Principles of Gravitational Lensing: Light Deflection as a Probe of Astrophysics and Cosmology

292 Pages·2018·9.55 MB·English

Arthur B. Congdon
Charles R. Keeton
Principles
of Gravitational
Lensing
Light Defl ection as a Probe
of Astrophysics and Cosmology
Springer Praxis Books
Astronomy and Planetary Sciences
Series editors
Martin A. Barstow
Leicester, United Kingdom
Ian Robson
Edinburgh, United Kingdom
Derek Ward-Thompson
Preston, United Kingdom
More information about this subseries at http://www.springer.com/series/4175
Arthur B. Congdon • Charles R. Keeton
Principles of Gravitational
Lensing
Light Deflection as a Probe of Astrophysics
and Cosmology
123
Arthur B. Congdon Charles R. Keeton
Monrovia, CA, USA Physics & Astronomy Department
Rutgers University
Piscataway, NJ, USA
Springer Praxis Books
ISSN 2366-0082 ISSN 2366-0090 (electronic)
Astronomy and Planetary Sciences
ISBN 978-3-030-02121-4 ISBN 978-3-030-02122-1 (eBook)
https://doi.org/10.1007/978-3-030-02122-1
Library of Congress Control Number: 2018960262
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Preface
Gravitational lensing has matured into a thriving area of astrophysics, with applica-
tions ranging from detecting extrasolar planets (microlensing), to constraining the
distribution of dark matter in galaxies (strong lensing), to determining cosmological
parameters (weak lensing). While some problems are best suited to one of the
three flavors of lensing (strong, weak, or micro), there are others for which it is
possible to combine constraints from the different lensing regimes. The distribution
of dark matter in clusters of galaxies offers one important example. For this reason,
proficiency in the entire subject is indispensable to the student and researcher alike.
In their influential 1992 monograph, Gravitational Lenses, P. Schneider, J.
Ehlers, and E.E. Falco presented the theory of gravitational lensing in a rigorous
and systematic way and discussed the observations and applications then known.
During the following quarter century, though, advances in instrumentation have
made observations not possible at that time become almost routine today. The
quality and the quantity of the data thus generated allow for sophisticated statistical
analysis, making even the subtle distortions in the observed shapes of galaxies due
to the large-scale structure of the universe detectable. Such weak lensing was in its
infancy in the early 1990s, and microlensing had yet to be born.
A recent textbook that incorporates current research methods and applications
of gravitational lensing at the undergraduate level is Gravitational Lensing by
Scott Dodelson. His aim is to present enough gravitational lens theory so that the
student can quickly confront the current literature and begin conducting research.
We aim to take a complementary approach and present a thorough discussion of the
principles of gravitational lensing. As the centenary of the observational discovery
of gravitational lensing approaches in May 2019, we can only hope that the present
effort is fitting tribute to researchers in the field: past, present, and future.
It is our hope that this book will prove useful to students and researchers alike.
For those with prior experience in lensing, this book may serve as reference material
or as a supplement for researchers who wish to explore aspects of lensing outside
their own expertise. For the student, we envision this book as the basis for a one-
semester course or seminar in lensing at the advanced undergraduate or beginning
v
vi Preface
graduate level. With this in mind, problems are included at the end of each chapter
(apart from the Introduction) to build familiarity with lensing calculations and show
how they connect with astrophysics research.
This book is organized as follow: We begin in Chap. 1 with a historical overview
to offer context and background to the development of gravitational lensing.
Gravitational lenses are introduced formally in Chap. 2. For the sake of gaining
hands-on experience, we offer an intuitive, Newtonian presentation and make the
necessary relativistic correction on the fly. The properties of gravitational lenses
with circular symmetry, many of which generalize to lenses with asymmetry, are
then discussed in some detail. Chapter 3 derives the fundamental equation for light
bending in its full, relativistic glory and introduces the homogeneous, isotropic
universe that is the backdrop for what follows in later chapters. We then turn to
the theory of multiple imaging by arbitrary mass distributions in Chap. 4.
The remaining chapters explore various applications of gravitational lensing.
Microlensing by stars and planets, in which multiple images cannot be spatially
resolved, is presented using complex variables in Chap. 5. Strong lensing, where
multiple images are resolved, is discussed in the context of galaxies in Chap. 6 and
clusters of galaxies in Chap. 7. Weak lensing, in which the gravitational field is too
weak to produce multiple images, is the subject of Chap. 7, where it is applied to
clusters, and Chap. 8, where it is applied to large-scale structure. Chapter 9 gives an
overview of lensing of the cosmic microwave background, in which we extend the
methods developed in the preceding chapters.
Strictly speaking, we assume background only in multivariable calculus and
introductory physics. Familiarity with intermediate classical mechanics, electro-
dynamics, and quantum mechanics would be useful, as much of the mathe-
matics encountered in those subjects applies to lensing. For those without such
experience, we include appendices on several topics that come up in the book:
variational calculus (Appendix A), complex variables (Appendix B), orthogonal
functions (Appendix C), Fourier analysis (Appendix D), and computational methods
(Appendix E).
Monrovia, CA, USA Arthur B. Congdon
Piscataway, NJ, USA Charles R. Keeton
November 2018
Acknowledgments
This book ultimately owes its existence to the productive and enjoyable collabora-
tion between PhD student, Arthur B. Congdon (ABC), and thesis advisor, Charles
R. Keeton (CRK), a decade ago. As the influence on astrophysical research of the
bending of light by gravity has only grown in the intervening years, we decided
to rekindle our partnership to bring the principles of gravitational lensing to as
broad an audience as possible. To this end, we have sought input from astronomers,
physicists, mathematicians, and students alike.
Our own expertise has been supplemented in important ways by many people.
In addition to being a friend and mentor to ABC for over 20 years, Allan Moser
has offered many insights, inspired both by his work as a physicist and by his close
interaction with the engineering world. Carl Droms continually reminded us that
what is considered rigorous to a physicist is often hand-waving to a mathematician.
His suggestions helped us to clarify our own thinking, which has translated into
clearer, more concise prose throughout the text. He also contributed a number
of figures that illustrate ideas presented in the text. Ted Burkhardt offered many
helpful pointers on making our book accessible to physicists as well as astronomers
and researchers as well as students. Several end-of-chapter problems he suggested
appear in the following pages. We greatly appreciate their input throughout the
duration of the writing process.
The following people read one or more chapter drafts and pointed out mathemati-
cal, scientific, stylistic, and typographical imperfections: Amir Babak Aazami, Sean
Brennan, Jerod Caligiuri, John Callas, Tim Jones, Erik Nordgren, Sean Perry, Catie
Raney, and Barnaby Rowe. Much of the material in this book served as the basis for
a graduate seminar on gravitational lensing taught by CRK at Rutgers University.
We are grateful to this inaugural group of students, who helped us translate our idea
of what a lensing textbook should be into one that can actually work in practice.
ABC wishes to thank his family for their ongoing support and encouragement
and Anna Göddeke for her steadfast friendship, both personal and intellectual,
throughout the course of the project. CRK acknowledges support from the US
National Science Foundation through grant AST-1716585.
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Light and Gravity in Newtonian Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Light Bending in General Relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Consequences of Light Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Gravitational Lensing as an Observational Science. . . . . . . . . . . . . . . . . . . 4
1.4.1 Strong Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.2 Microlensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.3 Weak Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Gravitational Lenses with Circular Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Deflection Angle: Newtonian Derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Theory of Axisymmetric Lenses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Thin Lens Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Lens Equation: Geometric Derivation . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Image Magnification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Axisymmetric Lens Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Point Mass Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.2 Singular Isothermal Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.3 Nonsingular Isothermal Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Einstein Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Supercriticality and Strong Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.1 Differentiable Lens Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.2 Discontinuous Lens Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.3 Divergent Convergence Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6 Magnification and Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
ix
x Contents
3 Light Deflection in Curved Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Review of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.1 Galilean Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.3 Four-Vectors in Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Geodesic Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.1 Contravariant and Covariant Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.2 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.3 Principle of Stationary Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Schwarzschild Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.1 Gravitational Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.2 Spacetime Interval Outside a Static, Spherically
Symmetric Mass Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.3 Circular Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Light Propagation in the Schwarzschild Metric . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.1 Deflection Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.2 Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5 Friedmann-Robertson-Walker Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5.1 Homogeneous, Isotropic Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5.2 Robertson-Walker Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5.3 Friedmann’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5.4 Cosmological Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4 Multiple Imaging in the Weak-Field Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1 Lens Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1.1 Lens Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.1.2 Fermat’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1.3 Convergence and Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 Amplification Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.1 Magnification for Constant Convergence and Shear . . . . . . . . . 96
4.2.2 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.3 Eigenvalues and Image Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3 Time Delay and Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4 Burke’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.5 Critical Curves and Caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5.1 Folds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5.2 Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.6 Surface Brightness and Extended Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.6.1 Conservation of Surface Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.6.2 Magnification of an Extended Source . . . . . . . . . . . . . . . . . . . . . . . . 107
4.7 Degeneracies in the Lens Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.7.1 Similarity Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.7.2 Mass-Sheet Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Contents xi
4.7.3 Source Position Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.7.4 Connection to Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.8 Multiplane Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5 Microlensing Within the Local Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.1 Microlensing by a Point Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.1.1 Light Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.1.2 Parallax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.1.3 Astrometric Microlensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Microlensing by Multiple Point Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.1 Complex Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.2.2 Binary Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.3 Microlensing of an Extended Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3.1 Single Lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3.2 Fold Caustic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4 Microlensing Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5.1 Probing Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.5.2 Finding Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.5.3 Characterizing Compact Objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6 Strong Lensing by Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.1 Singular Isothermal Lens Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.1.1 Spherical Lenses with External Shear . . . . . . . . . . . . . . . . . . . . . . . . 146
6.1.2 Elliptical Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2 Lenses with a Core of Finite Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2.1 Emergence of Radial Caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2.2 Central Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2.3 Caustic Metamorphosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.3 Ring Images of Extended Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.4 Perturbations Due to Small-Scale Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.4.1 Millilensing by Dark Matter Substructure . . . . . . . . . . . . . . . . . . . . 164
6.4.2 Microlensing by Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.5 Lens Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.5.1 Analytic Determination of Lens Parameters . . . . . . . . . . . . . . . . . 168
6.5.2 Statistical Determination of Lens Parameters . . . . . . . . . . . . . . . . 170
6.5.3 Modeling Lenses with Extended Sources . . . . . . . . . . . . . . . . . . . . 174
6.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181