Cosmological Reconstruction via Wave Mechanics

1. Cosmological Reconstruction via Wave Mechanics Peter Coles School of Physics & Astronomy University of Nottingham

3. Cosmological Reconstruction Problems • We observe redshifts and (sometimes) estimated distances in the evolved local Universe for some galaxies • Problem I. What is the real space distribution of dark matter • Problem II. What were the initial data that evolved into the observed data?

5. The Madelung Transformation } QuantumPressure

6. Growing mode Decaying mode The fluid approach • Cold Dark Matter evolves according to a Vlasov equation coupled to a Poisson equation for the gravitational potential • The Vlasov-Poisson system is hard, so treat collisionless CDM as a fluid… • Linear perturbation theory gives an equation for the density contrast • In a spatially flat CDM-dominated universe where: • Comoving velocity associated with the growing mode is irrotational:

7. Problems with the fluid approach • Linear theory only valid at early times when fluctuations in physical fluid quantities are small. • Perturbations grow and the system becomes non-linear in nature. • Linear theory predicts the existence of spatial regions with negative density … • There has to be a single velocity at each point.

8. A Particle Approach: The Zel’dovich approximation • Follows perturbations in particle trajectories: • Mass conservation leads to: • Zel’dovich approximation remains valid in the quasi-linear regime, after the breakdown of the fluid approach…

9. Problems with the Zel’dovich approximation • The Zel’dovich approximation fails when particle trajectories cross – shell crossing. • Regions where shell-crossing occurs are associated with caustics. • At caustics the mapping is no longer unique and the density becomes infinite. • Particles pass through caustics non-linear regime described very poorly.

10. Bernoulli Continuity ‘Modified potential’ The wave-mechanical approach • Assume the comoving velocity is irrotational: • The equations of motion for a fluid of gravitating CDM particles in an expanding universe are then: where and

11. The wave-mechanical approach • Apply the Madelung transformation to the fluid equations. • Obtain the Schrodinger equation: • the quantum pressure term • the De Broglie wavelength • It’s possible to add polytropic gas pressure too…using the Gross-Pitaevskii equation

12. The ‘free-particle’ Schrodinger equation • In a spatially flat CDM-dominated universe, the ‘potential’ in the linear regime; see Coles & Spencer (2003, MNRAS, 342, 176) • Neglecting quantum pressure, the Schrodinger equation to be solved is then the ‘free-particle’ equation: • Can be solved exactly: quantum-mechanical analogue of the Zel’dovich approximation.

13. Why bother? • An example: why is the density field so lognormal? • Very easy to see using this representation: Coles (2002, MNRAS, 329, 37); see also Szapudi & Kaiser (2003, ApJ, 583, L1).

14. Gravitational collapse in one dimension • Assume a sinusoidal initial density profile in 1D: where is the comoving period of the perturbation. • Free parameters are: 1. The amplitude of the initial density fluctuation. 2. The dimensionless number • Quantum pressure • DeBroglie wavelength

15. Gravitational collapse in one dimension Evolution of a periodic 1D self-gravitating system with

16. Relation to Classical Fluids • Write • Then, ignoring quantum pressure and having =1 • and define a velocity All trajectories on which A20 define a velocity field; the classical trajectories are streamlines of a probability flow

17. Streamlines and Solutions • Suppose such a streamline is a(t). • Any point (x,t) can be written • Then • Ignoring higher order terms • So Quantum Oscillation Classical Phase

18. The Trouble with  • The classical limit has 0… • BUT the “weight” oscillates wildly as this limit is approached. • For a finite computation, need a finite value of  • Also, system becomes “non-perturbative” • Quantum Turbulence! • Note  is dimensionally a viscosity; c.f. Burgers equation

19. Zel’dovich-Bernoulli Wave Mechanics and the Zel’dovich-Bernoulli method • In Eulerian space the Zel’dovich approximation becomes: • One method of doing reconstruction.. • The Zel’dovich-Bernoulli equation can be replaced by the ‘free-particle’ Schrodinger equation.. • ..detailed tests of this are in progress (Short & Coles, in prep).

20. Cosmic reconstruction • Gravity is invariant under time-reversal! • Unitarity means density is always well-behaved. • The reconstruction question: • Non-linear gravitational evolution is a major obstacle to reconstruction. • Non-linear multi-stream regions prevent unique reconstruction. • At scales above a few Mpc, multi-streaming is insignificant smoothing necessary. Given the large-scale structure observable today, can we reverse the effects of gravity and recover information about the primordial universe?

21. Further reconstruction • This is a very limited application of this idea. • Still one fluid velocity at each spatial position. • To go further we need to represent the distribution function and solve the Vlasov equation. • This needs a more sophisticated representation, e.g. coherent state (Wigner, Husimi)

22. The wave-mechanical approach • For a collisionless medium, shell-crossing leads to the generation of vorticity velocity flow no longer irrotational • Possible to construct more sophisticated representations of the wavefunction that allow for multi-streaming (Widrow & Kaiser 1993). Phase-space evolution of a 1D self-gravitating system with ,

23. Fuzzy Dark Matter • It is even possible that Dark Matter is made of a very light particle with an effective compton wavelength comparable to a galactic scale. • Dark matter then forms a kind of condensate, but quantum behaviour prevents cuspy cores. • The quantum of vorticity is also huge…

24. ..and another thing • Non-linear Schrödinger (Gross-Pitaevskii) equation In fluid description, this gives pressure forces arising from a polytropic gas.

25. Summary • The wave-mechanical approach can overcome some of the main difficulties associated with the fluid approach and the Zel’dovich approximation. • More sophisticated representations of the wavefunction can be used to allow for multi-streaming. • The quantum pressure term is crucial in determining how well the wave-mechanical approach performs. • The `free-particle’ Schrodinger equation can be applied to the problem of reconstruction.