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:

• 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

02

3

01

02

3

22

a

a

aa

x

x

xx

U

UUU

U

a

xU

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

sqx a

02

1 2

xaZeldovich-Bernoulli

• Assume an irrotational velocity flow :

01

02

1 2

xx

x

a

Va

01

02

1 2

xx

x

a

Pa

01

02

1 2

xx

x

a

a

A new method: The free-particle approximation

xU

Apply Madelung transformation again

22

22

2

2

x

x

P

Pa

i

22

2 xa

i

In the limit :

– negligible

– approaches Zeldovich- ccBernoulli equation

2P

0

Effective potential:

aV

2

3

Perform a Madelung transformation:

/exp1 i

Testing the free-particle approximation

• Use P3M 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

1200 hL

3128N

15625.1 h

12

2/D

10, dm 71.0h 84.00,8

Behaviour of the free-particle approximation

The role of quantum pressure

• Recall:

•

• Define a ratio:

P

Ca

2

2

1 x

C

P

22 DP

Point-by-point comparisons: Mpc14 hrsm

2/122/12

nb

nbr

Point-by-point comparisons: Mpc18 hrsm

One-point PDFs

Comparisons in Fourier space

22

2

ˆˆ

ˆˆ

nb

nbk

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

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