Master's Thesis in Theoretical Physics - Universiteit Utrecht

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wavelengths that are larger than the Hubble radius and are therefore called super-Hubblewavelengths. The solution of φ k in this limit is a constant plus a decaying mode. In otherwords, on super-Hubble scales with aH ≫ k the amplitude of field φ k is approximatelyfrozen in.During inflation the physical length scale increases due to the expansion of the universe, butthe Hubble radius remains almost constant. This means that quantum fluctuations withsub-Hubble wavelengths can become super-Hubble during inflation and their amplitudesare frozen in. In the Minkowski vacuum the amplitude of vacuum fluctuations scales as1l, where l is the physical length scale. An expansion of space implies that l increases andthat therefore the amplitude decreases. However, for super-Hubble modes the amplitude isfrozen in during inflation and therefore the amplitude is effectively amplified with respectto the Minkowski vacuum. The scalar perturbations form gravitational potential wells thatare the origin of the formation of large-scale structure. A measure for the amplitude ofscalar perturbations then is〈φ k φ k ′〉 = 1 k 3 P s(k)2π 3 δ 3 (k − k ′ ),where P s (k) is the spectrum of scalar perturbations. It is important that P s (k) is evaluatedat the first Hubble-crossing, which is the time that the quantum fluctuations become super-Hubble, so when k = aH. In a thorough calculation it can be shown thatP φ (k) = ∆ 2 Rwhere n s is the called the spectral index and( kk 0) ns −1,∣∆ 2 R = V ∣∣∣k=k024π 2 M 4 P ɛ , (3.20)is the amplitude of scalar perturbations at some pivot wavelength k 0 . V is here the inflatonpotential evaluated at the first Hubble-crossing and ɛ is the slow-roll parameter from Eq.(3.17). The power spectrum from Eq. (3.20) can actually be measured from the CMBR.The idea is that the quantum fluctuations create small inhomogeneities in the otherwisehomogeneous universe. These small inhomogeneities then form the origin of the formationof large scale structure in the universe. Through complicated but well understood effectsthese fluctuations leave their imprint on the spectrum of the CMBR. The amplitude ofperturbations ∆ 2 R can then be related to the fractional density perturbations δρ ρ , i.e.∆ 2 R ∼ ( δρρ) 2. (3.21)Furthermore since ρ ∝ T 4 , one finds that δρ ρ= 4 δT T. The fractional temperature perturbationsin the CMBR are , and we then findδTT≃10 −5∆ 2 R = (2.445 ± 0.096) × 10−9 , (3.22)measured by WMAP [12] at a reference wavelength k 0 = 0.002Mpc −1 . Because ∆ 2 is relatedRto the inflaton potential, see Eq. (3.20), this gives us constraints on the inflaton potential.For a quartic λφ 4 potential, we find that δρρ ∝ λ which gives λ ∼ 10 −12 . As we will see, incase of the SM Higgs boson the quartic self-coupling is of O(10 −1 ) and is therefore too big tobe the inflaton potential.
Figure 3.2: WMAP measurements of the spectral index n s and the tensor to scalar ratio r at a pivot wavelengthk = 0.002Mpc −1 . The scale-free Harrison-Zeldovich spectrum is excluded by two standard deviations, asis the λφ 4 inflaton potential.Now let us consider the other part of the power spectrum (3.20), namely the spectral indexn s . If the spectral index n s = 1, the spectrum is said to be scale invariant and this spectrumis called the Harrison-Zeldovich spectrum. However, inflation predicts that the spectralindex is not precisely 1, but there are small corrections to n s in terms of the slow-roll parameters.To be exact,n s = 1 − 6ɛ + 2η.The spectral index has been measured by the WMAP mission and it was found that n s =0.951 ± 0.016 [12]. The slow-roll parameters ɛ and η are expressed in terms of derivativesof the inflaton potential, see Eqs. (3.17) and (3.18), and the measurements of the spectralindex therefore constrain the form of the inflaton potential.There are more parameters that constrain the inflaton potential. The power spectrum inEq. (3.20) corresponds to the spectrum of scalar perturbations, i.e. density perturbations.There are however also tensor perturbations, leading to gravitational waves, and the correspondingspectrum of tensor perturbations can again be calculated. We would then findthat the power spectrum of tensor perturbations P T ∝ k n T. n T = 0 for a Harrison-Zeldovichspectrum, but inflation predicts that it is n T = −2ɛ. Often one calculates the ratio of thetensor to scalar perturbations, and one finds that r = −8n T = 16ɛ. The tensor to scalar ratior has also been measured by the WMAP mission, and so far it is found that r < 0.65 [12],which means that primordial gravitational waves that leave their imprint on the CMBRhave not been detected yet.In Fig.3.2 we show the 1 and 2 σ confidence contours of the spectral index n s and thetensor to scalar ratio r at a pivot wavelength k 0 = 0.02mpc −1 . These parameters have aspecific value for a choice of the inflaton potential, because they are expressed in termsof the slow-roll parameters. We have shown the results for a massive m 2 φ 2 potential, aquartic λφ 4 potential and the scale-free Harrison Zeldovich spectrum. Both the scale freeHarrison-Zeldovich spectrum and the spectrum for a quartic potential are excluded by over95% confidence level. As we will see in the next chapter, the Higgs potential is effectively aλφ 4 potential during inflation, and is therefore excluded by the WMAP measurements.In conclusion, we have found that there are constraints on the inflaton potential comingfrom measurements of the CMBR. These measurements require the inflaton to be a massivescalar particle with a λφ 4 inflaton potential. The quartic self-coupling should be λ ∼ 10 −12 .
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 22 and 23: VΦSlowrollregime0 MpΦFigure 3.1:
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
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 and 104:
Chapter 7Summary and outlookCosmolo
Page 105 and 106:
esulting action as the action of a
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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