Table Of ContentFundamental Theories of Physics 197
Horaţiu Năstase
Cosmology
and String
Theory
Fundamental Theories of Physics
Volume 197
Series Editors
Henk van Beijeren, Utrecht, The Netherlands
Philippe Blanchard, Bielefeld, Germany
Bob Coecke, Oxford, UK
Dennis Dieks, Utrecht, The Netherlands
Bianca Dittrich, Waterloo, ON, Canada
Detlef Dürr, Munich, Germany
Ruth Durrer, Geneva, Switzerland
Roman Frigg, London, UK
Christopher Fuchs, Boston, MA, USA
Domenico J. W. Giulini, Hanover, Germany
Gregg Jaeger, Boston, MA, USA
Claus Kiefer, Cologne, Germany
Nicolaas P. Landsman, Nijmegen, The Netherlands
Christian Maes, Leuven, Belgium
Mio Murao, Tokyo, Japan
Hermann Nicolai, Potsdam, Germany
Vesselin Petkov, Montreal, QC, Canada
Laura Ruetsche, Ann Arbor, MI, USA
Mairi Sakellariadou, London, UK
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Rainer Verch, Leipzig, Germany
Reinhard F. Werner, Hanover, Germany
Christian Wüthrich, Geneva, Switzerland
Lai-Sang Young, New York City, NY, USA
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ţ ă
Hora iu N stase
Cosmology and String
Theory
123
HoraţiuNăstase
Instituto deFísica Teórica,UNESP
São Paulo, Brazil
ISSN 0168-1222 ISSN 2365-6425 (electronic)
FundamentalTheories of Physics
ISBN978-3-030-15076-1 ISBN978-3-030-15077-8 (eBook)
https://doi.org/10.1007/978-3-030-15077-8
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To the memory of my mother,
who inspired me to become a physicist
Preface
This book deals with the intersection between string theory and cosmology, the
methods to obtain cosmology from string theory, i.e., “string cosmology.” While
therehavebeenmanydevelopmentsinthefieldwithinthelast15–20years,tomy
knowledge, there is no comprehensive review or book dedicated to all the lines of
researchinthefield.Thisisimportant,sincestringcosmologyisadevelopingfield,
anditisnotyetclearwhichlineofresearchwilleventuallybecomestandard.Since
there are people in both string theory and cosmology interested in the subject, as
wellasgraduatestudentsthathaveabasicknowledgeofboth,thebookintroduces
both general cosmology and string theory, before describing string cosmology
proper.
ThebookisbasedonagraduatecourseItaughtatmyinstitute,theInstitutefor
Theoretical Physics of UNESP, and for which the students indeed had this back-
ground:abitofcosmology,andabitofstringtheorybackground,butnotproficient
in either. I have added only 3 lectures at the end, denoted with an asterisk, which
correspondtomorerecent(butnolessimportant)developmentsinstringcosmology.
Thebookpreservesotherwisethecourseformat,withonelecturecorrespondingto
onechapter,andhavingalistof“importantpointstoremember”attheendofeach
chapter,aswellasexercises.Researchersthatwanttolearnaboutthesubjectarealso
an intended audience for this book, as it presents most approaches that have some
successso far in describing cosmology, so they can use it asa reference.
There are no comparable books out there, to my knowledge. There is an
excellentbookonstringinflationbyBaumannandMcAllister,InflationandString
Theory,butwhileithasagoodpresentation,itfocusesonlyoninflationaryaspects.
In terms of the introduction to cosmology, there is information exclusively about
inflation, assuming general cosmological knowledge, and the string theory intro-
ductionisveryquick.TheotherbookIamawareof,Gasperini’sStringCosmology,
deals with only some specific topics, mostly about inflation, and has only a small
introduction to inflation, being meant for advanced researchers.
São Paulo, Brazil Horaţiu Nǎstase
vii
Acknowledgements
I want to thank all the people who shaped me as a physicist throughout the years.
Firstandforemost,mymotherLigia,thefirstphysicistIknew,whoshowedmethe
wondersofphysics.Myhighschoolteacher,IosifSeverGeorgescu,putmeonthe
career path to physics and showed me that it is something I can do. My exchange
student advisor at the Niels Bohr Institute, Poul Olesen, introduced me to string
theory, my chosen field and the basis for this book. My Ph.D. advisor, Peter van
Nieuwenhuizen, taught me, besides supergravity and string theory, how to be a
completetheoreticalphysicistandtheimportanceofrigorandprecision.FromJuan
Maldacena, I learned a lot about the AdS/CFT correspondence, which forms the
basis for some of the research described here, as well as a lot about string theory
and cosmology.
ThisbookisbasedonacourseItaughtatourinstitute,IFTatUNESP;soIthank
allofthestudentswhoparticipatedinthecourseandhelpedmewiththeirquestions
and input.
I would like to thank all of my collaborations throughout the years, and in
particularthosewithwhomIworkedonanddiscussedresearchrelatedtothelines
describedinthisbook:JustinKhoury,AmandaWeltman,RogerioRosenfeld,Kurt
Hinterbichler, Kostas Skenderis, Robert Brandenberger. I want to thank my wife
Antonia for her understanding and encouragement while I worked on the book, in
the evenings at home. My students and postdocs, for my having less time to work
withthemwhileIworkedonthisbook.Finally,tomyeditorAngelaLaheeforher
encouragement and getting this book published.
ix
Contents
Part I Standard Cosmology
1 The Expanding Universe and the Big Bang . . . . . . . . . . . . . . . . . . 3
1.1 Observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Newtonian Approximation for the Expansion of Space . . . . . . . 5
1.3 Time Evolution of the Scale Factor for a Linear Equation
of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Universe with Several Matter Components. . . . . . . . . . . . . . . . 10
1.5 A Quick History of Experimental Observations and
Discoveries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 The Cosmological Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Relativistic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 The FLRW Metric and the Friedmann Equation . . . . . . . . . . . . 15
2.2 Types of Matter and Resulting Expansion of the Universe . . . . 19
2.3 Cosmological Evolution Formulas . . . . . . . . . . . . . . . . . . . . . . 21
3 The Propagation of Light and Measurements of Distance,
Luminosity and Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Propagation of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Measurements of Distance by Light. . . . . . . . . . . . . . . . . . . . . 29
3.3 Optical Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Stellar Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Measurements of Distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Mass Determination for Distant Bodies . . . . . . . . . . . . . . . . . . 37
4 Evidence for Dark Matter and the KCDM Model . . . . . . . . . . . . . 41
4.1 Evidences for Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Candidates for Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 WIMP Dark Matter Candidates . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 The KCDM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
xi
xii Contents
5 The Early Universe and Its Thermal History . . . . . . . . . . . . . . . . . 53
5.1 Defining the Hot Big Bang . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Equilibrium Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Thermodynamics of the Universe. . . . . . . . . . . . . . . . . . . . . . . 58
5.4 Particle Species in the Radiation Dominated (R.D). Era . . . . . . 61
5.5 Neutrino Decoupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 Big Bang Nucleosynthesis and Recombination . . . . . . . . . . . . . . . . 67
6.1 Recombination Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Big Bang Nucleosynthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Dependence of BBN Primordial Abundances
on Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.4 Observations and Comparison with BBN . . . . . . . . . . . . . . . . . 73
6.5 Summary of the Thermal History. . . . . . . . . . . . . . . . . . . . . . . 74
7 The Cosmic Microwave Background Radiation (CMBR)
Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.1 The CMBR and Its Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2 Kinematics of the CMBR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.3 Perturbation Theory for CMBR . . . . . . . . . . . . . . . . . . . . . . . . 83
7.4 Effect of Density Perturbations on the CMBR . . . . . . . . . . . . . 89
8 Problems to Be Solved by Inflation and How They Are Solved
in Inflationary Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.1 Problems with Standard Cosmology Before Inflation. . . . . . . . . 95
8.2 The Paradigm of Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.3 Inflationary Solutions to Cosmological Problems . . . . . . . . . . . 101
8.4 Inflation with a Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . 106
9 Slow-Roll Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9.1 Slow-Roll Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
9.2 General (Non-slow Roll) Formulas . . . . . . . . . . . . . . . . . . . . . 115
9.3 Slow Roll Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.4 Exact Solution for Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
9.5 Variants of Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
10 Reheating and Baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
10.1 Standard Reheating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
10.2 New Reheating and Preheating . . . . . . . . . . . . . . . . . . . . . . . . 127
10.3 Entropy Production and the Number of e-Folds . . . . . . . . . . . . 129
10.4 Baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
10.5 Relics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
