Master's Thesis in Theoretical Physics - Universiteit Utrecht

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drives inflation is called the inflaton.The first inflationary models suffered from some problems, such as the graceful exit andfine-tuning problems. The most accepted scenario is the chaotic inflationary model, wherea scalar field initially has a very large value and then slowly rolls down the potential wellto its minimum. We can define the slow-roll conditions that must generically be satisfiedin order for successful inflation to take place. These slow-roll conditions are determined bythe potential of the scalar field. So far there have not been any direct measurements thatconfirm whether or not inflation took place in the early universe, and we therefore also donot know the form of the scalar potential. This is also a hard nut to crack, since inflationconcerns the expansion of space and the only thing we can observe are photons and particles,which have in principle not much to do with inflation. However, cosmological inflationdoes make some predictions about our universe. One of the most important is that inflationpredicts a nearly scale invariant spectrum of density perturbations. This has actually beenmeasured from the CMBR and many cosmologists consider this to be the confirmation ofthe inflationary hypothesis. The deviations from a scale invariant spectrum can actuallybe expressed in terms of the slow-roll parameters, and since these parameters are derivedfrom the inflationary potential, measurements of the spectrum constrain the form of theinflaton potential. Inflation also predicts primordial gravitational waves that leave theirimprint on the CMBR, and cosmologists hope to see this very soon with the new Plancksatellite that was launched in May 2009. The question remains what the correct inflationarymodel really is. Chaotic inflation is possible in most scalar field models, and this is onthe one hand nice because the most general models are allowed, but on the other hand itleaves room for many exotic models.In this thesis we considered a specific type of inflationary models, the nonminimal inflationarymodels. In these models the scalar inflaton field is coupled to gravity, which in theaction formalism corresponds to the Ricci scalar R in the Einstein-Hilbert action. The termnonminimal coupling term ξRφ 2 effectively changes the gravitational constant G N suchthat it can become much smaller. On the other hand it acts as an additional mass termin the potential. We considered specifically nonminimal inflationary models with strongnegative coupling, ξ ≪ −1. In section 4.1 we considered the simplest model of a real scalarfield φ with a quartic potential with self-coupling λ that is nonminimally coupled to R. Wederived the dynamical equations for H and for the field φ and solved these analytically inthe slow-roll approximation and in the strong negative coupling limit. We saw that the fieldrolls down slowly to its minimum and that the large negative ξ reduces the initial valueof φ necessary for successful inflation, i.e. to solve the cosmological puzzles. The Hubbleparameter H has the special property that during inflation it is proportional to the fieldφ. Numerically we have solved the complete field equations and found that our analyticalresults are verified to great accuracy.Besides providing a successful inflationary model, the nonminimally coupled inflaton fieldhas the additional benefit that it lowers the constraint on the quartic self-coupling λ. Asmentioned, the measurements of the spectrum of density perturbations put constraints onthe slow-roll parameters, and since these are derived from the inflaton potential they thereforeconstrain the form of the potential. For minimally coupled models, the spectrum of densityperturbation is proportional to λ and the value of the self-coupling must therefore beλ ≃ 10 −12 . However, if we include a nonminimal coupling term, we find that the spectrumis proportional to √ λ/ξ 2 . In order to derive this, we first needed to perform a conformaltransformation of the metric that removes the nonminimal coupling term. We introducedthe general conformal transformation in section 4.2 and performed the specific conformaltransformation that removes the ξRφ 2 term in section ??. We found that we could write the
esulting action as the action of a minimally coupled scalar field with a modified potential.This new frame after the conformal transformation is called the Einstein frame. In the Einsteinframe we could easily derive the slow-roll parameters from the potential and foundthe proportionality of the spectrum of density perturbations to √ λ/ξ 2 . A strong negative ξtherefore reduces the constraint on λ, and in fact for large enough ξ we found that λ canhave a reasonable value of ∼ 10 −3 .The fact that λ does not have to be extraordinary small allows the Higgs boson to be theinflaton. The great advantage is that we do not need to introduce any additional exoticscalar particles that drive inflation, but can use the Higgs boson that is already an essentialpart of the Standard Model of particles. The only thing we have to add is an extra termin the action with the nonminimal coupling of the Higgs boson to R. We showed that in theEinstein frame the potential reduces to the ordinary Higgs potential for small values of theHiggs inflaton field φ. However, for large field values the potential becomes asymptoticallyflat and this makes sure that inflation is successful.In section 4.4 we introduced the two-Higgs doublet model. This is the simplest supersymmetricextension of the Standard Model and features a second Higgs doublet. In the unitarygauge there is in addition to a second real scalar field also a phase between the two fields.This phase can lead to CP violation through an axial vector current that could then beconverted in a baryonic current that is responsible for baryogenesis. We derived again thefield equations and solved these numerically for real Higgs fields. We found that one linearcombination of the two real scalar fields serves as the inflaton, whereas the other isan oscillating mode that quickly decays. Furthermore because there are more nonminimalcouplings and quartic self-coupling the initial value of the fields can be lowered even further.In chapter 5 we quantized our nonminimally coupled inflaton field. It is well known thatwe cannot simply combine quantum field theory and general relativity in a single theoryof quantum gravity because the theory is not renormalizable. However, for energies wellbelow the Planck scale of M P = 2.4 × 10 18 GeV we expect quantum gravitational effects notto play a role and we have no problem to do quantum field theory in a curved background.First we gave a useful derivation of the quantization of a scalar field in Minkowski spacein section 5.1. Although we showed that there is are infinite ways to quantize our fieldand therefore an infinite amount of choices for the vacuum, we found that there is only oneunique vacuum that is also the zero particle and lowest energy state. The reason is that theHamiltonian is constant in time. In an expanding universe the Hamiltonian is not constantbecause of the time dependent scale factor in the metric and therefore the choice of vacuumis not unique. By considering a simple example of a harmonic oscillator with a varyingfrequency with asymptotically constant regions we showed that the zero particle vacuumat one point in time is not the zero particle vacuum at a later time. Physically, particles arecreated by the expansion of the universe.Since it is very difficult, if not impossible to do quantum field theory if the vacuum andtherefore also the notion of a particle is not well-defined, we needed to find a natural choicefor the vacuum. First we quantized the nonminimally coupled inflaton field in section 5.2by considering the inflaton field as a the classical inflaton field that drives inflation plus asmall quantum fluctuation. In an expanding universe described by the FLRW metric wesaw that we could write the field equation for the fluctuations as a harmonic oscillator witha time dependent frequency. In the special case of quasi de Sitter space however thereare regions where the frequency of the harmonic oscillator is approximately constant andchanges slowly. In these adiabatic regions we can define a natural vacuum state knownas the Bunch-Davies vacuum that corresponds to zero particles and energy in the infinite
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INFLATION AND BARYOGENESIS THROUGH
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AbstractIn this thesis we consider
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6 The fermion sector 716.1 Fermion
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and particle physics. The Einstein
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2.2 An expanding universeWe are now
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Here, H 0 = H(t = 0) and we have ch
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Since the spatial background is hom
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into neutral hydrogen at a temperat
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horizon during inflation isl phys (
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VΦSlowrollregime0 MpΦFigure 3.1:
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wavelengths that are larger than th
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In the Standard Model, the only sca
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esearch of density perturbations in
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Finally, we can also vary the actio
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such that we get for Eqs. (4.18)Fro
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Φt252015105N8060402000 200 400 600
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4.2 The conformal transformationThe
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Eq. (4.44) is invariant under a con
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Note that we have succeeded in remo
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Figure 4.6: WMAP measurements of th
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Note that the matrices Ω and ˜Ξ
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and we can writeξ ++ =ξ −− =
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value of ξ. With the explicit expr
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0.600.800.550.75Φ1t0.50Φ2t0.700.4
Page 53 and 54: Chapter 5Quantum field theory in an
Page 55 and 56: with equation of motion0 = ¨φk +
Page 57 and 58: Eq. (5.18). If we now act with the
Page 59 and 60: (5.28) with α = 1 and β = 0, and
Page 61 and 62: 5.2 Quantization of the nonminimall
Page 63 and 64: If we now substitute the expansion
Page 65 and 66: the adiabatic regime, whereas the t
Page 67 and 68: the lowest energy state at a later
Page 69 and 70: wherey ≡ y ++ (x, x ′ ) = −(|
Page 71 and 72: Now we plug all these expansions in
Page 73 and 74: of the field and expand the action
Page 75 and 76: whereΦ k =( ) ( ) ( ) ( )φk,aa φ
Page 77 and 78: This again exactly the same express
Page 79 and 80: Chapter 6The fermion sectorIn the p
Page 81 and 82: In this representation the projecti
Page 83 and 84: These relations allow us to rewrite
Page 85 and 86: 6.2.1 Fermion propagator in curved
Page 88 and 89: We now again rescale the field χL/
Page 90 and 91: φψψFigure 6.1: Feynman diagram f
Page 92 and 93: Note that the fact that the scalar
Page 94 and 95: The next step is to get rid of the
Page 96 and 97: such that we can findχ(x ′ ) = e
Page 98 and 99: where in all the equations z = k∆
Page 100 and 101: We can now compare this equation to
Page 103: Chapter 7Summary and outlookCosmolo
Page 107: In future work we hope to study som
Page 111: Appendix AConventions• Throughout
Page 114 and 115: Finally, we use the Bianchi identit
Page 116 and 117: Note that we could also have calcul
Page 118: [24] A. O. Barvinsky, A. Y. Kamensh
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Master's Thesis in Theoretical Physics - Universiteit Utrecht

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