Thomson, Edward Andrew (2011) Schrodinger wave-mechanics

250 Pages·2011·5.66 MB·English

Thomson, Edward Andrew (2011) Schrodinger wave-mechanics and
large scale structure.
PhD thesis.

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Schr¨odinger Wave-mechanics and
Large Scale Structure
Edward Andrew Thomson
Astronomy and Astrophysics Group
Department of Physics and Astronomy
Kelvin Building
University of Glasgow
Glasgow, G12 8QQ
Scotland, U.K.
Presented for the degree of
Doctor of Philosophy
The University of Glasgow
October 2011
For Margaret, Gordon and Gordon.
“If we knew exactly the laws of nature and the
situation of the universe at the initial moment, we
could predict exactly the situation of the same
universe at a succeeding moment.”
- Henri Poincar´e
Abstract
In recent years various authors have developed a new numerical approach to cosmologi-
cal simulations that formulates the equations describing large scale structure (LSS) for-
mationwithinaquantummechanicalframework. ThismethodcouplestheSchr¨odinger
and Poisson equations. Previously, work has evolved mainly along two different strands
of thought: (1) solving the full system of equations as Widrow & Kaiser attempted,
(2) as an approximation to the full set of equations (the Free Particle Approximation
developed by Coles, Spencer and Short). It has been suggested that this approach can
be considered in two ways: (1) as a purely classical system that includes more physics
than just gravity, or (2) as the representation of a dark matter field, perhaps an Axion
field, where the de Broglie wavelength of the particles is large.
In the quasi-linear regime, the Free Particle Approximation (FPA) is amenable to
exact solution via standard techniques from the quantum mechanics literature. How-
ever, this method breaks down in the fully non-linear regime when shell crossing occurs
(confer the Zel’dovich approximation). The first eighteen months of my PhD involved
investigating the performance of illustrative 1-D and 3-D “toy” models, as well as a
test against the 3-D code Hydra. Much of this work is a reproduction of the work of
Short, and I was able to verify and confirm his results. As an extension to his work
I introduced a way of calculating the velocity via the probability current rather than
using a phase unwrapping technique. Using the probability current deals directly with
the wavefunction and provides a faster method of calculation in three dimensions.
After working on the FPA I went on to develop a cosmological code that did
not approximate the Schr¨odinger-Poisson system. The final code considered the full
Schr¨odinger equation with the inclusion of a self-consistent gravitational potential via
the Poisson equation. This method follows on from Widrow & Kaiser but extends
their method from 2D to 3D, it includes periodic boundary conditions, and cosmo-
logical expansion. Widrow & Kaiser provided expansion via a change of variables in
their Schr¨odinger equation; however, this was specific only to the Einstein-de Sitter
model. In this thesis I provide a generalization of that approach which works for any
flat universe that obeys the Robertson-Walker metric.
In this thesis I aim to provide a comprehensive review of the FPA and of the
Widrow-Kaiser method. I hope this work serves as an easy first point of contact to the
wave-mechanical approach to LSS and that this work also serves as a solid reference
point for all future research in this new field.
Acknowledgements
In undertaking this PhD I’ve had help from a few people within the department at
Glasgow and I use this page to gratefully acknowledge their effort. Firstly, my super-
visor Martin Hendry for providing me with the opportunity to undertake the project
as well as the support he has given along the way. Secondly, I thank Luis Teodoro for
helping to understand the theory behind wave-mechanics and for providing numerous
suggestions in order to build a robust code. He was a strong proponent of writing a
code that preserved the unitary nature of quantum mechanics and hence conserve key
physical values.
I’d like to give thanks to the following people for the help they provided along the
way (in no particular order):
Hamish Reid, my office mate, for interesting and thought provoking discussion over
the course of the project and also for helping to optimize my final code.
Tobia Carozzi for suggesting a method of finding vorticity in the velocity fields and for
strongly encouraging me to calculate total momentum for use as a diagnostic of the
code. He also helped to create the idea of particles with structure and the accompa-
nying version of wave-mechanics as presented in the final chapter.
David Sutherland for help with the interpretation of wave-mechanics in Chapter 3.
Hugh Potts for help with multi-dimensional fourier transforms.
I owe many thanks to Rebecca Johnston (Cambridge University), a fellow researcher
in LSS-wave-mechanics, for useful discussions on the subject matter.
I also owe thanks to Debbie Thomson for being a source of motivation and support.
Lastly, I thank my girlfriend Johanna for her support and encouragement:
Tack f¨or alla kladdkakor.
Contents
List of Figures xiii
1 Introduction 1
1.1 Large Scale Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Λ - CDM : Concordance model . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 A briefer history of time . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 On space and time . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Review of Numerical Simulations 36
2.1 N-Body simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.1 Direct Force Summation . . . . . . . . . . . . . . . . . . . . . . 40
2.1.2 Particle-Mesh N-Body . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Definition of symplectic . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Fluid dynamics and perturbation theory . . . . . . . . . . . . . . . . . 48
2.4 From old to new: wave-mechanical approach conceived . . . . . . . . . 49
3 Wave-mechanics 52
3.1 Introduction to wave-mechanics of LSS . . . . . . . . . . . . . . . . . . 52
3.1.1 Interpretation of the Schr¨odinger equation . . . . . . . . . . . . 53
3.1.2 Hydrodynamic form of wave-mechanics . . . . . . . . . . . . . . 56
3.2 Overview of wave-mechanics as applied to LSS . . . . . . . . . . . . . . 64
3.2.1 Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2.2 Widrow and Kaiser . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.3 Guenther . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.4 Widrow and Davies . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.5 Hu et. al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2.6 Harrison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
CONTENTS vi
3.2.7 Szapudi and Kaiser . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.8 Coles, Spencer, Short . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.9 Woo and Chiueh . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2.10 Johnston et. al. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Solving the Schr¨odinger equation . . . . . . . . . . . . . . . . . . . . . 78
3.3.1 Wave-mechanics and cosmological initial conditions . . . . . . . 79
3.4 Velocity calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.1 Phase Unwrapping . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.2 Phase-angle method . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4.3 Probability current method . . . . . . . . . . . . . . . . . . . . 84
3.5 Singularities as points of vorticity . . . . . . . . . . . . . . . . . . . . . 85
4 Free Particle Approximation 87
4.0.1 Linear Growth Factor . . . . . . . . . . . . . . . . . . . . . . . 89
4.1 1D Free Particle Approximation . . . . . . . . . . . . . . . . . . . . . . 91
4.1.1 1D Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 3D Free Particle Approximation . . . . . . . . . . . . . . . . . . . . . . 96
4.2.1 Toy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2.2 Real Cosmological Test . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.3 Results - Cosmological Test . . . . . . . . . . . . . . . . . . . . 100
4.2.4 Consistency checks . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.5 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3 Conclusion and evaluation of FPA . . . . . . . . . . . . . . . . . . . . . 107
4.3.1 From FPA to solving the full system . . . . . . . . . . . . . . . 108
5 Solving the full Schr¨odinger-Poisson system 111
5.1 Specific requirements of a Schr¨odinger solver . . . . . . . . . . . . . . . 113
5.2 Numerical method for solving the Schr¨odinger equation . . . . . . . . . 117
5.2.1 One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.2 Periodic boundaries . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.3 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2.4 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3 Solving the Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . 130
5.4 Computational algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.5 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.5.1 Cosmological Initial Conditions . . . . . . . . . . . . . . . . . . 133
CONTENTS vii
5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.6.1 Periodic Boundaries . . . . . . . . . . . . . . . . . . . . . . . . 135
5.6.2 2D Two-body gravitational interaction . . . . . . . . . . . . . . 143
5.6.3 Tophat collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.6.4 Cosmological simulation . . . . . . . . . . . . . . . . . . . . . . 150
5.7 Velocity & Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6 Conclusion 178
6.1 Future work: extending the full Schr¨odinger-Poisson system . . . . . . 183
6.1.1 Multiple fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.1.2 Including additional physics . . . . . . . . . . . . . . . . . . . . 185
6.1.3 Periodic boundaries via adhesive operators . . . . . . . . . . . . 186
6.1.4 Mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.1.5 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7 Epilogue: Vorticity and spin 191
7.1 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.1.1 Gravitoelectromagnetism . . . . . . . . . . . . . . . . . . . . . . 194
7.2 Spinning objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.2.1 N-body considerations . . . . . . . . . . . . . . . . . . . . . . . 204
7.2.2 Pauli equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.2.3 Quadrupole term and higher . . . . . . . . . . . . . . . . . . . . 210
A Translation 223
A.1 Translation of Madelung’s 1927 paper . . . . . . . . . . . . . . . . . . . 223
B Mathematical appendix 229
B.1 Madelung transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
B.2 Derivation of the initial velocity of the FPA . . . . . . . . . . . . . . . 231
B.3 Derivation of the Schr¨odinger and Poisson equations in the EdS model 234
B.3.1 Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . . 234
B.3.2 Schr¨odinger equation . . . . . . . . . . . . . . . . . . . . . . . . 234
List of Figures
1.1 This figure shows slices from observational data (2dF, Sloan) and from
theMillenniumSimulation. Qualitatively,allsliceslooksimilar; however
statistically there are small but notable differences which indicates that
currentsimulationsarenotyetadvancedenoughtoreproduceacomplete
picture of our Universe. (Virgo Consortium 2005b) . . . . . . . . . . . 7
1.2 This figure shows the evolved non-linear CDM power spectrum (think
black line) as generated by Smith et al, this power spectrum was calcu-
lated from the Λ-CDM theory. The input power spectrum is the initial
power spectrum at the start of the simulation, which is post Last Scat-
tering. The straight part of the dash line shown is what we expect
the inflation power spectrum to look like. The coloured lines and dat-
apoints are from power spectra generated from various runs of a Large
Scale Structure code used by Teodoro (Teodoro 2008). L0 is the length
of the simulation box (in h−1 Mpc) and N is the number of particles in
the simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3 This figure shows the angular power spectrum of the temperature fluctu-
ation of the CMB. The (lower) x-axis is the effective multipole number
(l) while at the top of the picture is the angular scale (degrees) (Wright
2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1 The graphs here replicate the results of Short, hence I show them in the
same format of log(2 + δ) against x/d at ‘times’: D = 1,59.06 (top)
and D = 117.16,174.98 (bottom); here ν = 1.0. Note: taking log(2+δ)
avoids taking log(0) for δ = 1. . . . . . . . . . . . . . . . . . . . . . . 93
−
4.2 These graphs show the one dimensional velocity that corresponds to
the over-densities of figure 4.1. The times are D = 1,59.06 (top) and
D = 117.16,174.98 (bottom). The axes are v/d and x/d . . . . . . . . . 94