1. Wave-mechanics and the adhesion approximation Chris Short School of Physics and Astronomy The University of Nottingham UK

2. Basics of structure formation Observations: Simplifications: –Newtonian gravity –Assume the universe is spatially flat –Assume the universe is dominated by collisionless CDM ?

3. The linearised fluid approach Equations of motion for a fluid of CDM particles: At early times linear perturbation theory tells us: –Density contrast has a growing mode –Comoving velocity flow associated with the growing mode is irrotational Continuity Euler Poisson

4. The Zeldovich approximation Follows perturbations in particle trajectories: Density field becomes singular when particle trajectories cross - shell-crossing Assuming no shell-crossing the Zeldovich approximation and Euler equation can be combined: Irrotational flow guaranteed up until shell-crossing Shell-crossing can generate vorticity Zeldovich-Bernoulli

5. Assume an irrotational velocity flow : A new method: The free- particle approximation Apply Madelung transformation again In the limit : – negligible – approaches Zeldovich- ccBernoulli equation Effective potential:Perform a Madelung transformation:

6. Testing the free-particle approximation Use P 3 M code HYDRA to do an N-body simulation with: – CDM particles –Cubic simulation box of side length Mpc –SCDM cosmology,, The testing process: –Generate initial density and velocity potential fields on a uniform grid with grid spacing Mpc –Construct the initial wavefunction –Evolve the initial wavefunction using the free-particle solution –CDM density field One dimensionless free parameter

7. Behaviour of the free-particle approximation

8. The role of quantum pressure Recall: Define a ratio:

9. Point-by-point comparisons: Mpc

11. One-point PDFs

12. Comparisons in Fourier space

13. Summary The free-particle approximation provides a new way of following the gravitational collapse of density fluctuations into the quasi-linear regime 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 Zeldovich- Bernoulli approximation in all tests shown –guarantees a density field that is everywhere positive –is quick and easy to implement