Master's Thesis in Theoretical Physics - Universiteit Utrecht

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horizon during inflation isl phys (t) = 1 H 0(e H 0t − 1). (3.5)Thus, the particle horizon increases exponentially with time. The Hubble radius howeveris constant during inflation (since H = ȧ/a = H 0 is constant) and is equal to R H = 1/H 0 .So we conclude that during de Sitter inflation the particle horizon grows exponentially withrespect to the Hubble radius. If this happens the particle horizon can grow to many orders ofmagnitude beyond the Hubble radius. So particles that cannot communicate now, becausetheir separation distance is much larger than R H , were able to communicate in the pastbecause the particle horizon is much larger than the distance between the particles. Thissolves the horizon puzzle.The flatness puzzle is solved by inflation as well. The FLRW equations can also be derivedfor the FLRW metric with k ≠ 0 from Eq. (2.4). This changes the first of the FLRW equationstoH 2 = 8πG Nρ − k 3 a 2 , (3.6)which we can rewrite as1 = ρ −kρ crit H 2 a 2 ≡ Ω total − Ω k , (3.7)where the critical energy density, the density necessary for a flat universe with k = 0, isdefined asρ crit = 3H2 t8πG N. (3.8)H t is the Hubble parameter today and is approximately 73±3 kms −1 Mpc −1 . Measurementsof the energy density of the universe giveΩ k =kH 2 = 0.01 ± 0.02, (3.9)a2 so the energy density in our universe is at this moment very close to the critical density.Now we want to find how the magnitude of the curvature term in Eq. (3.7) at earlier times.By again using Table 2.1, we find that during matter era Ha ∝ t − 1 3 and during radiationera Ha ∝ t − 1 2 . Since k is constant, we conclude that the curvature term k 2 /(Ha) 2 was muchsmaller in the early universe. This means that if we live in a universe with some k ≠ 0,the energy density in the early universe must have been extremely fine tuned close to thecritical density. Although a fine tuning problem is not a problem per se, it is highly unlikelythat the energy density of the universe was so extremely close to the critical density suchthat we now measure a curvature term which is close to zero.Inflation solves this puzzle by noting that during (de Sitter) inflation the factor Ha ∝ e H0t .The curvature term now scales as k/(Ha) 2 ∝ e −2H0t , so at earlier times this term was muchbigger. This also means that the energy density could have been far from the critical densityand we do not have a fine tuning problem. A useful quantity to define now is the number ofe-folds N(t),( ) ∫ a(t) tN(t) = ln = H(t ′ )dt ′ , (3.10)a iwhere a i ≡ a(t = 0). In one e-fold the scale factor has increased by a factor e, which inthe case of de Sitter inflation happens when t = 1/H 0 . The flatness puzzle is solved whenΩ k ≃ 1 at some early time (before inflation), and one can calculate that the number of e-foldsnecessary for this is N(t) ≃ 70. This solves the flatness puzzle.Finally, the cosmic relics puzzle is solved by inflation. The solution is comparable to the0
solution to the homogeneity puzzle. Suppose we have some initial energy density of cosmicrelics ρ relics . During inflation, the size of the universe increases by an enormous factor(the scale factor increases by a factor e N , which is huge). Therefore, any preexisting energydensity of cosmic relics is diluted to practically nothing and we will never observe cosmicrelics such as magnetic monopoles.3.4 Inflationary modelsInflation is basically an epoch during which the scale factor accelerates, thus the universeundergoes an accelerated expansion. Formally, inflation occurs whenSince äa = Ḣ + H2 , this also meansä > 0. (3.11)ɛ = − Ḣ < 1. (3.12)H2 From Eq. (2.8) we see that we need to haveρ + 3p < 0. (3.13)This means that we need some special field with p < − 1 3ρ, i.e. negative pressure. This ispossible within certain particle physics models. An essential ingredient is a scalar field.As we know from the Standard Model, in scalar field models a phenomenon known as theHiggs mechanism can occur. The potential of a scalar field φ can be such that there are twominima and the scalar field is initially in the minimum with the lowest value (at φ = 0).However, as time increases and the temperature drops, the second minimum at φ ≠ 0. cantake a lower value than the initial minimum. We then say that the scalar field is in thefalse vacuum state and it wants to be in the true vacuum state. A phase transition will takeplace, in which the scalar field moves from the initial symmetric vacuum phase at φ = 0 tothe non-symmetric vacuum phase at φ ≠ 0. This is a process known as spontaneous symmetrybreaking. Going back to the statement that we need negative pressure for successfulinflation, first realize that negative pressure is not that unusual if you think of it as anattractive force. Basically, the scalar field is pulled out of its false vacuum state into thetrue vacuum, thus the scalar field experiences negative pressure.Alan Guth used precisely the mechanism of spontaneous symmetry in his original article[3] on "old" inflation. He considers a first order phase transition, which means that thereis a potential barrier between the false vacuum and the true vacuum. The idea is that thescalar field wants to move to the true vacuum, but it cannot because of the barrier, and itis trapped in the false vacuum state with some high temperature. The true vacuum willbecome lower and lower as the temperature drops, until finally at some low temperaturethe scalar field will finally move to the true vacuum. The latent heat is released and thisgenerates a huge entropy production which solves the flatness and horizon puzzles. Theflaw in this original model is that the bubbles of the true vacuum phase that are formedcannot overtake the rapid expansion of the universe. In other words, the universe expandsfaster that the bubbles grow, so the bubbles never coalesce and the universe never reachesthe true vacuum state. Thus, inflation never ends, a problem known as the graceful exitproblem.Linde[6] and Albrecht and Steinhardt[7] proposed a new inflationary model, not surprisinglyknown as new inflation. In this scenario the universe will undergo a second orderphase transition or a crossover, such that no bubbles are formed and the universe will as
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 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
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
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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
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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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