Contact: Pr. Philippe Amram


  • Introduction to cosmology
  • General Relativity
  • Friedmann-Lemaı̂tre models
  • Properties of Friedmann-Lemaı̂tre models
  • Project (choose 1 project from a list of 4 proposals)


Motivation:the goal of this optional lecture is to introduce general relativity and the subsequent main models of the Universe and tests in observational cosmology.

I. The main observations of the Universe.

  1. Radiation and observations.The observables and their interpretation.
  2. Distribution of the observed matter in the Universe. The baryonic matter: direct and indirect observations
  3. The dark matter and the dark energy
  4. The three pillars of cosmology :galaxies recession, cosmological radiation, elementary particles and primordial nucleosynthesis

II.General relativity

  1. Tensors calculations
  2. Operators,frame transformations, Christoffel symbols, differential
  3. Curve spaces Metric tensors, geodesic tensors, curvature tensors, Einstein tensors
  4. Einstein equations
  • The principles (equivalence, cosmological, Mach,general relativity,covariance).
  • Tensor
  • Energy-momentum
  • Complete formulation of the Einstein equations

III. The Friedmann --Lemaître models

  1. Expression of the four-dimension metric dimension
  2. Christoffel symbols. Ricci tensor.Scalar curvature.
  3. Energie-momentum tensor and Einstein equations
  4. Expression of the general relativity equations in the Robertson-Walker metric
  5. Unvarying models at null cosmological constant

IV. Properties of the Friedmann--Lemaître models

  1. Observational tests and evolution of the Universe distances, age, Hubble constant, spectral shifts, deceleration parameters, scaling factors, time, energy density, temperature, entropy.
  2. The standard model : successes and difficulties
  3. Inflationary models

V. Introduction to the standard model

  1. The new cosmological tests :anisotropy and cosmological background, baryonic acoustic oscillations, far supernova and gravitational shear
  2. The cosmological parameters and precision cosmology. Acceleration of the Universe.
  3. The Universe contents : baryons, dark matter, dark energy
  4. Structure of the Universe: Amplitude of fluctuations, spectral index, reionization epoch.
  5. Formation of the first stars and galaxies.