Master's Thesis in Theoretical Physics - Universiteit Utrecht

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VΦSlowrollregime0 MpΦFigure 3.1: Chaotic inflationary potential. The field has an initial value of O(M P ) and slowly rolls down thepotential well.a whole move to the true vacuum state. In order for this not to happen almost instantaneously,the potential needs to be very flat. Moreover, the initial conditions of the field mustbe φ 0 ≃ 0, ˙φ0 ≃ 0 and the field must slowly roll down the potential well. This leads to theso-called slow-roll conditions, which we will discuss in the next section. The flatness of thepotential and the initial conditions for the field form the naturalness and fine-tuning problemsfor the new inflationary scenario.One of the simplest scenario for inflation is chaotic inflation, proposed by Linde[8]. Theidea of chaotic inflation is shown in Fig. 3.1. In this scenario, the universe was initiallyin a chaotic state where the value of the scalar field could take any value. This happenedin the Planck era where quantum effects dominate the universe. In particular, the valueof the field could be very large (i.e O(M P )). Below the Planck era quantum effects becomesubdominant and the scalar field behaves classically. This means that the field slowly rollsdown the potential well and inflation is a success. The chaotic inflationary scenario allowsfor many forms of the potential, in particular a m 2 φ 2 or a λφ 4 potential.3.5 Inflationary dynamicsIn this section we will show more explicitly how inflation could happen in scalar field models.Assume again the simple scalar field action from Eq. (2.33) and the energy density andpressure from Eqs. (2.35). The scalar field we will now call the inflaton, the field that drivesinflation. Using these expressions we find that for successful inflation we need˙φ 2 ≪ V (φ). (3.14)In chaotic inflationary models the inflaton field starts at some large initial value and rollsdown the potential well. In most of these models the potential energy is much greater than1the kinetic energy, i.e. ˙φ 2 2≪ V (φ). Also, it turns out that in many models the inflatonslowly rolls down the potential well. Taking the scalar field equation (2.36) into account,this means that ¨φ ≪ 3H ˙φ. The equation for H 2 and the field equation the becomeH 2 ≃ V (φ)3M 2 p(3.15)3H ˙φ ≃ −V ′ (φ). (3.16)
In general, we can define the so-called slow-roll parameters ɛ and η byɛ = 1 2 M2 p( V′) 2(3.17)V ′′Vη = M 2 pV , (3.18)where V ′ ≡ dV /dφ and the Planck mass was defined in Eq. (3.1). The slow-roll conditionsare thenɛ ≪ 1, η ≪ 1. (3.19)The qualitative argument for these slow-roll conditions is the following: in order for inflationto take place the potential energy must dominate the kinetic energy. This means thatin a chaotic inflationary scenario (see Fig. 3.1) the scalar field should not roll down thepotential well ’too fast’. In order to achieve this, the slope of the potential well should not betoo steep. Since the slow-roll parameters in Eq. (3.17) and (3.18) contain derivatives of thepotential, they therefore tell us something about the steepness of the slope. The slow-rollconditions (3.19) are such that the slope is not too steep.Note the two different definitions for ɛ in Eq. (3.17) and (2.10). One can show that theseare equal in the slow-roll approximation. Take the time derivative of Eq. (3.15) to get anequation for Ḣ, eliminate the ˙φ term by substituting Eq. (3.16) and divide by H 2 . You willrecover Eq. (3.17).In a simple inflationary scenario with a real scalar field in a quartic potential V (φ) = λφ 4 , wecan easily see that the slow-roll conditions (3.19) are satisfied when the fields φ ∼ O(10M P ).In chaotic inflationary scenarios this is a typical initial value for the field φ.3.6 Constraints on inflationary modelsSo far we have only considered characteristics of general inflationary models. We havehowever not yet discussed constraints on inflationary models. Although inflation is a hypotheticalera that happened in the very early universe, we fortunately can constrain inflationarymodels. These constraints follow from cosmological perturbation theory, see forexample Refs. [9][10][11]. The idea is that in the early universe there are small quantumfluctuations of the scalar field and metric. During inflation some of these fluctuations arefrozen in and inflation actually amplifies the vacuum fluctuations. The scalar metric perturbationsare responsible for the large scale structure formation in the universe and leavetheir imprint on the spectrum of the CMBR. This effect can be measured and constrains theinflationary model. Let us sketch in a few steps how this works.Consider a simple free scalar field theory with field equation−□φ = ¨φ + 3H ˙φ −∇ 2a 2 φ = 0.Suppose now that the spatial derivative term is large compared to the damping term withH, so we can neglect the second term. We now see that we have a wave equation with planewave solutions. If we see the field φ as a small quantum fluctuation and Fourier transformthe field, we see that for sub-Hubble wavelengths with aH ≪ k the Fourier-transformedfield φ k obeys a harmonic oscillator equation and fluctuates.If the spatial derivative terms are small such that we can neglect the third term, we findthat the term 3H ˙φ leads to damping of φ with a characteristic decay rate proportional toH. This means that ˙φ ∝ Hφ, such that we find that H 2 ≫ k2a 2 in this limit. These are
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 10 and 11: and particle physics. The Einstein
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 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
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
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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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