This course offers a comprehensive introduction to the Standard Model (SM), the fundamental theoretical framework describing electromagnetic, weak, and strong interactions. Students will explore the structure, symmetries, and phenomenology of the SM, along with its experimental verification.
Main Topics
Structure and symmetries of the Standard Model.
Quantum Electrodynamics (QED) refresher.
Quantum Chromodynamics (QCD) and strong interactions.
Electroweak theory and unification.
Higgs mechanism and mass generation.
Flavour physics, mixing, and CP violation.
Experimental tests and validation of the SM.
Skills Acquired
Understand the theoretical foundations of the SM and its construction.
Describe the role of symmetries and conservation laws in particle physics.
Explain how mass arises via the Higgs mechanism.
Connect theoretical predictions with experimental results.
Reading List
T.-P. Cheng & L.-F. Li, Gauge Theory of Elementary Particle Physics.
I. Aitchison & A. Hey, Gauge Theories in Particle Physics.
This course introduces Quantum Field Theory (QFT) in the operator formalism, providing the essential tools to describe interactions among elementary particles. Beginning with the quantization of scalar and spinor fields, students progress to Quantum Electrodynamics (QED), symmetry principles, and an introduction to non-abelian gauge theories. The course blends conceptual foundations with practical computational techniques, including Feynman diagrams, scattering amplitudes, and renormalization.
Main Topics
Foundations of QFT – Motivation, connection between quantum mechanics and special relativity, spin–statistics, locality, and causality.
Scalar Fields – Canonical quantization, propagators, vacuum energy, Casimir effect, and interactions in scalar theories (λϕ⁴, Yukawa).
Spinor Fields – Dirac and Weyl equations, chirality, Majorana fermions, fermion quantization, and Feynman rules for fermions.
Quantum Electrodynamics (QED) – Gauge invariance, quantization of the electromagnetic field, QED Feynman rules, key scattering processes (Bhabha, Compton), and loop corrections.
Renormalization – Power counting, divergences, dimensional regularization, and electron self-energy.
Non-Abelian Gauge Theories – Lagrangians for gauge fields and fermions, gauge invariance, running couplings, and asymptotic freedom.
Reading List
An Introduction to Quantum Field Theory, M.E. Peskin and D.V. Schroeder. Addison-Wesley Pub. Co. (1995).
The Quantum Theory of Fields, Vols. I and II, S. Weinberg. Cambridge Univ Press (1995). Curso 2018-2019.
Quantum Field Theory in a Nutshell, A. Zee. Princeton University (2003).
Quantum Field Theory, M. Srednicki. Cambridge Univ Press (2007).
Modern Quantum Field Theory, T. Banks. Cambridge Univ Press (2008).
Quantum Field Theory, C. Itzykson and J.B. Zuber. McGraw Hill (1980).
Field Theory: a Modern Primer, P. Ramond. Benjamin (1981).
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)
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.
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.
This course provides a comprehensive introduction to modern cosmology, covering the theoretical framework, observational evidence, and open questions about the origin, evolution, and large-scale structure of the Universe.
Main Topics
Cosmological principles and the Friedmann–Lemaître–Robertson–Walker models.
Thermal history of the Universe: from the hot Big Bang to the matter and radiation eras.
Inflation and the generation of cosmic perturbations.
Baryogenesis and Big Bang nucleosynthesis.
Cosmic microwave background and large-scale structure formation.
Formation of the first stars and galaxies.
Observational cosmology: distance measurements, standard candles, and large surveys.
Gravitational waves in cosmology.
Computational methods for simulating cosmic evolution.
Open problems: dark matter, dark energy, and alternatives to the standard model.
Skills Acquired
Interpret and apply theoretical models of cosmic evolution.
Relate observational data to cosmological parameters.
Analyze the interplay between theory, simulations, and astronomical observations.
This course offers an in-depth study of gravity as described by Einstein’s General Theory of Relativity. Students will explore gravity as a geometric property of spacetime, learn to derive and interpret exact solutions to Einstein’s equations, and understand their cosmological and astrophysical relevance.
Main Topics
Foundations of General Relativity: equivalence principles, covariance, and differential geometry.
Einstein’s field equations: vacuum and matter solutions, weak field limits, and exact metrics like Schwarzschild, Reissner-Nordström, and Kerr.
Experimental tests of General Relativity and classical Solar System tests.
Introduction to black hole physics, including singularities, event horizons, and black hole mechanics.
Gravitational waves: theory, emission, and modern detection methods.
Astrophysical applications: black hole mergers and supermassive black holes in galaxies.
Skills Acquired
Understand gravity as a geometric effect on spacetime structure.
Derive and analyze key exact solutions of Einstein’s equations.
Relate theory with experimental and observational tests.
Gain insight into current research topics like gravitational waves and black hole astrophysics.
Reading List
Lecture Notes on General Relativity by Daniel Baumann (all sections except Section 7 on Cosmology).
Additional Recommended Books:
Sean M. Carroll, Spacetime and Geometry
James Hartle, Gravity
Bernard Schutz, A First Course in Relativity
Ray d’Inverno, Introducing Einstein’s Relativity
M. P. Hobson et al., General Relativity
Anthony Zee, Einstein Gravity in a Nutshell
Robert M. Wald, General Relativity
Steven Weinberg, Gravitation and Cosmology
P. A. M. Dirac, General Theory of Relativity
Misner, Thorne & Wheeler, Gravitation
Eric Poisson & Clifford Will, Gravity
Yvonne Choquey-Bruhat, Introduction to General Relativity
Course Outline:
Special Relativity: Lorentz transformations, spacetime, relativistic kinematics and dynamics
Gravity as Geometry: equivalence principle, curved spacetime
