Table Of ContentSPRINGER BRIEFS IN PHYSICS
Kaloian Lozanov
Reheating
After
Inflation
123
SpringerBriefs in Physics
Series Editors
BalasubramanianAnanthanarayan,CentreforHighEnergyPhysics,IndianInstitute
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EgorBabaev,PhysicsDepartment,UniversityofMassachusettsAmherst,Amherst,
MA, USA
MalcolmBremer,HHWillsPhysicsLaboratory,UniversityofBristol,Bristol,UK
Xavier Calmet, Department of Physics and Astronomy, University of Sussex,
Brighton, UK
FrancescaDiLodovico,DepartmentofPhysics,QueenMaryUniversityofLondon,
London, UK
Pablo D. Esquinazi, Institute for Experimental Physics II, University of Leipzig,
Leipzig, Germany
Maarten Hoogerland, University of Auckland, Auckland, New Zealand
Eric Le Ru, School of Chemical and Physical Sciences, Victoria University
of Wellington, Kelburn, Wellington, New Zealand
Dario Narducci, University of Milano-Bicocca, Milan, Italy
James Overduin, Towson University, Towson, MD, USA
Vesselin Petkov, Montreal, QC, Canada
Stefan Theisen, Max-Planck-Institut für Gravitationsphysik, Golm, Germany
Charles H.-T. Wang, Department of Physics, The University of Aberdeen,
Aberdeen, UK
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USA
Andrew Whitaker, Department of Physics and Astronomy, Queen’s University
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Kaloian Lozanov
fl
Reheating After In ation
123
Kaloian Lozanov
MaxPlanckInstitute for Astrophysics
Garching,Bayern, Germany
ISSN 2191-5423 ISSN 2191-5431 (electronic)
SpringerBriefs inPhysics
ISBN978-3-030-56809-2 ISBN978-3-030-56810-8 (eBook)
https://doi.org/10.1007/978-3-030-56810-8
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2020
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Acknowledgments
I would like to thank my Ph.D. supervisors, Mustafa Amin, who has taught me
everythingIknowaboutReheating,andAnthonyChallinor,whointroducedmeto
thefieldofCosmology.Withoutthem,thewritingofthesenoteswouldhavenotbeen
possible. I am also immensely grateful to Eiichiro Komatsu for the opportunity to
continuemyresearchintheareaofReheatingandtogiveashortlectureseriesonit.
v
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Inflation and Initial Conditions for Reheating . . . . . . . . . . . . . . . . . 5
2.1 Standard Cosmology, Its Puzzles and Why We Need
Inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Dynamics of Inflation and Setting the Scene for Reheating. . . . . . 12
2.2.1 Homogeneous Dynamics of Inflation . . . . . . . . . . . . . . . . 12
2.2.2 Slow-Roll Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 End of Slow-Roll Inflation. . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.4 Cosmological Perturbations from Inflation. . . . . . . . . . . . . 17
3 Preheating: The Decay of the Inflaton Condensate. . . . . . . . . . . . . . 29
3.1 Perturbative Treatment of Reheating . . . . . . . . . . . . . . . . . . . . . . 30
3.1.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.2 Bose Condensation of Decay Products
in the Perturbative Limit . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Parametric Resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.1 Floquet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Narrow Resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.3 Broad Resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.4 Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Stochastic Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.1 Metric Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.2 Expansion of Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.3 Multi-field Preheating . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Tachyonic Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Instant Preheating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
vii
viii Contents
4 Non-linear Reheating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Back-Reaction: The End of Preheating. . . . . . . . . . . . . . . . . . . . . 50
4.1.1 Back-Reaction at the Background Level . . . . . . . . . . . . . . 50
4.1.2 Re-scattering and Non-linearity. . . . . . . . . . . . . . . . . . . . . 51
4.2 Non-linear Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.1 Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.2 Non-linear Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Turbulent Scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.1 Two Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.2 Perturbative Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.3 Non-perturbative Effects. . . . . . . . . . . . . . . . . . . . . . . . . . 58
5 Reheating and High-Energy Physics Models. . . . . . . . . . . . . . . . . . . 61
5.1 Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4 Non-minimal Couplings to Gravity . . . . . . . . . . . . . . . . . . . . . . . 66
5.5 Non-conventional Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6 Miscellaneous. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6 Observational Implications and Signatures of Reheating . . . . . . . . . 69
6.1 Expansion History of Reheating and the CMB. . . . . . . . . . . . . . . 70
6.2 Relics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2.1 Baryon Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2.2 Magnetic Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3 Metric Fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3.1 Gravitational Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3.2 Non-Gaussianities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Appendix: Gravitational Waves from Preheating.. .... .... ..... .... 83
References.... .... .... .... ..... .... .... .... .... .... ..... .... 85
Notation and Conventions
We use natural units in which 2¼c¼k ¼(cid:1) ¼1. In these units, the reduced
pffiffiffiffiffiffiffiffiffi B 0
Planck mass is given by m ¼1= 8pG.
Pl
Greek indices l, m and so on go over the four space-time coordinates
xl ¼½x0;x1;x2;x3(cid:2)T with x0 for the time coordinate.
Minkowski metric is given by g ¼diag½1;(cid:3)1;(cid:3)1;(cid:3)1(cid:2).
lm
Latin labels i, j, k and so on go over the three spatial coordinates.
Spatial vectors are written in boldface.
Summation over repeated indeces is assumed unless otherwise stated.
The Ricci tensor, defined in terms of the Christoffel symbols, is
R (cid:4)@ Ck (cid:3)@ Ck þCk Cq (cid:3)Cq Ck ; ð1Þ
lm k lm m lk kq lm lk mq
and the Ricci scalar is R¼glmR .
lm
R
The spatial Fourier transform of a field fðxÞ is f ¼ fðxÞe(cid:3)ik(cid:5)xd3x=ð2pÞ3 and the
R k
inverse transform is fðxÞ¼ f eik(cid:5)xd3k.
k
ix
Chapter 1
Introduction
Over the past decades, the understanding of our cosmic history has improved
spectacularly. Precise measurements of the temperature anisotropies in the cosmic
microwave background (CMB) [1, 2] have revealed a homogeneous and isotropic
universe on large scales with tiny (if any) spatial curvature, (cid:2) , and nearly scale-
K
invariant primordial curvature perturbations. The observations provide compelling
evidenceforaninflationaryphaseintheearlyuniverse[3–8],implyingaconnection
betweengravityandquantummechanics.Measurementsoftherelativeabundances
oflight-elementsarealsoinexcellentagreementwithourexpectations[9].Thepre-
dictionsofbig-bangnucleosynthesis(BBN),basedonthewell-understoodphysics
ofnuclearreactions,pointtowardsahotanddenseuniverse,inlocalthermalequilib-
riumatlatetimes.Theytellusthatthefirstlight-elementsstartedformingatcosmic
timet (cid:2)1satequilibriumtemperatureT ∼ E ∼1MeV.
BBN BBN BBN
Connectingthesetworemarkableepochs,however,couldbechallenging.Since
theenergyscaleattheendofinflationcanbeashighas E ∼1016GeV,withthe
inf
durationofinflationcorrespondingto(cid:3)t (cid:2)10−36s,thereisahugerangeofenergy
inf
(andtime)scaleswhichispoorlyunderstoodandpoorlyobservationallyconstrained.
Current cosmological experiments cannot probe the period between inflation and
BBN. This is because typically the effects from the interesting post-inflationary
dynamicsareonsubhorizonscalesduetocausalityandarewashedoutbythelater
non-linearevolutionofstructure.Thethermalstateoftheuniverse,requiredforBBN,
also hides information about earlier times. Collider experiments can shed light on
some intermediate-energy phenomena, e.g., the electro-weak symmetry breaking
and QCD phase transitions, E ∼102GeV and E ∼102MeV, respectively,
EW QCD
buttheywillnotbeabletocovertheentireenergyrangeintheforeseeablefuture.
Nevertheless,itiscrucialthatwetrytounderstandtheperiodbetweeninflation
andBBNbetter,forboththeoreticalandobservationalreasons.Inthestandardlore,
the universe at the end of inflation is cold and dark, virtually empty of particles
anddominatedbytheapproximatelyhomogeneousinflatonfield.Theenergyofthe
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K.Lozanov,ReheatingAfterInflation,
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