UNITEXT for Physics Oliver Piattella Lecture Notes in Cosmology UNITEXT for Physics Series editors Paolo Biscari, Dipartimento di Fisica, Politecnico di Milano, Milan, Italy Michele Cini, Dipartimento di Fisica, University of Rome Tor Vergata, Rome, Italy Attilio Ferrari, Università di Torino, Turin, Italy Stefano Forte, Università di Milano, Milan, Italy Morten Hjorth-Jensen, Department of Physics, University of Oslo, Oslo, Norway Nicola Manini, Department of Physics, Università degli Studi di Milano, Milan, Italy Guido Montagna, Università di Pavia, Pavia, Italy Oreste Nicrosini, Dipartimento di Fisica Nucleare e Teorica, Università di Pavia, Pavia, Italy Luca Peliti, Dipartimento di Scienze Fisiche, Università di Napoli, Naples, Italy Alberto Rotondi, Pavia, Italy UNITEXT for Physics series, formerly UNITEXT Collana di Fisica e Astronomia, publishes textbooks and monographs in Physics and Astronomy, mainly in English language, characterized of a didactic style and comprehensiveness. The books published in UNITEXT for Physics series are addressed to graduate and advanced graduate students, but also to scientists and researchers as important resources for their education, knowledge and teaching. More information about this series at http://www.springer.com/series/13351 Oliver Piattella Lecture Notes in Cosmology 123 Oliver Piattella Núcleo Cosmo-UFES and Department of Physics Federal University of Espírito Santo Vitória, Espírito Santo Brazil ISSN 2198-7882 ISSN 2198-7890 (electronic) UNITEXT for Physics ISBN 978-3-319-95569-8 ISBN 978-3-319-95570-4 (eBook) https://doi.org/10.1007/978-3-319-95570-4 Library of Congress Control Number: 2018947829 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. 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This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To Giuseppina, Lilli e Dorotea Together we are gold Preface Considerate la vostra semenza: fatti non foste a viver come bruti ma per seguir virtute e canoscenza (Consider well the seed that gave you birth: you were not made to live as brutes, but to follow virtue and knowledge) Dante Alighieri, Divina Commedia, Canto XXVI These lecture notes are based on the handwritten notes which I prepared for the cosmology course taught to graduate students of PPGFis and PPGCosmo at the Federal University of Esprito Santo (UFES), starting from 2014. This course covers topics ranging from the evidence of the expanding universe to Cosmic Microwave Background anisotropies. In particular, Chap. 1 commences with a bird’s-eye view of cosmology, showing its tremendous evolution during the last fifty years and the open problems on which cosmologists are working today. From Chap. 2 on starts the conventional content, hopefully exposed in a not too conventional manner. The main topics are as follows: the expansion of the universe, relativistic cosmology and Friedmann equations (Chap. 2); thermal history, Big Bang Nucleosynthesis, recombination and cold relic abundance (Chap. 3); cosmo- logical perturbation theory and perturbed Einstein equations (Chap. 4); perturbed Boltzmann equations (Chap. 5); primordial modes of perturbations (Chap. 6); ran- dom variables and stochastic character of cosmological perturbations (Chap. 7); inflationary paradigm (Chap. 8); evolution of perturbations (Chap. 9); anisotropies in the Cosmic Microwave Background sky (Chap. 10). A selection of extra topics is offered in Chap. 11, whereas Chap. 12 collects some important and well-known physical and mathematical results. Scattered throughout the text are many exercises. I chose not to put them at the end of a chapter or at the end of the book because I want the reader to stop and do some work in the moment in which this is needed. Most of the exercises are not extensions of the material covered, but consist in developing calculations necessary to the topics addressed. The idea here is, since these are lecture notes, not only to provide a book for consultation but to put the reader-student to work. vii viii Preface When I prepared these lecture notes, I heavily relied on the following three famous textbooks in cosmology: • S. Dodelson, Modern Cosmology, (Dodelson 2003), • V. Mukhanov, Principles of Physical Cosmology, (Mukhanov 2005), • S. Weinberg, Cosmology, (Weinberg 2008), but also on many other books and papers, which are cited throughout the text. I tried not to simply develop the calculations there contained, but to provide an original presentation. I hope to have succeeded, but of course the final word is up to the reader. I want now to make three recommendations to the reader-student. First, read the original papers in order to make contact with the original ideas in their primeval forms and to appreciate the geniuses of their authors. Second, the technological advance has provided us with sophisticated instru- ments such as Inspire, http://inspirehep.net/, a powerful tool for researching papers. One of the features I like most about it is the possibility to check which papers have cited the one in which we are interested, thus allowing us to rapidly get up to date on a given topic. I sometimes joke about this by rephrasing a sentence which we all have many times read in papers and books when referring to a certain work: see … and references therein. Thanks to the above-mentioned citation tool we can now add also the sentence see … and references thereout. These notes will be up to date when published, but cosmology is a very vivacious research field so they will probably become outdated in a few years. My recommendation is thus to take advantage of the references thereout tools and keep yourself always updated. Third and final one, in preparing these notes I have intensively employed the CLASS code. This is very user-friendly, so it was not as hard as it may sound. My point in doing so is that analytic calculations are very stimulating and very useful in order to understand the physics ruling the cosmological phenomena, so they have an enormous didactical value. On the other hand, observation needs precise cal- culations, and these can be done only numerically. So, my recommendation here is the following: do not be afraid of numerical codes and learn how to use them proficiently. I thank all my students and colleagues who have helped me, through questions and suggestions, in the challenging but rewarding task of writing these notes. Extra thanks to Rodrigo Von Marttens, whose help with CLASS has been crucial in order for me to rapidly grasp how to run the code. Special thanks are due to my editor Aldo Rampioni and his assistant Kirsten Kley-Theunissen for their kind help and encouragement throughout the realisation of the project. Vitória, Brazil Oliver Piattella February 2018 Preface ix References Dodelson, S.: Modern cosmology. Academic Press, Amsterdam, Netherlands (2003) Mukhanov, V.: Physical foundations of cosmology. Cambridge University Press, Cambridge, UK (2005) Weinberg, S.: Cosmology. Oxford University Press, Oxford, UK (2008) Contents 1 Cosmology............................................ 1 1.1 The Expanding Universe and its Content ................. 1 1.1.1 Olbers’s Paradox ............................ 3 1.1.2 The Accelerated Expansion of the Universe and Dark Energy ............................ 4 1.1.3 Dark Matter ................................ 4 1.2 Cosmological Observations ........................... 7 1.2.1 The Cosmic Microwave Background .............. 7 1.2.2 Redshift Surveys ............................ 8 1.2.3 Gravitational Waves Observatories ............... 9 1.2.4 Neutrino Observation ......................... 9 1.2.5 Dark Matter Searches ......................... 10 1.3 Redshift ......................................... 10 1.4 Open Problems in Cosmology ......................... 10 1.4.1 Cosmological Constant and Dark Energy ........... 11 1.4.2 Dark Matter and Small-Scale Anomalies ........... 12 1.4.3 Other Problems ............................. 13 References ............................................ 13 2 The Universe in Expansion ............................... 17 2.1 Newtonian Cosmology .............................. 17 2.2 Relativistic Cosmology .............................. 18 2.2.1 Friedmann–Lemaître–Robertson–Walker Metric ...... 19 2.2.2 The Conformal