AA 471/671: Relativity and Cosmology

Course Coordinator: Suman Majumdar

Course Instructors: Siddharth S. Malu and Suman Majumdar

Course Structure:


  1. Special Relativity + Relativistic dynamics: Michaelson-Morley Experiment, Galilean vs. Lorentz transformations, Lorentz invariance, the metric, scalars in special relativity, relativistic dynamics, acceleration in special relativity (2-3 hrs)

  2. General Relativity: The need for GR, Vectors and Tensors, Geodesics, Equations of Geodesic and Geodesic Deviation, Curvature - definition and expression, heuristic approach to a general equation to relate curvature and matter from weak-field limit / Newtonian approach, the Einstein Equations (3 hrs)

  3. The Schwarzschild Metric: as the vacuum, static solution to GR equations, Orbits in the Schwarzschild metric, Classical Tests of GR - deviation of light rays near a massive object, anomalous perihelion shift of mercury, the idea of comoving co-ordinates, FRWL metric, the Friedmann equations from the Einstein equations (3-4 hrs)


  1. Introduction: Our place is not special, Olbers’ paradox, Cosmological principle, Expansion of the universe, cosmological redshift, scale factor, Hubble’s law, Steady state cosmology, content of the universe, quick revision of the metric in curved space-time and FRW metric. (1 hr)

  2. Comoving coordinates, Comoving distance and proper distance, Luminosity distance, Methods to measure distances, Cosmic dynamics: Friedmann equation: Newtonian derivation, modifications due to the GR, Friedmann equation: relationship between scale factor and time. (1 hr)

  3. Friedmann equations: Thermodynamics and Fluid Equations and Equation of state, revisiting redshift in the context of Friedmann equation, Omegas, Critical density, components/parameters of the universe. (1 hr)

  4. Friedmann equations: Single component universes: evolution of the scale factor, distances and energy density etc., What is the key integral here? (1 hr)

  5. Friedmann equations: Multi-component universe, evolution of the scale factor, distances and energy density etc., which component dominates when?, benchmark/concordance model of cosmology (1 hr)

  6. Cosmography: Age of the universe, horizons, revisiting distances, luminosity and angular diameter distance, Standard candle and Cosmological distance ladder, what will we gain by measuring distances (estimating cosmological parameters)?

    Problems with SN Ia data (d_L vs z plots) for different set of cosmological parameters (1 hr)

  7. Matter, Visible matter, Dark matter in galaxies, Dark matter in clusters, Gravitational lensing, candidates for dark matter, methods of detecting dark matter (1 hr)

  8. CMBR: history of the discovery, characteristics, evolution with redshift (detailed derivation), Physics of recombination and decoupling, Last scattering surface, estimation of no. density of different species of particles, Thermal history of the universe. (2 hrs)

  9. CMBR: temperature fluctuations, what causes the fluctuations? Ways to measure it? and anisotropies: dipole and quadrupole, CMB power spectrum: what are the different peaks? When were they discovered? (1 hr)

  10. Brief on Big Bang Nucleosynthesis. (1 hr)

  11. Structure formation:

    1. Linear perturbation: (3 hrs)

      1. Fluid equations, GTR analogues, application of fluid equations for adiabatic sound waves in gas

      2. Newtonian approach for deriving the growth of small scale fluctuations in the matter and radiation with time

      3. Jeans length scale and Jeans mass

      4. Growing and decaying modes of fluctuations, effect of domination of different cosmological components on these modes

    2. Spherical collapse (1 hr)

    3. Statistical description of the structures: Fourier analysis of all of the things mentioned before: (3 hrs)

      1. Density fluctuations,

      2. Growing modes

      3. Power Spectrum, transfer function and window function, sigma_8

      4. Redshift space distortions

      5. N-point statistics

      6. What is the nature of the initial fluctuations? How to generate the primordial fluctuations in a computer (Zeldovich approximation)?

      7. How the perturbations evolve from the radiation dominated to the matter dominated era? Radiation-Matter equality epoch

      8. Halo mass function and its evolution

  12. 21-cm Cosmology: (1 hr) (if time permits)


  1. General Relativity: An introduction for physicists by Hobson, Efstathiou and Lasenby

  2. Introduction to General Relativity, Black Holes and Cosmology by Yvonne Choquet-Bruhat

  3. Cambridge DAMTP Notes on GR: http://www.damtp.cam.ac.uk/user/hsr1000/lecturenotes_2012.pdf

  4. Modern General Relativity Lecture Notes: http://eagle.phys.utk.edu/guidry/astro421/lectures/grLect.pdf

  5. Introduction to Cosmology by Barbara Ryden

  6. Video Lectures on Introduction to Cosmology by Barbara Ryden -- https://www.youtube.com/watch?v=ndSD9U34-gM and associated links

  7. Video lectures on Introduction to Astronomy and Cosmology by Somnath Bharadwaj -- https://nptel.ac.in/courses/nptel_download.php?subjectid=115105046


  1. Modern Cosmology by Scott Dodelson

  2. Cosmological Physics by John Peacock

  3. Class Notes on Cosmology by John Peacock -- https://www.roe.ac.uk/japwww/teaching/cos5_1213/cos5_full.pdf

  4. Class Notes on Astrophysical Cosmology by John Peacock -- https://www.roe.ac.uk/~jap/teaching/cos4notes/cos4_0102.ps

  5. An Introduction to Modern Cosmology by Andrew Liddle

  6. Cosmology by Michael Rowan-Robinson

  7. Extragalactic Astronomy and Cosmology by Peter Schneider

  8. Principles of Physical Cosmology by P. J. E. Peebles

  9. The Large Scale Structure of the Universe by P. J. E. Peebles

Skills required to succeed in this course:

  1. Basic computing skills in python/C/C++ etc

  2. Basic plotting skills using python/gnuplot/matlab etc

  3. LaTeX to write reports for the assignments/projects (Overleaf is an easy way out!)

  4. Integration (both analytical and numerical)! Cosmology is all about measuring distances and to measure/predict/estimate distances you have to do some integrations!

Grading principle:

Grades will be decided based on your participation in the class, solving the assignments, performance in the class tests, and mid-semester and end-semester exams. Please pay attention to the lectures, go through the course material after every lecture, so that you can avoid piling up of tasks/assignments.