Master's Thesis in Theoretical Physics - Universiteit Utrecht

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value of ξ. With the explicit expression for the Ricci scalar in Eq. (4.94) the field equationbecomes,∑[k=1,2 ξ ik φ k¨φ i + 3H ˙φi +M 2 p − 2 ∑ k,l,m=1,2 φ ∗ k ξ −2∑](δ lm − 6ξ lm )( ˙φ∗lφ˙m ) =kl(δ lm − 6ξ lm )φ m l,m=1,2∑[k=1,2 ξ ik φ kM 2 p − 2 ∑ k,l,m=1,2 φ ∗ k ξ 6 ∑ξ lm [φ ∗ ∂Vlkl(δ lm − 6ξ lm )φ m l,m=1,2∂φ ∗ + ∂V]φ m ] − 4V − ∂Vm ∂φ l ∂φ ∗ (4.95) ,iwhere V = V (φ 1 ,φ 2 ). We have written the potential terms on the righthand side of theequation of motion. In the slow-roll approximation these potential terms dominate over thekinetic terms. As in the single field case, we assume the slow-roll conditions (4.12). Thisbasically means we neglect the terms proportional to ¨φ and ˙φ2 but keep the term 3H ˙φi andthe potential terms. Thus in the slow-roll approximation the field equation becomes3H ˙φi ≈∑k=1,2 ξ ik φ kM 2 p − 2 ∑ k,l,m=1,2 φ ∗ k ξ kl(δ lm − 6ξ lm )φ m[6 ∑l,m=1,2ξ lm [(φ ∗ ∂Vl∂φ ∗ m+ ∂V ]φ m )] − 4V − ∂V∂φ l ∂φ ∗ .iThen we substitute this expression into the energy constraint equation (4.91) and also usethe slow-roll approximation to findH 2≈1{3(M 2 p − 2 ∑ 2i, j=1,2 ξ i j φ ∗ i φ V +j) M 2 p − 2 ∑ k,l,m=1,2 φ ∗ k ξ (4.96)kl(δ lm − 6ξ lm )φ m×[(M 2 p − 2 ∑ )(φ ∗ k ξ klφ l −∑ξ i j [φ ∗ ∂V ∂V)ik,l=1,2i, j=1,2∂φ ∗ + φ j ] − 8V ∑φ ∗ i∂φ ξ i jξ jk φ k]}.j i i, j,k=1,2If we now substitute the potential (4.72) and take the large negative nonminimal couplinglimit ξ ii ≪ 0, we find a complicated expression for H 2 and the equations of motion. However,we can solve these equations numerically, which we will do in the next section.4.4.2 Numerical solutions of the field equationsIn Fig. 4.7 we have numerically calculated solutions for the fields φ 1 and φ 2 from the fieldequation (4.95). As a matter of simplicity we have chosen the fields and nonminimal couplingsto be real. All the nonminimal couplings have typical values ξ ii ∼ −10 3 and thequartic couplings λ i ∼ 10 −3 . One interesting observation is that inflation does not happenimmediately, i.e. the fields do not roll down from their initial value to their minimumstraight away. Instead, they interact (through the interaction terms with couplings ξ 12 , ξ 21and λ 3 ) and start to oscillate rapidly. This is shown in Fig. 4.8. After a short time the oscillationsare damped out and the fields acquire a "fixed" value. Then the fields do roll down totheir minimum and inflation works. The numerical solutions for φ 1 and φ 2 in Fig. 4.7 showthat during inflation the fields are proportional to each other. This suggest that we can takeone linear combination of the two fields that is the inflaton. The other linear combination isan oscillating mode that quickly decays. This could then be a source for particle productionand perhaps baryogenesis.In Fig. 4.9a we show the growth of the number of e-folds in time. For the typical nonminimalmodel as mentioned above, the maximum number of e-folds that is reached at the endof inflation is N ≃ 1200. The condition necessary to solve the flatness and horizon puzzlesis N ≥ 70, so this condition is easily satisfied. We could therefore relax the initial values ofthe fields or the values of the nonminimal couplings. Thus, our nonminimal couplings couldbe much smaller, or the initial values of the fields could be lower. The Hubble parameter
0.50.80.40.6Φ1t0.30.20.1Φ2t0.40.20.00 1. 10 6 2. 10 6 3. 10 6 4. 10 6 5. 10 6 6. 10 6 7. 10 60.0(a) Numerical solution of φ 1 (b) Numerical solution of φ 20 1. 10 6 2. 10 6 3. 10 6 4. 10 6 5. 10 6 6. 10 6 7. 10 6ttFigure 4.7: Numerical solutions of the fields φ 1 and φ 2 in the two-Higgs doublet model. The nonminimalcouplings are real and have typical values in the order of 10 3 . For the simulations above, ξ 11 = 10 3 , ξ 22 =1.2 × 10 3 , ξ 12 = ξ 21 = 10 3 . The quartic couplings also have typical values 10 −3 . In this case λ 1 = 0.8 × 10 −3 ,λ 2 = 1.2 × 10 −3 and λ 3 = 2 × 10 −3 . The initial values for φ 1 and φ 2 are 0.4 and 0.8 respectively, in units ofM P . The time t is given in units of MP −1 . After some complicated interaction between the two fields at earlytimes (shown in Fig. 4.8), the fields reach a certain value and slowly roll down to their minimum. During theinflationary period the fields are proportional to each other, which suggests that we can find linear combinationsof the two fields such that one of the linear combinations is the actual inflaton, whereas the other combinationis an oscillating field that rapidly decays and could lead to particle creation.is also shown in Fig. 4.9b. Again we see that the Hubble parameter is decreasing linearlyin time and is proportional to both φ 1 and φ 2 . This also suggests that we have only onelinear combination of the fields that is the inflaton. This linear combination is then alsoproportional to H. At the end of inflation, the potential term in the equation for H 2 (4.91)is close to zero and the term containing the nonminimal coupling dominates. We againsee the behavior typical to nonminimal inflationary models, the Hubble parameter makessharp drops to almost zero.Our conclusion from our numerical simulations is that inflation also works in a nonminimallycoupled two-Higgs doublet model. However, only one particular combination of thetwo fields is the actual inflaton. The other linear combination is an oscillating mode thatquickly decays and could be a source for particle production and baryogenesis. In this thesiswe will focus our attention on the Higgs boson as the inflaton and leave the oscillating modeaside for now. In future work we hope to take a closer look at the oscillating mode and thereheating process.We can comment on our own numerical results. First of all, we have shown numerical resultsfor the fields φ 1 and φ 2 with the potential (4.72). As we have mentioned earlier thissection, we could also have a kinetic term that couples derivatives of φ 1 and φ 2 . This wecould then diagonalize by defining new fields φ + and φ − , which again gave us a hermiteannonminimal coupling matrix. The only difference is in the potential term, that now alsocontains terms φ + φ 3 − and φ −φ 3 + . We have numerically verified that these extra potentialterms do not change our results significantly, although we will not show the results here.The numerical results look the same as those in Figs. 4.7, 4.8 and 4.9, but some of thenumerical factors will be different.A second comment on our own results is that we have calculated the numerical results forreal nonminimal couplings and fields. However, the nonminimal coupling ξ 12 is in principlea complex quantity, and the fields are complex as well (although only their relative phase isa physical degree of freedom). Of course we would have to include this fact into our numer-
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 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 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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