LANCS-TH/9720, FERMILAB-PUB-97/292-A, CERN-TH/97-383, OUTP-98-39-P hep-ph/9807278 (After final proof-reading, March 1999) Particle Physics Models of Inflation and the Cosmological Density Perturbation ;1 David H. Lyth† and Antonio Riotto ∗ †Department of Physics, Lancaster University, Lancaster LA1 4YB. U. K. E-mail: [email protected] ∗ CERN, Theory Division, CH-1211, Geneva 23, Switzerland. E-mail: [email protected] Abstract This is a review of particle-theory models of inflation, and of their predictions for the primordial density perturbation that is thought to be the origin of structure in the Uni- verse. It contains mini-reviews of the relevant observational cosmology, of elementary field theory and of supersymmetry, that may be of interest in their own right. The spectral index n(k), specifying the scale-dependence of the spectrum of the curvature perturbation, will be a powerful discriminator between models, when it is measured by Planck with accuracy ∆n 0.01. The usual formula for n is derived, as well as its less familiar extension to the∼ case of a multi-component inflaton; in both cases the key ingredient is the separate evolution of causally disconnected regions of the Universe. Primordial gravitational waves will be an even more powerful discriminator if they are observed, since most models of inflation predict that they are completely negligible. We treat in detail the new wave of models, which are firmly rooted in modern particle the- ory and have supersymmetry as a crucial ingredient. The review is addressed to both astrophysicists and particle physicists, and each section is fairly homogeneous regarding the assumed background knowledge. To appear in Physics Reports 1On leave of absence from Theoretical Physics Department, University of Oxford,U.K. Contents 1 Introduction 2 2 Observing the density perturbation (and gravitational waves?) 8 2.1Theprimordialquantities............................ 9 2.2Theobservablequantities............................ 11 3 The slow-roll paradigm 14 3.1 The slowly rolling inflaton field . .................... 14 3.2Theslow-rollpredictions............................. 16 3.2.1 Thespectrum............................... 16 3.2.2 Thespectralindex............................ 18 3.2.3 Errorestimatesfortheslow-rollpredictions.............. 19 3.3Beyondtheslow-rollprediction......................... 19 3.4 The number of e-foldsofslow-rollinflation................... 21 3.5Gravitationalwaves............................... 23 3.6Beforeobservableinflation............................ 24 4 Calculating the curvature perturbation generated by inflation 25 4.1Thecaseofasingle-componentinflaton.................... 26 4.2Themulti-componentcase............................ 29 4.3Thecurvatureperturbation........................... 30 4.4Calculatingthespectrumandthespectralindex............... 31 4.5 When will becomeconstant?......................... 33 4.6Workingouttheperturbationgeneratedbyslow-rollinflation........R 34 4.7Anisocurvaturedensityperturbation?..................... 35 5 Field theory and the potential 36 5.1Renormalizableversusnon-renormalizabletheories.............. 36 5.2Thelagrangian.................................. 38 5.3Internalsymmetry................................ 41 5.3.1 Continuosanddiscretesymmetries................... 41 5.3.2 Spontaneouslybrokensymmetryandvevs............... 42 5.3.3 Explicitelybrokenglobalsymmetries.................. 44 5.3.4 The restoration of a spontaneously broken internal symmetry . 45 5.4Thetruevacuumandtheinflationaryvacuum................. 45 5.5Supersymmetry.................................. 46 5.6Quantumcorrectionstothepotential...................... 47 5.6.1 GaugecouplingunificationandthePlanckscale........... 48 5.6.2 Theone-loopcorrection......................... 49 5.7Non-perturbativeeffects............................. 52 5.7.1 Condensationanddynamicalsupersymmetrybreaking........ 52 5.7.2 Anon-perturbativecontributiontothepotential........... 52 5.8Flatnessrequirementsonthetree-levelinflationpotential.......... 53 5.9Satisfyingtheflatnessrequirementsinasupersymmetrictheory....... 54 1 5.9.1 Theinflatonamatterfield....................... 55 5.9.2 Theinflatonabulkmodulusorthedilaton.............. 56 6 Forms for the potential; COBE normalizations and predictions for n 56 6.1Single-fieldandhybridinflationmodels.................... 57 6.2Monomialandexponentialpotentials...................... 58 6.3 The paradigm V = V0 + ........................... 59 6.4Theinvertedquadraticpotential........................··· 59 6.5Invertedhigher-orderpotentials......................... 61 6.6Anotherformforthepotential......................... 63 6.7Hybridinflation.................................. 64 6.8Hybridinflationwithaquadraticpotential.................. 64 6.9Massesfromsoftsusybreaking......................... 66 6.10Hybridthermalinflation............................. 67 6.11Invertedhybridinflation............................. 68 6.12Hybridinflationwithacubicorhigherpotential............... 68 6.13Mutatedhybridinflation............................. 69 6.14Hybridinflationfromdynamicalsupersymmetrybreaking.......... 71 6.15 Hybrid inflation with a loop correction from spontaneous susy breaking . 72 6.16 Hybrid inflation with a running mass . .................... 74 6.16.1Generalformulas............................. 74 6.16.2Thefourmodels............................. 76 6.16.3Observationalconstraints........................ 79 6.17Thespectralindexasadiscriminator...................... 79 7 Supersymmetry 80 7.1Introduction.................................... 80 7.2Themotivationforsupersymmetry....................... 81 7.3Thesusyalgebraandsupermultiplets..................... 81 7.4Thelagrangianofglobalsupersymmetry.................... 83 7.5Spontaneouslybrokenglobalsusy........................ 86 7.5.1 The F and D terms........................... 86 7.5.2 Tree-level spontaneous susy breaking with an F term........ 87 7.5.3 Dynamicallygeneratedsuperpotentials................. 87 7.5.4 Quantummodulispaces......................... 88 7.6Softsusybreaking................................ 89 7.6.1 Soft susy breaking from a D term................... 90 7.6.2 Gauge-mediatedsusybreaking..................... 91 7.7 Loop corrections and running . .................... 93 7.7.1 One-loopcorrections........................... 94 7.7.2 TheRenormalizationGroupEquations(RGE’s)........... 95 7.8Supergravity................................... 95 7.8.1 Specifyingasupergravitytheory.................... 96 7.8.2 The scalar potential and spontaneously broken supergravity . 97 7.9Supergravityfromstringtheory......................... 99 2 7.9.1 A single modulus t ............................ 99 7.9.2 Three moduli tI ............................. 100 7.9.3 Thedilaton................................ 101 7.9.4 Horava-Witten M-theory . .................... 102 7.10Gravity-mediatedsoftsusybreaking...................... 102 7.10.1Generalfeatures............................. 103 7.10.2Gravity-mediatedsusybreakingfromstringtheory.......... 104 7.10.3Formalismforgravity-mediatedsupersymmetrybreaking...... 105 8 F-term inflation 107 8.1Preservingtheflatdirectionsofglobalsusy.................. 107 8.2 The generic F -termcontributiontotheinflatonpotential.......... 107 8.2.1 Theinflatonmass............................ 108 8.2.2 Thequarticcouplingandnon-renormalizableterms.......... 108 8.3Preservingflatdirectionsinstringtheory................... 109 8.3.1 Arecipeforpreservingflatdirections................. 109 8.3.2 Preservingtheflatnessinweaklycoupledstringtheory........ 110 8.3.3 Caseofalinearsuperpotential..................... 111 8.3.4 Generating the F term from a Fayet-Iliopoulos D-term....... 111 8.3.5 Simpleglobalsusymodelsofinflation................. 113 8.4Modelswiththesuperpotentiallinearintheinflaton............. 114 8.5Amodelwithgauge-mediatedsusybreaking.................. 115 8.6 The running inflaton mass model revisited . ............. 116 8.6.1 Thebasicscenario............................ 116 8.6.2 Directionsformodel-building...................... 117 8.6.3 Running with a gauge coupling . .................... 117 8.7AvariantoftheNMSSM............................ 119 9 D-term inflation 121 9.1Keepingthepotentialflat............................ 122 9.2Thebasicmodel................................. 122 9.3Constructingaworkablemodelfromstringtheory.............. 125 9.4 D-terminflationandcosmicstrings....................... 132 9.5 A GUT model of D-terminflation....................... 133 10 Conclusion 136 3 1 Introduction We do not know the history of the observable Universe before the epoch of nucleosynthesis, but it is widely believed that there was an early era of cosmological inflation [202, 176, 194, 195]. During this era, the Universe was filled with a homogeneous scalar field φ, called the inflaton field, and essentially nothing else. The potential V (φ) dominated the energy density of the Universe, decreasing slowly with time as φ rolled slowly down the slope of V . The attraction of this paradigm is that it can set the initial conditions for the subsequent hot big bang, which otherwise have to be imposed by hand. One of these is that there be no unwanted relics (particles or topological defects which survive to the present and contradict observation). Another is that the initial density parameter should have the value Ω = 1 to very high accuracy, to ensure that its present value has at least roughly this value. There is also the requirement that the Universe be homogeneous and isotropic to high accuracy. All of