Leptogenesis in a spatially flat Milne-type universe. Ion I. Cotaescu Abstract The quantum electrodynamics on a spatially flat (1+3)-dimensional Friedmann- Lematre-Robertson-Walker space-time with a Milne-type scale factor is considered focusing on the amplitudes of the allowed effects in the first order of perturbations. The definition of the transition rates is reconsidered obtaining an appropriate angular behavior of the probability of the pair creation from a photon which has a similar rate as the leptons creation from vacuum. Pacs: 04.20.Cv, 04.62.+v, 11.30.-j arXiv:1602.06810 Keywords: Milne, FLRW, spatially flat, leptogenesis, transition amplitudes, transition rates. 1
Contents Introduction 3 Milne’s and Milne-type universes 5 Free fields on M 8 First order QED amplitudes 14 Rates and probabilities 21 Graphical analysis 31 Concluding remarks 37 2
Introduction • In general relativity, the standard quantum field theory (QFT) based on perturbations is inchoate since one payed more attention to alternative non- perturbative methods as, for example, the cosmological particle creation [1, 2, 3, 4, 5, 6, 7, 8]. • The manifolds of actual interest in the actual cosmology are the spatially flat FLRW manifolds which are symmetric under translations and, consequently, there are quantum modes expressed in terms of plane waves with similar properties as in special relativity. • These manifolds are useful for studying the behavior of the quantum matter in the presence of classical gravity turning back to the perturbation methods of the quantum field theory where significant results were obtained by many authors [9, 10, 11, 12, 13, 14, 15, 16]. 3
• Inspired by these results we built the QED in Coulomb gauge on the de Sitter expanding universe [17], analyzing the processes in the first order of perturbations that are allowed on this manifold since the energy and momentum cannot be conserved simultaneously [9, 10, 11, 17]. • Recently we completed this approach with the integral representation of the fermion propagators we need for calculating Feynman diagrams in any order of perturbations [18]. Thus we have an example of a complete QED on the de Ssitter background. • Looking for another example of manifold where the QED could be constructed without huge difficulties we observed that there exists an expanding space-time where the free field equations can be analytically solved [19]. This is the (1+3) -dimensional spatially flat FRLW manifold whose expansion is given by a Milne-type scale factor, proportional with the proper (or cosmic) time, t . 4
Milne’s and Milne-type universes The general metric in spherical coordinates of the (1+3) -dimensional FLRW manifolds, dr 2 � � ds 2 = dt 2 − a ( t ) 2 1 − κr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 (1) with a Milne-type scale factor, a ( t ) = ωt , depending on parameter ω , is produced by the sources ω 2 + κ ω 2 + κ 3 p = − 1 ρ = ω 2 t 2 , ω 2 t 2 , (2) 8 πG 8 πG Genuine Milne universe: ω = 1 and κ = − 1 → ρ = p = 0 Spatially flat Milne-type univese (M): κ = 0 and arbitrary ω such that 3 1 p = − 1 1 ρ = t 2 , t 2 . (3) 8 πG 8 πG 5
In this manifold we define the usual FLRW chart whose coordinates x µ (labeled by the natural indices µ, ν, ... = 0 , 1 , 2 , 3 ) are the proper time x 0 = t and the Cartesian space coordinates, x i ( i, j, k... = 1 , 2 , 3 ), for which we x = ( x 1 , x 2 , x 3 ) . may use the vector notation � This chart, denoted by { t, � x } , is related to the conformal flat one, { t c , � x } , where we have the same space coordinates but the conformal time t c ∈ ( −∞ , ∞ ) defined as � a ( t ) = 1 dt ω ln( ωt ) → a ( t c ) = e ωt c . t c = (4) The corresponding line elements read ds 2 = g µν ( x ) dx µ dx ν = dt 2 − ( ωt ) 2 d� x · d� x = e 2 ωt c ( dt 2 c − d� x · d� x ) . (5) 6
The expansion of M that can be better observed in the chart { t, � x } , of ˆ x i = ωtx i , where the line element ’physical’ space coordinates ˆ 1 − 1 x dt � � ds 2 = dt 2 + 2 � t 2 � x · � x · d� t − d� x · d� ˆ x ˆ ˆ ˆ ˆ x , ˆ (6) lays out an expanding horizon at | � x | = t and tends to the Minkowski space- ˆ time when t → ∞ and the gravitational sources vanish. In M we introduce the local orthogonal non-holonomic frames defined by α = e µ the vector fields e ˆ α ∂ µ and the associated co-frames given by the 1- ˆ α = ˆ forms ω ˆ e ˆ µ dx µ , labeled by the local indices, ˆ α µ, ˆ ν, ... = 0 , 1 , 2 , 3 . Here we use exclusively the diagonal tertrad gauge which preserves the symmetry of M as a global one, ω 0 = dt = e ωt c dt c , e 0 = ∂ t = e − ωt c ∂ t c , (7) ω i = ωtdx i = e ωt c dx i . e i = 1 ωt ∂ i = e − ωt c ∂ i , (8) 7
Free fields on M The massive Dirac field ψ of mass m which satisfy the field equation ( E D − m ) ψ = 0 where E D = iγ 0 ∂ t + i 1 ωtγ i ∂ i + 3 i 1 tγ 0 − m . (9) 2 The term of this operator depending on the Hubble function ˙ a a = 1 t can be removed at any time by substituting ψ → ( ωt ) − 3 2 ψ . The fundamental solutions of the Dirac equation can be derived in the chiral representation (with diagonal γ 5 ) where we have to look for solutions of the form x ) = [2 πa ( t )] − 3 2 e i� p · � x U p ( t ) u σ U � p,σ ( t, � (10) x ) = [2 πa ( t )] − 3 2 e − i� p · � x V p ( t ) v σ V � p,σ ( t, � (11) 8
depending on the diagonal matrix-functions u + p ( t ) , u − U p ( t ) = diag � p ( t ) � , (12) v + p ( t ) , v − V p ( t ) = diag � p ( t ) � , (13) whose matrix elements are functions only on t and p = | � p | , determining the time modulation of the fundamental spinors. The spin part is encapsulated in the spinors of the momentum-helicity basis that in the chiral representation of the Dirac matrices read [29] � � � � u σ = 1 v σ = c ξ σ ( � p ) − η σ ( � p ) √ √ (14) ξ σ ( � p ) η σ ( � p ) 2 2 p ) = iσ 2 ξ ∗ where ξ σ ( � p ) and η σ ( � σ are the Pauli spinors of the helicity basis corresponding to the helicities σ = ± 1 2 as given in the Appendix A. 9
The fundamental spinors are solutions of the free Dirac equation whether the modulation functions u ± p ( t ) and v ± p ( t ) satisfy the first order differential equations i∂ t ± 2 σp � � u ± p ( t ) = m u ∓ p ( t ) , (15) ωt i∂ t ∓ 2 σp � � v ± p ( t ) = − m v ∓ p ( t ) , (16) ωt in the chart with the proper time. The solutions of these systems must satisfy the charge-conjugation symmetry [19], p ( t ) � ∗ , v ± u ∓ p ( t ) = � (17) and the normalization conditions p | 2 + | u − p | 2 = | v + p | 2 + | v − p | 2 = 1 . | u + (18) 10
that determine the definitive form of the fundamental spinors, � K σ − i p � e i� p · � x � ω ( im t ) ξ σ ( � p ) mt U � p,σ ( x ) = (19) 3 K σ + i p ω ( im t ) ξ σ ( � p ) π [2 πωt ] 2 � K σ − i p � e − i� p · � x � ω ( − im t ) η σ ( � p ) mt V � p,σ ( x ) = , 3 − K σ + i p ω ( − im t ) η σ ( � p ) π [2 πωt ] 2 (20) according to the identity (77). The fundamental spinors (19) and (20) form the momentum-helicity basis in which the general solutions of the Dirac equation can be expanded as x ) = ψ (+) ( t, � x ) + ψ ( − ) ( t, � ψ ( t, � x ) � � d 3 p p,σ ( x ) b † ( � = [ U � p,σ ( x ) a ( � p, σ ) + V � p, σ )] . (21) σ 11
After quantization, the particle ( a , a † ) and antiparticle ( b , b † ) operators satisfy the canonical anti-commutation relations [19], p, σ ) , a † ( � p ′ , σ ′ ) } = { b ( � p, σ ) , b † ( � p ′ , σ ′ ) } { a ( � = δ σσ ′ δ 3 ( � p ′ ) . p − � (22) Then ψ becomes a quantum free field that can be used in perturbation for calculating physical effects. The free Maxwell field A µ can be written easily in the conformal chart taking over the well-known results in Minkowski space-time since the free Maxwell equations are conformally invariant. The electromagnetic gauge does not have this property such that we are forced to adopt the Coulomb gauge with A 0 ( x ) = 0 as in Refs. [21, 17], remaining with the free Maxwell equations 1 ( ∂ 2 t c − ∆) A i ( x ) = 0 , (23) � g ( x ) 12
which can be solved in momentum-helicity basis where we obtain the expansion � � � � d 3 k k,λ ; i ( x ) α ( � k,λ ; i ( x ) ∗ α † ( � A i ( x ) = µ � k, λ ) + µ � k, λ ) , (24) λ in terms of the modes functions, 1 1 e − ikt c + i� x ε i ( � k · � µ � k,λ ; i ( t c , � x ) = √ k, λ ) , (25) (2 π ) 3 / 2 2 k depending on the momentum � k ( k = | � k | ) and helicity λ = ± 1 of the ε λ ( � polarization vectors � k ) in Coulomb gauge (given in Appendix A). Hereby we obtain the mode functions in the FLRW chart 1 1 ω e i� ( ωt ) − i k x ε i ( � k · � µ � k,λ ; i ( t, � x ) = √ k, λ ) . (26) (2 π ) 3 / 2 2 k 13
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