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Cosmological Reconstruction via Wave Mechanics. Peter Coles School of Physics & Astronomy University of Nottingham. Cosmological Reconstruction Problems. We observe redshifts and (sometimes) estimated distances in the evolved local Universe for some galaxies

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Cosmological Reconstruction via Wave Mechanics

Peter Coles

School of Physics & Astronomy

University of Nottingham

cosmological reconstruction problems
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?
the fluid approach

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


  • Comoving velocity associated with the growing mode is irrotational:
problems with the fluid approach
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.
a particle approach the zel dovich approximation
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…
problems with the zel dovich approximation
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.



‘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


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
the free particle schrodinger equation
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.
why bother
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).
gravitational collapse in one dimension
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
gravitational collapse in one dimension1
Gravitational collapse in one dimension

Evolution of a periodic 1D self-gravitating system with

relation to classical fluids
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

streamlines and solutions
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

the trouble with
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
wave mechanics and the zel dovich bernoulli method


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).
cosmic reconstruction
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?

further reconstruction
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)
the wave mechanical approach
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 ,

fuzzy dark matter
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…
and another thing
..and another thing
  • Non-linear Schrödinger (Gross-Pitaevskii) equation

In fluid description, this gives pressure forces arising from a polytropic gas.

  • 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.