Univ. Belgrade
Dr. P. Jovanović
Gravitation and Cosmology
(S2, elective, 6 ECTS)
Learning Outcomes:Understanding the basic theoretical concepts and predictions of general relativity and cosmology, as well as ability to practically use the underlying mathematical apparatus for scientific investigations in extragalactic astronomy and cosmology.
Knowledge and Understanding:
At the end of the semester, students will acquire a detailed knowledge about general relativity as standard theory of gravity and foundation for modern cosmology. They will also have a deep insight into the nature of gravity and connection between the spacetime geometry and matter-energy distribution. Besides, the students will acquire the knowledge about the most important theoretical predictions of general relativity and their experimental confirmations, as well as about modern cosmology which is based on general relativity and which is in good agreement with the observed cosmic microwave background radiation, accelerating expansion of the universe, abundances of light elements and the distribution of galaxies at large scales.
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 gravitation and cosmology. Moreover, they will be capable of studying different effects of strong gravity in the vicinity of black holes (such as e.g. gravitational redshift and orbital precession), calculating cosmological distances and constraining cosmological parameters using several techniques.
PrerequisitesKnowledge of mathematical analysis, geometry and calculus.
Program1. General relativity (GR) as geometric theory of gravitation: 1.1. Basic mathematical concepts and tools: spacetime as 4-dimensional pseudo-Riemannian manifold. Einstein summation convention. General coordinate transformations. Curved spacetime and curvilinear coordinates. Tangent and dual bases. Covariant and contravariant transformation rules. Scalars (invariants), vectors and tensors. Tensor operations and Ricci calculus. 1.2. Spacetime in absence of gravity: Minkowski spacetime. Lorentz transformations. Reference frames. Events. Invariance of the spacetime interval. 1.3. Introduction to GR: Principle of general covariance and equivalence principle. Metric tensor as gravitation field. Affine connections and Christoffel symbols. Geodesic equations. Equations of motion of freely falling particles in gravitational field. Weak gravitational field. Parallel transport around a closed curve. Riemann-Christoffel curvature tensor and Bianchi identities. Ricci curvature tensor and Ricci scalar. Einstein tensor. Stress-energy tensor as source of gravity. Perfect fluid. 1.4. Calculus of variations and variational principle: Lagrangian. Euler-Lagrange equations. Hamilton’s principle of stationary action. 1.5. Einstein field equations (EFE): Einstein-Hilbert action. Cosmological constant. EFE in vacuum and in the presence of matter. Schwarzschild solution of EFE. 1.6. Experimental tests of GR: deflection of light. Perihelion precession. Gravitational red shift. Black holes. Gravitational waves. 2. Cosmology based on GR: 2.1. Big Bang theory: stages in cosmic evolution. 2.2. Cosmological principle: homogeneity and isotropy of the universe. Construction of Friedmann-Lemaître-Robertson-Walker (FLRW) metric. 2.3. Cosmic scale factor: cosmological redshift. Hubble parameter. Deceleration parameter. 2.4. Friedmann equations (FE): derivation from EFE for FLRW metric and perfect fluid. Cosmological density parameters. Mass-energy budget of the universe (baryonic matter, dark matter and dark energy). 2.5. Models of Friedmannian cosmology: Friedmann-Einstein, Einstein-de Sitter, de Sitter and standard ΛCDM cosmological model. 2.6. Inflation: cosmological problems (flatness, horizons and monopoles). Inflaton scalar field. Einstein-Hilbert action, EFE and FE with inflaton. Slow-roll inflation. 2.7. Observational cosmology: comoving coordinates and cosmological distances (proper, comoving, luminosity and angular diameter distance). Hubble-Lemaître law. Supernovae of type Ia (SN Ia). Cosmic microwave background radiation (CMBR). Baryon acoustic oscillations (BAO).
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.
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 two questions and providing the detailed explanation about the topics to which they are related. Problems and questions will cover both general relativity and cosmology.
Adopted TextbooksWeinberg, S., 1972, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley-VCH Harrison, E. R., 2000, Cosmology. The science of the universe, Cambridge University Press, Cambridge, UK Weinberg, S., 2008, Cosmology, Oxford University Press Inc., New York, USA Peebles, P.J.E., 1993, Principles Of Physical Cosmology, Princeton University Press, Princeton, New Jersey, USA
Recommended readingsDavis, T. M., Lineweaver, C. H. 2004, Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe, PASA, 21, 97 (arXiv:astro-ph/0310808) Hogg, D. W., 2000, Distance measures in cosmology, arXiv:astro-ph/9905116v4