Master's Thesis in Theoretical Physics - Universiteit Utrecht

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into neutral hydrogen at a temperature T ∼ 3000 K. Before this moment the photons weretightly coupled to electrons via Compton scattering. However, as soon as the neutral hydrogenformed the photons decoupled from the electrons and the universe became transparent.These photons from the time of decoupling we can still see today as a constant backgroundradiation, also known as the Cosmic Microwave Background Radiation (CMBR). Since weare essentially looking back into a time where the universe was much smaller than today,the CMBR provides valuable information about the early universe.3.2 Cosmological puzzlesThe previous section is in principle a good description of the history of our universe and ithas proved to be extremely useful for cosmologists. However, cosmologists could not explaincertain puzzles within this standard history of the universe. As observations of our universebecame better and better, these problems worsened up to a point where they could not beignored. Let us explain the puzzles.Homogeneity puzzle: This puzzle concerns one of the two main points of most cosmologicalmodels, namely the homogeneity of the universe on the largest scales. From theobservation of the CMBR we can derive that the inhomogeneities are of the order of 10 −4 onthe Hubble length scale, which is roughly the size of our visible universe. However, gravityis an attractive force, and when stars, planets, galaxies and clusters formed we expect thisto create many more inhomogeneities.Horizon puzzle: To state this puzzle, first we have to define some important cosmologicaldistances. The first is the Hubble radius, defined asR H ≡ c H , (3.2)where I have explicitly shown the speed of light c, which in our convention equals 1. TheHubble radius is a way to see whether particles are causally connected. If particles areseparated by distances larger than the Hubble radius, they cannot communicate now. Thisdoes not necessarily mean that the particles were also out of causal contact at earlier times.To see this we can actually calculate the distance that light has traveled in a certain periodof time by looking at the invariant line element in Eq. (2.5). For light ds 2 = 0, so we canintegrate this expression to find the comoving distance at time t,∫ tcdt ′l c (t) =0 a(t ′ ) , (3.3)where the time t = 0 is the time at which the light signal has been sent. This is not yetthe physical distance, because physical distances have increased by a factor a(t) due to theexpansion of the universe. Thus the physical distance is∫ tcdt ′l phys (t) = a(t)0 a(t ′ ) , (3.4)This physical distance is called the particle horizon, since it is the maximum distance fromwhich light could have traveled from particles to an observer. In other words, if two particlesare separated by a distance greater than l phys , they were never in causal contact. Sothe situation can occur where particles cannot communicate now because their separationdistance is larger than R H , but they were in causal contact before because the distance thatseparates the particles is smaller than the particle horizon.
If we consider the evolution of the universe with a radiation and matter era, we can calculateboth the Hubble radius and particle horizon. We use Table 2.1 and Eqs. (3.2) and(3.4) and find that during radiation era R H = 2t and l phys = 2t = R H . Thus in a radiationdominated universe the Hubble radius and the particle horizon are equal. During matterera R H = 3 2 t and l phys = 3t = 2R H , so the particle horizon is twice as big as the Hubbleradius in a matter dominated universe. If we assume that our universe only went throughan era of radiation and matter domination, we can safely say that the particle horizon isapproximately equal to the Hubble radius and at most twice the Hubble radius. Taking thisinto account, plus the fact that R H is roughly 13 billion lightyears, we find approximately3000 causally disconnected regions in the CMB map of the universe. To be more precise,these are regions that are not in causal contact now because their separation distance islarger than R H , and were therefore also never in causal contact because the particle horizonis approximately equal to the Hubble radius. However when we look at the temperaturefluctuations of the CMBR, we see that these are of the order of δT/T ∼ 10 −5 . The horizonpuzzle is: How can these temperature differences be so extremely small for regions thatwere never in causal contact?Flatness puzzle: Some people do not consider this to be a real problem, but it is an interestingpuzzle nonetheless. Our universe is on the largest scales described by the FLRWmetric in Eq. (2.4) and observations from the CMBR support a flat universe with k = 0.But the FLRW metric was derived by assuming the most general metric for an expandinguniverse, and our universe could just as well have been described by a metric with k ≠ 0which has the same number of symmetries. How can we explain that our universe is soflat?Cosmic relics puzzle: In the very early universe many cosmic relics could be created atextreme energy scales or phase transitions. Examples of cosmic relics are particles suchas gravitons, gravitinos and the lightest supersymmetric particle LSP. Other examplesare topological defects such as domain walls, cosmic strings and the notorious magneticmonopole. The puzzle is that all of these cosmic relics, even the ones that are very longlived, have not been observed.3.3 Inflation as solution of the puzzlesIn 1981 Alan Guth[3] proposed a solution to the puzzles in the previous section. He addeda new era to the history of the universe known as cosmological inflation. This inflationaryera happened prior to the radiation era but below the Planck scale. The idea is that theuniverse undergoes a period of rapid accelerated expansion in which the universe grows tomany orders of magnitude the size of the present universe. We explain now how inflationsolves the puzzles:The homogeneity puzzle is solved by noting that the inhomogeneities are formed becauseof the attractive gravitational interaction. In an inflationary universe the exponential expansionof the universe tends to pull the gravitationally interacting particles apart, suchthat the inhomogeneities do not form. When inflation ends, the universe is almost perfectlyhomogeneous. As we will see at the end of the chapter, quantum fluctuations can still causesmall inhomogeneities in the homogeneous universe at the end of inflation. Eventuallythese small inhomogeneities will then form the large scale structures that we see today inour universe.The horizon puzzle is solved in the following way. During (de Sitter) inflation the scalefactor increases exponentially in time, i.e. a ∝ e H0t (see Table 2.1). We find that the particle
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 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
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
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whereΦ k =( ) ( ) ( ) ( )φk,aa φ
Page 77 and 78:
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
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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