Master's Thesis in Theoretical Physics - Universiteit Utrecht

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and particle physics. The Einstein equations that follow from Einstein’s theory of generalrelativity allow for a solution that leads to an accelerated expansion of our universe. A specialtype of matter with negative pressure is needed for this particular solution, and Guthrealized that this is possible within certain scalar field models. The inflationary hypothesiscould solve many of the problems that cosmologists faced at the time, and it lead to arevolution in cosmology. Today there are literally hundreds of inflationary models and newmodels are being build all the time. Many of these models involve exotic new particles andare in that sense not very appealing.In this thesis our main focus is on a particular inflationary model, which we will call nonminimalinflation. In this model a scalar field is coupled to gravity which can effectivelyreduce Newton’s gravitational constant and lead to inflation. The main advantage of thismodel is that in the limit of strong coupling the only scalar particle in the Standard Model,the Higgs boson, is an appropriate candidate for the inflaton, the scalar field that drivesinflation. Since there is no need to introduce any exotic new particles, the beauty of thismodel is evident.The outline of this thesis is the following. In chapter 2 we will introduce some basic cosmology.Then in chapter 3 we outline the basics of cosmological inflation. In chapter 4 we willintroduce the nonminimal inflationary model and we will see that inflation is successful ifthe scalar field is nonminimally coupled to gravity. Furthermore we will see that observationsof the spectrum of density perturbations constrain the quartic self-coupling of theinflaton field in the minimal coupling case to a tiny value, but the constraint can be relaxedby several orders of magnitude if the nonminimal coupling is sufficiently strong. This allowsthe Higgs boson to be the inflaton, proposed by Bezrukov and Shaposhnikov in [4]. We alsointroduce the two-Higgs doublet model in chapter 4, and show that nonminimal inflationworks in this model as well. The extra degrees of freedom and couplings in the two-Higgsdoublet model allow for CP violation and this might lead to baryogenesis.Chapter 5 is all about combining quantum field theory and general relativity. Although itis well known that quantum gravity is a non-renormalizable theory, quantum gravitationaleffects become important only at energies above the reduced Planck scale of M P = 2.4×10 18GeV where the universe was so small that quantum mechanics and gravity become equallyimportant. At lower energies we do not see these effects and we can ’just’ do quantum fieldtheory on a curved background. Quantization of the nonminimally coupled inflaton field inan expanding universe is however not so trivial. We will see that there is no unique vacuumchoice and that in general particle number is not well defined in an expanding universe,which we also mentioned above in the heuristic picture of particle creation. Fortunatelywe will see that we can make a natural vacuum choice during inflation, the Bunch-Daviesvacuum with zero particles. With this specific vacuum we can uniquely quantize the nonminimallycoupled inflaton field and do quantum field theory. We calculate the propagatorfor the scalar inflaton field and eventually extend our results to calculate the propagator forthe two-Higgs doublet model in quasi de Sitter space.In chapter 6 we will calculate the effect of the Higgs boson as the inflaton on the StandardModel fermions. The Higgs boson interacts with the fermions and gives a mass tothe fermions through the Higgs mechanism. As a final application we use the masslessfermion propagator and the propagator for the two-Higgs doublet model in quasi de Sitterspace to calculate the one-loop self-energy for the massless fermions. This one-loop fermionself-energy effectively generates a mass for the massless fermions.
Chapter 2Cosmology2.1 The Cosmological Principle and Einstein’s equationsMost cosmological models are based on the assumption that the universe is the same everywhere.This might seem strange to us, since we clearly see other planets and the sunaround us, with empty space everywhere in between. However, on the largest scales, whenwe average over all the local density fluctuations, the universe is the same everywhere.This statement is formulated more precisely by the cosmological principle, which statesthat the universe is isotropic and homogeneous on the largest scales. Isotropy means thatno matter what direction you look, the universe looks the same. The observation of the cosmicmicrowave background radiation (CMBR) supports the idea of isotropy. The COBE andWMAP missions found that the deviations in the CMBR from a perfect isotropic universeare of the order of 10 −5 [5, 2].Homogeneity is the idea that the metric is the same on every point in space. Stated otherwise,when we look at the universe from our planet earth and do the same on a planet 1billion lightyears away, we should still observe the same universe. Again, the same universemeans that the universe looks the same on the largest scales. Since we have been assumingfor quite a while that our planet is not the center of the universe, we should also observe anisotropic universe anywhere else.Thus, cosmologists describe the homogeneous and isotropic universe on the largest scales.Since the universe is electrically neutral on these scales, the infinite ranged electromagneticforce does not play a role. Therefore, the dynamics of the universe are determined bythe gravitational interaction, described by the Einstein equationsG µν = 8πG N T µν , (2.1)where G µν is defined in Eq. (B.10), G N is Newton’s constant and T µν is the energy momentumor stress-energy tensor. This tensor describes the influence of matter on the dynamicsof the universe. For the isotropic and homogeneous universe we can describe the matter andenergy content as a perfect fluid, a fluid which is isotropic in its rest frame. For a perfectfluid the energy momentum tensor takes the formT µν = (ρ + p)u µ u ν + pg µν , (2.2)where ρ and p are the energy density and pressure in the rest frame and u µ is the four velocityof the fluid. Choosing our frame to be the rest frame of the fluid where u µ = (1,0,0,0),we find that the energy momentum tensor isT µν = (ρ + p)δ 0 µ δ0 ν + pg µν. (2.3)3
Page 1: INFLATION AND BARYOGENESIS THROUGH
Page 5: AbstractIn this thesis we consider
Page 8 and 9: 6 The fermion sector 716.1 Fermion
Page 12 and 13: 2.2 An expanding universeWe are now
Page 14 and 15: Here, H 0 = H(t = 0) and we have ch
Page 16 and 17: Since the spatial background is hom
Page 18 and 19: into neutral hydrogen at a temperat
Page 20 and 21: horizon during inflation isl phys (
Page 22 and 23: VΦSlowrollregime0 MpΦFigure 3.1:
Page 24 and 25: wavelengths that are larger than th
Page 26 and 27: In the Standard Model, the only sca
Page 28 and 29: esearch of density perturbations in
Page 30 and 31: Finally, we can also vary the actio
Page 32 and 33: such that we get for Eqs. (4.18)Fro
Page 34 and 35: Φt252015105N8060402000 200 400 600
Page 36 and 37: 4.2 The conformal transformationThe
Page 38 and 39: Eq. (4.44) is invariant under a con
Page 40 and 41: Note that we have succeeded in remo
Page 42 and 43: Figure 4.6: WMAP measurements of th
Page 44 and 45: Note that the matrices Ω and ˜Ξ
Page 46 and 47: and we can writeξ ++ =ξ −− =
Page 48 and 49: value of ξ. With the explicit expr
Page 50 and 51: 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
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5.2 Quantization of the nonminimall
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If we now substitute the expansion
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the adiabatic regime, whereas the t
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the lowest energy state at a later
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wherey ≡ y ++ (x, x ′ ) = −(|
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Now we plug all these expansions in
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of the field and expand the action
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whereΦ k =( ) ( ) ( ) ( )φk,aa φ
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This again exactly the same express
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Chapter 6The fermion sectorIn the p
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In this representation the projecti
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These relations allow us to rewrite
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6.2.1 Fermion propagator in curved
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We now again rescale the field χL/
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φψψFigure 6.1: Feynman diagram f
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Note that the fact that the scalar
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The next step is to get rid of the
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such that we can findχ(x ′ ) = e
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where in all the equations z = k∆
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We can now compare this equation to
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Chapter 7Summary and outlookCosmolo
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esulting action as the action of a
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In future work we hope to study som
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Appendix AConventions• Throughout
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Finally, we use the Bianchi identit
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Note that we could also have calcul
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[24] A. O. Barvinsky, A. Y. Kamensh
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Master's Thesis in Theoretical Physics - Universiteit Utrecht

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