Univ. Belgrade
Dr. P. Jovanović
Gravitational Lenses
(S3, elective, 6 ECTS)
Learning Outcomes:Understanding the basic theoretical concepts, practical techniques and applications of the theory of gravitational lensing, and ability to use the acquired skills for scientific investigations in extragalactic astronomy and observational cosmology.
Knowledge and Understanding:
At the end of the semester, students will acquire a detailed knowledge about the theory and the most important applications of gravitational lensing, especially in observational cosmology. They will learn how the observed time delays and statistics of strong lensing by galaxies can be used for constraining the Hubble constant, and cosmological density parameters, as well as how the strong lensing by clusters of galaxies can be used for detection of the most distant galaxies in the universe. Besides, the students will be trained to study and model the observed microlensing light curves caused by stars and extrasolar planets. Moreover, they will acquire the basic knowledge about mass reconstruction by weak lensing and its applications for studying the distributions of both visible and dark matter.
Applying Knowledge and Understanding:At the end of the course, students will be able to understand and solve a wide range of problems from the theory of gravitational lensing and observational cosmology. Specifically, they will be capable of calculating different cosmological distances and constraining cosmological parameters, such as e.g. estimation of Hubble constant from the measured time delays of lensed quasars. Moreover, they will know how to use the observed images of such a quasar to estimate both amount and distribution of mass in its lensing galaxy. Besides, the students will be trained for detecting signatures of extrasolar planets in the observed microlensing light curves.
PrerequisitesKnowledge of mathematical calculus and geometry, as well as basic programming skills.
Program1. Brief introduction to observational cosmology: 1.1. Cosmological principle, Friedmann-Lemaître-Robertson-Walker metric, perfect fluid, cosmological equation of state, Friedmann equations, cosmological parameters, standard ΛCDM cosmological model. 1.2. Distance measures in cosmology: comoving distance, angular diameter distance, luminosity distance, comoving volume. 1.3. Determination of cosmological parameters from SN Ia, CMBR and BAO. 2. Theory of gravitational lensing: 2.1. Basic principles (light bending in gravitational field) and types of gravitational lenses: strong (macro, micro) and weak lensing. 2.2. Geometrically thin lens: light deflection angle, lens equation, Einstein radius. 2.3. Point-like lenses: image positions and magnification, Paczyński light curves, binary lenses. 2.4. Extended lenses: surface mass density (convergence), deflection (lensing) potential, simple lens models: Singular Isothermal Sphere (SIS), Softened Isothermal Sphere, Isothermal Ellipsoid. 2.5. Fermat potential: lensing time delay and its dependence on Hubble constant. 2.6. Lensing optical depth: dependence on cosmological parameters, statistics of strong lenses. 2.7. Shear: simple lens models with shear, lens mapping, distortion matrix, magnification, critical curves and caustics. 2.8. Microlensing: simple models (point-like microlens, straight-fold caustic and magnification map), timescales, light curves, influence on extended sources and on AGN emission in different spectral bands. 2.9. Weak lensing: shape distortions, reduced shear, mass reconstruction: Kaiser–Squires and finite-field inversions. 3. Applications of gravitational lensing: 3.1. Macrolensing: detection of distant galaxies (natural telescopes), constraining the cosmological parameters. 3.2. Microlensing: detection of extrasolar planets, investigation of physics and geometry in the vicinity of the central supermassive black holes of AGN. 3.3. Weak lensing: detection of dark matter.
Description of how the course is conductedSee the next paragraph
Description of the didactic methodsThe course consists of theory classes with slide presentations of lectures and practical exercises. Every theory class is accompanied by examples, so that concepts do not remain too abstract and that students can practice with actual calculations. Practical exercises will consist of solving problems on the blackboard and writing scripts in Python programming language, if there is the need for graphical illustration of the theoretical models, their practical applications or their comparison with the observations.
Description of the evaluation methodsLearning assessment will be achieved through the final exam which will consist of written and oral part. Written part will require solving several problems, while oral part will require answering to two questions and providing the detailed explanation about the topics to which they are related. Problems and questions will cover both observational cosmology and gravitational lensing.
Adopted TextbooksSchneider, P., Kochanek, C., Wambsganss, J., 2006, Gravitational Lensing: Strong, Weak and Micro, Saas-Fee Advanced Course 33, Springer Schneider, P., Ehlers, J., Falco, E.E.: Gravitational Lenses, Springer, 1992 Peebles, P.J.E., 1993, Principles Of Physical Cosmology, Princeton University Press, Princeton, New Jersey, USA
Recommended readingsNarayan, R., Bartelmann, M. 1996.: Lectures on Gravitational Lensing, arXiv:astro-ph/9606001v2 Hogg, D. W., 2000, Distance measures in cosmology, arXiv:astro-ph/9905116v4