wave-mechanics and the adhesion approximation chris short school of physics and astronomy the...
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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:
• 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
1~