Academic Team

Course Content

Type: Optional– PEC&AC Track – 3rd Trimester

This course covers advanced mathematical and theoretical tools for modern gravitational physics. It combines in-depth study of differential geometry, Hamiltonian formulations of General Relativity, and black hole thermodynamics, with an introduction to extensions of Einstein’s theory.

Main Topics

• Advanced differential geometry: differential forms, Lie groups and algebras, Yang–Mills fields, spinors in curved space, first-order formalism of General Relativity, and elements of supergravity.
• Hamiltonian formalism of GR: ADM mass, time, and conserved quantities.
• Conserved charges in gauge and gravitational theories: Komar integrals, Abbott–Deser approach, Witten’s positive mass theorem.
• Black hole thermodynamics: laws of black hole mechanics, cosmic censorship, singularity theorems, horizon topology, and no-hair results.
• Extensions of GR: scalar–tensor theories, f(R)f(R)f(R) gravity, Lovelock theories, topological terms, and 3D gravity.

Reading List

• Spacetime and Geometry, Carroll, Cambridge University Press (2019)
• Gravitation; Misner, Thorne, Wheeler, and Freeman (1970)
• General Relativity; Wald, The University Chicago Press (1984)
• Gravitation and Cosmology; Weinberg, Addison Wesley (1978)
• Gravity: Newtonian, Post Newtonian, Relativistic; Poisson and Will, Cambridge University Press (2014)
• The large scale structure of spacetime; Hawking and Ellis, Cambridge University Press (1973)

Teaching Plan Information

Course Schedule

Academic Team

Course Content

Type: Optional– PEC&AC Track – 3rd Trimester

Module A. Introduction. The Homogeneous Universe
The standard Cosmological Model. Cosmological parameters.
Problems of the Big Bang model.
The Inflationary solution.

Module B: The Non-Homogeneous Universe: Perturbations in the Matter Distribution
Linear theory of perturbation evolution in Friedmann universes.
Non-linear perturbation theory: Zeldovich approximation and N-body simulations.

Module C: Background Radiation
Propagation of a radiation field in an inhomogeneous Friedmann Universe.
Origin of anisotropies in the Background Radiation: Sachs-Wolfe effect, integrated Sachs-Wolfe effect.
Acoustic oscillations on the last scattering surface. Adiabatic and isocurvature modes.
Recent observations and implications for structure formation models. Cosmological parameters.
Sunyaev-Zeldovich effect: hot gas in clusters and secondary anisotropies in the background radiation.

Module D: Large-Scale Structure
Observations of matter distribution.
Statistical methods in Cosmology.
Measurement of the correlation function and power spectrum from galaxy catalogs.
Luminosity functions and mass functions. Comparison between simulations and observations.
Baryon acoustic oscillations in the galaxy distribution. Spectroscopic and photometric surveys.
Determination of cosmological parameters from large-scale structure.

Module E: Gravitational Lenses
History. Basic concepts.
Equations of gravitational lensing effect.
Strong and weak lenses.
Applications in Cosmology.

Reading List

• Modern Cosmology – S. Dodelson. Academic Press (2003).
  The Cosmic Microwave Background – R. Durrer. Cambridge University Press (2008).
• The Early Universe – E.W. Kolb, M.S. Turner. Addison-Wesley (1990).
• The Primordial Density Perturbation: Cosmology, Inflation and the Origin of Structure – D.H. Lyth, A. Liddle. Cambridge U.P. (2009).
• Statistics of the Galaxy Distribution – V. J. Martínez and E. Saar. Chapman & Hall (2002).
• Structure Formation in the Universe – T. Padmanabhan. Cambridge U. P. (1993).
• Cosmological Physics – J.A. Peacock. Cambridge University Press (1998).
• Large Scale Structure of the Universe – P.J.E. Peebles. Princeton U.P. (1980).
• Cosmology – S. Weinberg. Oxford U.P. (2008).
• Extragalactic Astronomy and Cosmology – P. Schneider. Springer-Verlag (2006).
• Dark Energy – Yun Wang. Wiley-VCH (2010).
• Practical Statistics for Astronomers – J.V. Wall and C.R. Jenkins. Cambridge Univ. Press.
• Measuring our Universe from Galaxy Redshift Surveys, O. Lahav and Y. Suto (2004)
• Lectures on Gravitational Lensing, Ramesh Narayan & Matthias Bartelmann (1996), astro-ph/9606001
• Gravitational Lensing in Astronomy, Joachim Wambsganss (1998), Living Reviews,
• Cosmological Parameters from Observations of Galaxy Clusters, S. Allen (2011)
• The evolution of the large-scale structure of the universe: beyond the linear regime, F. Bernardeau (2013)
• Dark Energy and CMB (S. Dodelson et al., 2013)
• Distance Probes of Dark Energy (Kim, Padmanabhan et al., 2013)
• Cosmological Perturbation theory (classic paper by Ma and Bertschinger)
• Planck cosmological results
• Supernova Ia cosmology
• Dark energy review

All the course information is available on its website.

Teaching Plan Information

Course Schedule

Academic Team

Course Content

Type: Optional– PEC&AC Track – 3rd Trimester

This course has been taught in the postgraduate programmes at Madrid Autónoma University, Durham University, the University of Valencia (2018), and The Max Planck Institute München (2009). It has also been taught (in different formats) in a number of postgraduate schools: Taller de Altas Energías (TAE 2014, 2015, 2017, 2018, 2019, 2021), 1st International School on Particle Physics and Cosmology (UIMP 2019), Higgs Centre School of Theoretical Physics (2016), STFC HEP School (2015, 2016, 2017).

Some notes for the course (pdf).
Exercises (pdf).
Extra exercise (pdf).

1.- Introduction

In this session, we present the observational evidence that points towards the existence of dark matter. We introduce the concept of dark matter halo and think about its properties (slides).

2.- Cosmology 101

A brief reminder of Early Universe Cosmology. Emphasis is put on how to compute the abundance of a given species in equilibrium. The Boltzmann equation that describes the evolution of the number density is derived (slides).

3.- Dark Matter Production (Freeze-out)

We apply the Boltzmann equation to the case of a non-relativistic DM particle that “freezes-out”, and define WIMPs (slides).

4.- Dark Matter Production (WIMP models and Freeze-in)

We study some specific particle realisations of WIMPs (concentrating on simplified models) and study special cases, such as resonant annihilation and co-annihilations. We then introduce a new paradigm, where dark matter with very small couplings “freezes-in” (slides).

5- Axions

We review the misalignment mechanism for the production of axions. The cosmological implications are reviewed. We then comment on axion detection.

6- Direct Detection

We show the basics of direct dark matter detection. We derive the equation for the observed detection rate and comment on uncertainties associated to nuclear physics and astrophysical parameters of the dark matter halo (slides).

EXERCISES

Exercises for the course (solutions available upon request) (pdf).

Extra exercise (pdf)

Reading List

Group Theory:
1) H. Georgi: “Lie Algebras in Particle Physics”,
2) H.F. Jones: “Groups, Representations and Physics”
3) J.F. Cornwell: “Group Theory in Physcis”, Vols. 1,2

Differential Geometry:
1) Nash, Sen: “Topology and Geometry for Physicists”
2) M. Nakahara: “Geometry, Topology and Physics”

All the course information is available on its website.

Teaching Plan Information

Course Schedule