QFT in the Early Universe

What all this is about

The temperature fluctuations in the Cosmic Microwave Background (CMB) are a snapshot of the primordial density perturbations that gave rise to structure formation.  But… where were these initial fluctuations coming from? In this course, we will discuss the limitations of the hot Big Bang model in addressing this question and its answer within the inflationary paradigm.

We will extend the canonical quantization in Minkowski spacetime to extreme cosmological environments where familiar concepts like “vacuum state” or “particle” become generically ill-defined.  Using the resulting formalism, we will present a detailed derivation of the primordial power spectra. In particular, we will show that the quantum-mechanical treatment of inflation leads to an almost scale invariant spectrum of classical density perturbations in excellent agreement with observations and to a (still-undiscovered) gravitational wave background.

The success on the inflationary paradigm will motivate the search of specific realizations within particle physics scenarios.  We will discover that, contrary to other studies in cosmology, the physics of inflation remains highly speculative. To parametrize our ignorance, we will make use of one of the boldest frameworks in particle physics: effective field theories.

We will discuss the transition from the cold and empty post-inflationary Universe to the hot and highly entropic one we observe today. This transition involves rich physics such as Bose enhancement, turbulence and thermalization.

Time permitting, we will discuss some selected topics in the Early Universe with special emphasis on its potential connection with inflation.

Have a look to the preliminary program here


The course is intended to be self-consistent. However, a basic knowledge of Quantum Field Theory (Lagrangians, symmetries, canonical quantization…), General Relativity (concept of proper distance, Einstein equations …) and basic cosmology (FRW metrics, Friedmann equations, hot Big Bang model… ) is highly recommended. Familiarity with some concepts from quantum optics (coherent states, squeezed states,…) could be also helpful but not strictly necessary.                                        
Units and notation

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Lecture notes                                         

Exercise 1 GW from symmetry principles
Exercise 2 Integrating out d.o.f
Exercise 3 Higgs inflation
Exercise 4 ADM formalism
Exercise 5 Action for curvature pert.
Exercise 6 Inflationary observables in SR
Exercise 7 Decoherence without decoherence
Exercise 8 The onset of the hBB
Exercise 9 Reading



–  Cosmological Inflation and Large-Scale Structure, Andrew R. Liddle,  David H. Lyth, Cambridge University Press, 2000.
– Physical Foundations of Cosmology, Viatcheslav Mukhanov, Cambridge University Press, 2005.
– Introduction to Quantum Effects in Gravity, Viatcheslav Mukhanov and Sergei Winitzki, Cambridge University Press, 2007. Also online
–  Inflation and String Theory, Daniel Baumann and Liam McAllister, Cambridge University Press.
Decoherence and the appearance of a classical world in quantum theory, E. Joos et al.,Springer-Verlag Berlin Heidelberg 1996.– Classical Theory of Gauge Fields, Valery Rubakov, Translated by Stephen S. Wilson, Princeton University Press.
–  Cosmic Strings and Other Topological Defects,  A. Vilenkin, E. P. S. Shellard, Cambridge Monographs on Mathematical Physics).
– Quantum Fields in Curved Space, N.D. Birrell, P.C.W. Davies, Cambridge University Press, 1982.
– Vacuum Quantum effects in Strong Fields, A.A. Grib, S.G. Mamayev and V.M. Mostepanenko, Friedmann Laboratory publishing, 1994.
–  Dynamics of the Standard Model, Eugene Golowich, John F. Donoghue, Barry R. Holstein, Cambridge Monographs on Particle Physics.
– Supersymmetry and String Theory: Beyond the Standard Model, Michael Dine, Cambridge University Press, 2007. 

Online lecture notes

– TASI Lectures on Inflation, Daniel Baumann, arXiv:0907.5424.
– Lectures on Inflation, Leonardo Senatore, arXiv:1609.00716.
– Inflationary Perturbations: the Cosmological Schwinger Effect, Jerome Martin, arXiv:0704.3540.
– Inflation and Reheating, Juan Garcia-Bellido, Lectures.
– Inflation and the Theory of Cosmological Perturbations, Antonio Riotto, arXiv:hep-ph/0210162
– Les Houches Lectures on Cosmology and Fundamental Theory, Juan Maldacena, Lectures
– TASI Lectures on Effective Field Theory, Witold Skiba arXiv:1006.2142.
–  Introduction to Effective Field Theory, C. Burgess,   arXiv:hep-th/0701053.

 Selected articles and reviews

–  Non-Gaussian features of primordial fluctuations in single field inflationary models,    J. M. Maldacena, astro-ph/0210603.
–  Semiclassicality and Decoherence of Cosmological Perturbations, David Polarski and Alexei A Starobinsky, gr-qc/9504030.
– Towards the theory of reheating after Inflation, Lev Kofman, Andrei Linde, Alexei Starobinsky, arXiv:hep-ph/9704452.