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

• 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 Madelung Transformation

222

2

*

2

22

)/exp(

)(2

1

0)(

Vt

i

i

Vt

t

Quantum Pressure

} v

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 universewhere:

• Comoving velocity associated with the growing mode is irrotational:

2/3)()(

aaDaaD Growing mode

Decaying mode

)X()(),X( iaDa

dad /XU

UXU

• 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.

Problems with the fluid approach

1

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…

)Q(UQ),Q(X aa

1)1(

1),X( 3

1

iia

a

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.

XQ

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

UXU

0)(

0)(21

Xx

2X

U

UU

a

Va

Bernoulli

Continuity

b /

Up

aa

aaaV

222

2

‘Modified potential’

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

)/exp( Ui

PVa

i 2X

2

2

2

X2

2P

dB

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.

0V

2X

2

2

a

i

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).

• 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 dimension

DX

Xi 2

cos)( 0

D

ie aDR /22/1 eRP

edB R/1

Gravitational collapse in one dimension

1

14

DX /1

1

5.0 5.05.05.0

30

100.1001.0

eR

70

102.1001.0

eR

DX /

Evolution of a periodic 1D self-gravitating system with )/2cos()( 0 DXXi

Relation to Classical Fluids

• Write• Then, ignoring

quantum pressure and having =1

• and define a velocity

)exp(iSA

0

02

1

22

2

Sm

A

t

A

VSmt

S

Sm

v 1

All trajectories on which A20 define a velocity field; the classical trajectories are streamlines of a probability flow

Streamlines and Solutions

• Suppose such a streamline is a(t).

• Any point (x,t) can be written

• Then • Ignoring higher order

terms

• So

axt )(

taStaStxS ,][)]([),(

)(exp])[exp(

)(exp])[exp(),(exp

amiaiS

SiaiStxiS

)exp(])[exp( aimaiSA

Classical PhaseQuantum Oscillation

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

)/exp( iS

2dRs

• 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).

Wave Mechanics and the Zel’dovich-Bernoulli method

)2(

0)(21

22

2X

aaaa

ap

U

UU

Zel’dovich-Bernoulli

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

• 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• 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 ,

0)( Xvi)/exp()( 22

0 LXXi

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

• Non-linear Schrödinger (Gross-Pitaevskii) equation

222

2

Vmt

i

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

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.