Introduction to Gravitational Lensing: With Python Examples

Acknowledgments
Contents
About the Author
Part I Generalities
	1 A Brief History of Gravitational Lensing
		1.1 Corpuscular Theory of Light
		1.2 The Einstein Revolution
		1.3 How to Prove the Deflection of Light?
		1.4 The Eddington Expeditions
		1.5 Following Intuitions
		1.6 First Observational Discoveries
		1.7 The First Microlensing Observations
		1.8 The Detection of Weak Lensing
		References
	2 Light Deflection
		2.1 Deflection of a Light Corpuscle
		2.2 Deflection of Light According to General Relativity
			2.2.1 Fermat Principle and Light Deflection
				Deflection in the Perturbed Minkowski's Space–Time
				Effective Refractive Index
				Deflection Angle
				Born Approximation
			2.2.2 Deflection of Light in the Strong Field Limit
		2.3 Deflection by an Ensemble of Point Masses
		2.4 Deflection by an Extended Mass Distribution
		2.5 Python Applications
			2.5.1 Light Deflection by a Black-Hole
			2.5.2 Light Deflection by an Extended Mass Distribution
		References
	3 The General Lens
		3.1 Lens Equation
		3.2 Lensing Potential
		3.3 First Order Lens Mapping
			3.3.1 First Order Lensing of a Circular Source
		3.4 Magnification
		3.5 Lensing to the Second Order
			3.5.1 Complex Notation
		3.6 Time Delay Surface
			3.6.1 Gravitational and Geometrical Time Delays
			3.6.2 Multiple Images and Magnification
			3.6.3 Examples
				Axially Symmetric Lenses: One-Dimensional Case
				Axially Symmetric Lenses: Two-Dimensional Case
				Elliptical Potentials
			3.6.4 General Considerations
		3.7 Python Applications
			3.7.1 Implementing a Ray-Tracing Algorithm
			3.7.2 Derivation of the Lensing Potential
			3.7.3 Lensing Maps
			3.7.4 Critical Lines and Caustics
			3.7.5 Shear and Flexion
			3.7.6 Full Ray-Tracing Simulation and Time Delay Surface
			3.7.7 Lensing by Numerically Simulated Mass Distributions
		References
Part II Applications
	4 Microlenses
		4.1 The Point-Mass Lens
			4.1.1 Deflection Angle and Lensing Potential
			4.1.2 Lens Equation
			4.1.3 Multiple Images
			4.1.4 Critical Lines, Caustics, and Magnification
			4.1.5 Source Magnification
			4.1.6 Microlensing Cross Section
		4.2 Microlensing Light-Curve
			4.2.1 Light-Curve Fitting
		4.3 Microlensing Parallax
			4.3.1 Orbital Parallax
			4.3.2 Satellite Parallax
			4.3.3 Terrestrial Parallax
		4.4 Astrometric Microlensing
		4.5 Photometric Microlensing: Optical Depth and Event Rates
			4.5.1 Optical Depth
				Optical Depth of an Exponential Disk
			4.5.2 Event Rate
		4.6 Results from MACHO Searches
		4.7 Multiple Point Masses
			4.7.1 Generalities
				Deflection Angle
				Lens Equation
				Critical Lines
			4.7.2 Binary Lenses
				Lens Equation
				Critical Lines and Caustics
				Multiple Images
				Image Magnifications and Light-Curves
		4.8 Planetary Microlensing
			4.8.1 Perturbations of the Central Caustic
			4.8.2 Perturbations of the Planetary Caustic
			4.8.3 Perturbations of the Resonant Caustic
			4.8.4 Perturbations of the Inner and Outer Images
			4.8.5 Analysis of the Light-Curve in a Planetary Caustic Crossing Event
			4.8.6 Planetary Microlensing Detections
		4.9 Python Applications
			4.9.1 Standard Microlensing Light-Curve
			4.9.2 Fitting the Standard Light-Curve
			4.9.3 Distribution of Microlensing Event Timescale
			4.9.4 Astrometric Microlensing Effect
			4.9.5 Critical Lines and Caustics of a Binary Lens
			4.9.6 Solving the Lens Equation of the Binary Lens
			4.9.7 Light-Curve in a Binary Microlensing Event
		References
	5 Extended Lenses
		5.1 Circular, Axially Symmetric Lenses
			Critical Lines and Caustics
			Einstein Radius
			Tangential and Radial Magnification of the Images
		5.2 Power-Law Lens
			5.2.1 Lenses with 1 2
			5.2.3 Singular Isothermal Sphere
		5.3 Softened (Non-singular) Isothermal Lenses
		5.4 Elliptical Lenses
			5.4.1 Singular Isothermal Ellipsoid
				Convergence
				Lensing Potential
				Deflection Angle
				Shear
				Critical Lines
				Caustic and Cut
				Multiple Images
				Distortion and Parity of the Images
			5.4.2 Softened (Non-singular) Elliptical Models
			5.4.3 Pseudo-Elliptical Models
		5.5 Other Profiles
			5.5.1 The Navarro–Frenk–White Model
			5.5.2 The Dual Pseudo-Isothermal Mass Distribution
		5.6 External Perturbations
		5.7 Multiple Mass Components
		5.8 Time Delays
		5.9 Mass-Sheet Degeneracy
		5.10 Multiple Lens Planes
		5.11 Python Applications
			5.11.1 Numerical Solution of the Lens Equation
				Multiple Images by a SIE Lens
			5.11.2 Triangle Mapping
			5.11.3 SIS Lens in an External Shear
			5.11.4 Multiple Lens Planes
		References
	6 Lensing by Galaxies and Clusters
		6.1 Strong Lensing by Galaxies and Galaxy Clusters
			6.1.1 Scale of the Lensing Events
			6.1.2 Strong Lensing Cross-Section
			6.1.3 The Quest for Strong Lensing Galaxies
			6.1.4 Strong Lensing by Galaxy Clusters
			6.1.5 Lens Inversion
				Parametric Reconstruction Algorithms
				Simultaneous Reconstruction of Source and Lens
				Complex Parametric Models
				Free-Form Reconstruction Algorithms
		6.2 Weak Lensing by Galaxy Clusters
			6.2.1 The Principle
			6.2.2 Ellipticity Measurements
			6.2.3 Tangential and Cross Component of the Shear
			6.2.4 Aperture Mass Densitometry
			6.2.5 The Kaiser and Squires Inversion Algorithm
			6.2.6 Challenges in Shear Measurements
				Intrinsic Source Ellipticity
				Effects of the Point-Spread-Function
			6.2.7 Redshift Dependence of the Signal
			6.2.8 Limitations of the Methods
		6.3 Applications of Lensing by Galaxies and Galaxy Clusters
			6.3.1 The Nature of Dark Matter
			6.3.2 The Interplay Between Dark Matter and Baryons
			6.3.3 Cosmic Telescopes
			6.3.4 Cosmological Applications
		6.4 Python Applications
			6.4.1 Parametric Strong Lensing Mass Reconstruction
				Simulating a Lens
				Lens Modeling
				Using More Constraints
				Optimization in the Source Plane
			6.4.2 Parametric Weak Lensing Mass Measurement
				Weak Lensing Measurements
				Fit of the Tangential Shear Profile
			6.4.3 The Kaiser-Squires Inversion Algorithm
		References
	7 Lensing by Large-Scale Structure
		7.1 Light Propagation Through an In-homogeneous Universe
			7.1.1 Deflection of Light
			7.1.2 Effective Convergence
			7.1.3 Limber's Equation and the Convergence Correlation Function
			7.1.4 Effective Lensing Potential, Lensing Jacobian, Shear
		7.2 Cosmic Shear
			7.2.1 Shear Correlation Functions
			7.2.2 Shear in Apertures and Aperture Mass
			7.2.3 E- and B-modes
			7.2.4 Cosmic Shear as a Cosmological Probe
		7.3 Lensing of Cosmic Microwave Background
			7.3.1 Lensing of the CMB Temperature
			7.3.2 Lensing of the CMB Polarization
			7.3.3 Reconstruction of the Lensing Potential
		7.4 Python Applications
			7.4.1 Effective Shear and Potential
			7.4.2 Power Spectrum
			7.4.3 Correlation Functions
		References
Part III Appendixes
	8 Python Mini-Tutorial
		8.1 Installation
		8.2 Documentation
		8.3 Running Python
		8.4 Your First Python Code
		8.5 Variables
		8.6 Strings
		8.7 Lists
		8.8 Tuples
		8.9 Dictionaries
		8.10 Blocks and Indentation
		8.11 IF/ELIF/ELSE
		8.12 While Loops
		8.13 For Loops
		8.14 Functions
		8.15 Classes
		8.16 Inheritance
		8.17 Modules
		8.18 Importing Packages
	9 Cosmology Primer
		9.1 The Friedmann–Lemaitre–Robertson–Walker Metric
		9.2 Redshift
		9.3 The Friedmann Equations
		9.4 Cosmological Parameters
		9.5 Cosmological Distances
		9.6 The Friedmann Models
			9.6.1 Single Component Models
			9.6.2 Multiple Component Models
		9.7 Structure Formation
			9.7.1 Linear Growth of Density Perturbations
			9.7.2 Density Power Spectrum
			9.7.3 Non-linear Evolution
		9.8 Mass Function
		9.9 Dark Energy Models
		References
Index