM particles: U 3U U x U x 0 a 2a x 1 U 0 a 3 2 x 2 0 2a Euler Continuity Poisson • At early times linear perturbation theory tells us: – Density contrast has a growing mode a – Comoving velocity flow associated with the growing mode is irrotational U x The Zeldovich approximation • Follows perturbations in particle trajectories: x q as • Density field becomes singular when particle trajectories cross - shell-crossing • Assuming no shell-crossing the Zeldovich approximation and Euler equation can be combined: 1 2 x 0 a 2 Zeldovich-Bernoulli • Irrotational flow guaranteed up until shell-crossing • Shell-crossing can generate vorticity A new method: The freeparticle approximation • Assume an irrotational velocity flow U x: 1 1 2 2 xx VP 0 0 aa 2 2 x 1 x 0 a In the limit 0 : 2 P Perform negligible a Madelung – approaches Zeldovichtransformation: ccBernoulli equation 1 exp i / – Apply Madelung Effective potential: transformation 3 again V 2a 2 2 i x P a 2 2 22 2 i P x x a 2 2 Testing the free-particle approximation • Use P3M code HYDRA to do an N-body simulation with: 3 – N 128 CDM particles – Cubic simulation box of side length L 200h 1 Mpc – SCDM cosmology dm, 0 1 , h 0.71 , 8, 0 0.84 • The testing process: – Generate initial density and velocity potential fields on a uniform grid with grid spacing 1.5625h 1 Mpc – Construct the initial wavefunction – Evolve the initial wavefunction using the free-particle solution 2 – CDM density field 1 • One dimensionless free parameter D / 2 Behaviour of the free-particle approximation The role of quantum pressure • Recall: 1 2 x P a 2 C • P 2 D2 • Define a ratio: P C Point-by-point comparisons: 1 rsm 4h Mpc r nb 2 1/ 2 nb 2 1/ 2 Point-by-point comparisons: 1 rsm 8h Mpc One-point PDFs Comparisons in Fourier space k ˆ 2 ˆ ˆ nb 2 ˆ nb 2 Summary • The free-particle approximation provides a new way of following the gravitational collapse of density fluctuations into the quasi-linear regime ~ 1 • Behaviour of the free-particle approximation depends strongly upon the value of the free parameter • The free-particle approximation – out-performs linear perturbation theory and the ZeldovichBernoulli approximation in all tests shown – guarantees a density field that is everywhere positive – is quick and easy to implement
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