Master's Thesis in Theoretical Physics - Universiteit Utrecht

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Note that for D = 4, the factor Γ(2 − D 2) is singular and the propagator seems to be infinite(apart from the singularity at x = x ′ ). However, we will see that the propagator is finitewhen we expand the propagator up to linear order in D −4. The first part without the sumsis finite and we do not need to expand here. For the second part, we first expand Γ(2 − D 2 )around D = 4,Γ(2 − D 2 ) = − 2D − 4 − γ E + O((D − 4)),where γ E ≡ −ψ(1) is the Euler constant and ψ(z) ≡ d(lnΓ(z))dzis the digamma function. For thenon-singular Gamma functions in the sums, we use the following expansionΓ(a + b(D − 4)) = Γ(a) + bΓ ′ (a)(D − 4) + O((D − 4) 2 ) = Γ(a)(1 + ψ(a)(D − 4)) + O((D − 4) 2 ).Furthermore, we need to expand ν D to linear order in (D − 4), which gives√ ( ) D − 1 − ɛ 2(D − 1)(D − 2ɛ) 1 m 2ν D =+2(1 − ɛ) (1 − ɛ) 2 ξ +(1 − ɛ) 2 H 2√ ( ) 3 − ɛ 2 (Rξ + m2=+2(1 − ɛ) H 2 (1 − ɛ) 2 + 3 − ɛ2(1 − ɛ) 2 − 3 + 4 − 2ɛ )(1 − ɛ) 2 ξ (D − 4) + O(D − 4) 2√(= ν 2 1 3+(1 − ɛ) 2 2 − ɛ )2 + (−7 + 2ɛ)ξ (D − 4) + O(D − 4) 2[= ν 1 + 1 (1 1 32 ν 2 (1 − ɛ) 2 2 − ɛ ) ]2 + (−7 + 2ɛ)ξ (D − 4) + O(D − 4) 2= ν + 1 32 νC(D − 4) + O(D − 4)2 2, C =− ɛ 2+ (−7 + 2ɛ)ξ( 1 2 (3 − ɛ))2 − 3(4 − 2ɛ)ξIn the second line I have used that R ≡ R D=4 = 3(4 − 2ɛ)H 2 . In the third line I have usedthe expression for ν ≡ ν D=4 from Eq. (5.76). In the last line I have implicitly defined theconstant C. When ɛ ≪ 1 and Ξ ≫ 1, this constant is of order unity. Now we can expand thedifferent gamma functions that appear in the sums of the propagator (5.99)Γ( 3 2 + ν D + n) = Γ( 3 2 + ν + n)(1 + 1 2 νC(D − 4)ψ(3 + ν + n)) + O(D − 4)22Γ( 3 2 − ν D + n) = Γ( 3 2 + ν + n)(1 − 1 2 νC(D − 4)ψ(3 − ν + n)) + O(D − 4)22Γ( D − 12Γ( D − 12Γ( 1 2 + ν D) = Γ( 1 2 + ν)(1 + 1 2 νC(D − 4)ψ(1 + ν)) + O(D − 4)22Γ( 1 2 − ν D) = Γ( 1 2 − ν)(1 − 1 2 νC(D − 4)ψ(1 − ν)) + O(D − 4)22+ ν D + n) = Γ( 3 2 + ν + n)(1 + (1 + 1 2 νC(D − 4))ψ(3 + ν + n)) + O(D − 4)22− ν D + n) = Γ( 3 2 − ν + n)(1 + (1 − 1 2 νC(D − 4))ψ(3 − ν + n)) + O(D − 4)22Γ(3 − D 2 + n) = Γ(1 + n)(1 − 1 (D − 4)ψ(1 + n)) + O(D − 4)22Γ( D 2 + n) = Γ(1 + n)(1 + 1 2 (D − 4)ψ(1 + n)) + O(D − 4)2 ,and also( y) n−D2 +2 ( y) n 1( y)= (1 −4 4 2 ln (D − 4)) + O(D − 4) 2 .4
Now we plug all these expansions into the sums of Eq. (5.99). One thing we can immediatelysee is that the zeroth order expansions cancel against each other in the sums, and thereforethe terms that are summed over are only linear in D − 4. This is the reason why we wrotethe scalar propagator as in (5.99). Our final result for the finite propagator (up to linearorder in (D − 4)) is) 1−D2i∆(x, x ′ ) ∞ = [(1 − ɛ)2 HH ′ ] D 2 −1Γ( D ( y(4π) D 2 2 − 1) 4+ (1 − ɛ)2 HH ′ ∞∑ Γ( 3 2 + ν + n)Γ( 3 2 − ν + n) 1( y) n(4π) 2 n=0 Γ( 1 2 + ν)Γ( 1 2 − ν) Γ(1 + n)Γ(2 + n) 4[ ( y)× ln + ψ( 3 ]4 2 + ν + n) + ψ(3 2 − ν + n) − ψ(1 + n) − ψ(2 + n) + O(D − 4).For convenience I will defineΨ n = ψ( 3 2 + ν + n) + ψ(3 − ν + n) − ψ(1 + n) − ψ(2 + n).2Next we use the relation xΓ(x) = Γ(x + 1) to get∞∑Γ( 3 2 + ν + n)Γ(3 2 − ν + n) = (1 2 + ν)(1 2 − ν)Γ(1 2 + ν)Γ(1 2 − ν)n=0The propagator then takes the form+( 1 2 + ν)(1 2 − ν)(3 2 + ν)(3 2 − ν)Γ(1 2 + ν)Γ(1 − ν) + ....2= Γ( 1 [(2 + ν)Γ(1 2 − ν) 1 4 − ν2 ) + ( 1 4 − ν2 )( 9 ]4 − ν2 ) + ...[∞∑= Γ( 1 (( ) n∏ 2k + 1 2 ) ]2 + ν)Γ(1 2 − ν) − ν 2 .2n=0) 1−D2k=0(5.100)i∆(x, x ′ ) ∞ = [(1 − ɛ)2 HH ′ ] D 2 −1Γ( D ( y(4π) D 2 2 − 1) 4+ (1 − ɛ)2 HH ′ ∞∑ ( y) (( ) n n∏ 2k + 1 2 )Ψn(4π) n=0[ ]2− ν 2 + O(D − 4). (5.101)4k=02Now we take another look at the expression for ν from Eq. (5.76) and define a quantitys ≡ξR + m2H 2 , s ≪ 1. (5.102)s ≪ 1 because the mass term precisely cancels the main contribution coming from the ξRterm during inflation. This happens because the classical inflaton field φ 0 ∝ H duringinflation, see Eq. (4.19). We can make an expansion for s ≪ 1 in the expression for ν. Thisgives(1 3 − ɛν =1 − ɛ 2 + s ). (5.103)3 − ɛNow we also expand for ɛ ≪ 1, which is also true during inflation. We getν = 3 2 + s + ɛ + O(sɛ). (5.104)3
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INFLATION AND BARYOGENESIS THROUGH
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AbstractIn this thesis we consider
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6 The fermion sector 716.1 Fermion
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and particle physics. The Einstein
Page 12 and 13:
2.2 An expanding universeWe are now
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Here, H 0 = H(t = 0) and we have ch
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Since the spatial background is hom
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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 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: wherey ≡ y ++ (x, x ′ ) = −(|
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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