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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:
U
3U
 U   x U 
  x  0
a
2a

  x  1   U   0
a
3
2
 x  2  0
2a
Euler
Continuity
Poisson
• At early times linear perturbation theory tells us:
– Density contrast has a growing mode    a
– Comoving velocity flow associated with the growing mode is
irrotational U  x
The Zeldovich approximation
• Follows perturbations in particle trajectories:
x  q  as
• Density field becomes singular when particle
trajectories cross - shell-crossing
• Assuming no shell-crossing the Zeldovich
approximation and Euler equation can be combined:
 1
2
  x  0
a 2
Zeldovich-Bernoulli
• Irrotational flow guaranteed up until shell-crossing
• Shell-crossing can generate vorticity
A new method: The freeparticle approximation
• Assume an irrotational velocity flow U  x:
 1 1 2 2
   xx 
 VP
 0 0
aa 2 2

  x  1    x   0
a
In the limit   0 :
2
P


Perform negligible
a Madelung
–
approaches Zeldovichtransformation:
ccBernoulli equation
  1    exp  i /  
–
Apply Madelung
Effective
potential:
transformation
3 again
V 
2a

   2 2
i
    x  P 
a  2

  2 
 22  2
i P    x  x
a 2 2
Testing the free-particle
approximation
• Use P3M code HYDRA to do an N-body simulation with:
3
– N  128 CDM particles
– Cubic simulation box of side length L  200h 1 Mpc
– SCDM cosmology  dm, 0  1 , h  0.71 ,  8, 0  0.84
• The testing process:
– Generate initial density and velocity potential fields on a
uniform grid with grid spacing   1.5625h 1 Mpc
– Construct the initial wavefunction
– Evolve the initial wavefunction using the free-particle solution
2
– CDM density field     1
• One dimensionless free parameter D   / 2
Behaviour of the free-particle
approximation
The role of quantum pressure
• Recall:
 1
2
  x  P
a 
2

C
• P  2  D2
• Define a ratio:

P
C
Point-by-point comparisons:
1
rsm  4h Mpc
r 
 nb

2 1/ 2
nb

2 1/ 2
Point-by-point comparisons:
1
rsm  8h Mpc
One-point PDFs
Comparisons in Fourier space

 k  
 ˆ
 
2
ˆ  ˆ
nb
2
 ˆ
nb
2



Summary
• The free-particle approximation provides a new way
of following the gravitational collapse of density
fluctuations into the quasi-linear regime  ~ 1
• 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 ZeldovichBernoulli approximation in all tests shown
– guarantees a density field that is everywhere positive
– is quick and easy to implement