Introduction to Gravitational Lensing : With Python Examples

This book introduces the phenomenology of gravitational lensing in an accessible manner and provides a thorough discussion of the related astrophysical applications. It is intended for advanced undergraduates and graduate students who want to start working in this rapidly evolving field. This includes also senior researchers who are interested in ongoing or future surveys and missions such as DES, Euclid, WFIRST, LSST. The reader is guided through many fascinating topics related to gravitational lensing like the structure of our galaxy, the searching for exoplanets, the investigation of dark matter in galaxies and galaxy clusters, and several aspects of cosmology, including dark energy and the cosmic microwave background. The author, who has gained valuable experience as academic teacher, guides the readers towards the comprehension of the theory of gravitational lensing and related observational techniques by using simple codes written in python. This approach, beyond facilitating the understanding of gravitational lensing, is preparatory for learning the python programming language which is gaining large popularity both in academia and in the private sector. 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