Herrera Martín, Antonio (2018) Wave dark matter as a gravitational lens for electromagnetic and ...

189 Pages·2017·21.74 MB·English

Herrera Martín, Antonio (2018) Wave dark matter as a gravitational lens for
electromagnetic and gravitational waves. PhD thesis.

http://theses.gla.ac.uk/9027/

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Enlighten:Theses
http://theses.gla.ac.uk/
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Wave Dark Matter as a Gravita-
tional Lens for Electromagnetic and
Gravitational Waves
Antonio Herrera Mart´ın, M.Phys.
Astronomy and Astrophysics Group
School 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
April 2018
This thesis is my own composition except where indicated in the text.
No part of this thesis has been submitted elsewhere for any other degree
or qualification.
Copyright (cid:13)c 2018 by Antonio Herrera Martin
The oldest and strongest emotion of mankind is fear, and the oldest and strongest
kind of fear is fear of the unknown. - H. P. Lovecraft
And if he left off dreaming about you, where do you suppose you’d be? -LewisCarroll,
Through the Looking-Glass
Abstract
The majority of the matter in the known universe is believed to be in the form of Dark
Matter, and its widely accepted description is done by Cold Dark Matter (CDM).
Nevertheless, its exact properties and composition are still unknown, and it is one of
the most active areas of research in Cosmology.
The use of Cold Dark Matter has been successful to describe the general behaviour
of Dark Matter at large scales. However, it has encountered problems explaining phe-
nomena at other regimes as on the scale of galaxy halos. Therefore, other models have
been proposed over time which are able to retain the reasonable success of CDM on
large scales and extent it to other regimes where CDM has problems to explain the
observed data. One of such models is Scalar field Dark Matter (SFDM). Its properties
allow it to produce similar results at large scales and solve the problems encountered at
galacticscales. Nevertheless, thedifficultytoobtaindirectobservationsofDarkMatter
makes it difficult to give a definitive comparison between the models. Therefore, it is
important to study dark matter through different methods of analysis that would allow
to increase the validity of its scope, and these methods are constantly being researched.
In this work, a particular density profile known as Wave Dark Matter is implemented
as a gravitational lens to study its behaviour in the cases where it produces strong
lensing of light and of gravitational waves. Analytical functions for the description of a
solitoncoreandasolitoncore+NFWtailareappliedtoasub-sampleof6galaxiesfrom
The Sloan Lens ACS Survey to constrain the lensing parameters and their relation with
the profile. Furthermore, by considering the soliton core to be the main contributor to
the mass profile, this is implemented as a lens for the case of the wave approximation
4
and further to describe the major effects of the lens on gravitational waves.
It was found that the soliton core is too compact and dense in order to reproduce
the observed values of the data for the lensed galaxies. However, adding a NFW
tail alleviates the problem and reaches radii and masses within the range reported
in the literature, although the size of the NFW tail cannot be properly constrained.
Meanwhile for gravitational waves, it was found that the lensing parameters of the
soliton core, if they are expected to describe a galaxy, will be such that they are more
likely to be observed spaceborne gravitational wave detectors.
In summary, therefore, a wave dark matter soliton in combination with a NFW tail is
able to represent a galaxy, and the combination of ligh and gravitational waves should
give new insight on the validity of the profile as a description of Dark Matter galactic
haloes.
Contents
List of Tables v
List of Figures vii
Preface xvi
Acknowledgements xviii
1 Introduction 1
1.1 Modern Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 The early universe . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Standard model of cosmology . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Homogeneity and isotropy . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Friedmann Equation . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.3 ΛCDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.4 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 Scalar Field Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Gravitational lensing 21
2.1 The bending of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Lens equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Point like lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.1 Lensing Potential and Fermat potential . . . . . . . . . . . . . . 25
CONTENTS ii
2.3.2 Caustics and critical curves . . . . . . . . . . . . . . . . . . . . 25
2.4 Extended mass distribution . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Most common lensing profiles . . . . . . . . . . . . . . . . . . . . . . 27
2.5.1 SIS profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.2 NFW profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.3 Burkert profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Types of lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6.1 Strong lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6.2 Weak lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6.3 Microlensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Bayesian inference for data analysis 33
3.1 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1 Bayes’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Hypothesis space . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.3 Marginalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Model comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Markov Chain Monte Carlo (MCMC) . . . . . . . . . . . . . . . . . . . 42
3.5 Nested sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5.1 Multinest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Lensing of scalar field dark matter 48
4.1 The reduced lens equation . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Ultra light Dark Matter profiles . . . . . . . . . . . . . . . . . . . . . . 50
4.2.1 BEC or general SFDM profile . . . . . . . . . . . . . . . . . . . 51
4.2.2 ψDM soliton profile . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Gravitational lensing with a SFDM profile . . . . . . . . . . . . . . . . 52
4.4 Gravitational lensing with a WaveDM profile . . . . . . . . . . . . . . . 54
4.5 A more complete density profile . . . . . . . . . . . . . . . . . . . . . . 58
CONTENTS iii
4.5.1 Surface mass density . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5.2 Gravitational Lensing Mass . . . . . . . . . . . . . . . . . . . . 64
5 Galaxy analysis 67
5.1 Matching properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.1 Galaxy sample . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.2 Soliton core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2.3 Complete profile . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2.4 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . 90
6 Wave optics for gravitational lensing 92
6.1 Diffraction integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1.1 The lensing system . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1.2 The amplification factor . . . . . . . . . . . . . . . . . . . . . 94
6.1.3 Geometrical optics approximation . . . . . . . . . . . . . . . . . 95
6.2 Lens models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.1 The point mass lens . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.2 Singular Isothermal Sphere(SIS) - lens . . . . . . . . . . . . . . 97
6.2.3 NFW lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2.4 ψDM - soliton lens . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3 Comparison of lens properties . . . . . . . . . . . . . . . . . . . . . . . 102
6.3.1 NFW and Soliton . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7 Gravitational lensing of gravitational waves 106
7.1 Gravitational lensing effects . . . . . . . . . . . . . . . . . . . . . . . . 106
7.1.1 Effects on the lensed wave . . . . . . . . . . . . . . . . . . . . . 107
7.1.2 Effects on parameter estimation . . . . . . . . . . . . . . . . . . 110
7.1.3 The soliton lens revisited . . . . . . . . . . . . . . . . . . . . . . 114
7.2 Lensing of a gravitational wave . . . . . . . . . . . . . . . . . . . . . . 121
CONTENTS iv
7.2.1 Applying the amplification factor . . . . . . . . . . . . . . . . . 122
7.2.2 Lensed signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8 Concluding remarks and future work 134
Bibliography 137
A Galaxy posteriors 152
A.1 Posteriors for M = 1011.5 . . . . . . . . . . . . . . . . . . . . . . . . . 153
s
A.2 Posteriors for M < 1011.5 . . . . . . . . . . . . . . . . . . . . . . . . . 156
s
B Lensing figures 160