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ENFPC 2010

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  • 1. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic Approach to Warm InflationLeandro Alexandre da Silva1Rio de Janeiro State UniversityDepartment of Theoretical PhysicsXXXI Encontro Nacional de F´ısica de Part´ıculas e Campos01/09/20101in collaboration with R.O. Ramos
  • 2. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm Inflation1 Stochastic Approach to Inflation2 Warm Inflation3 Stochastic approach to Warm Inflation
  • 3. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic Approach to InflationWhen?Stochastic Inflation: ∼ 1987Why?Exponentially rapid expansion → “freeze” of inflaton quantumfluctuations on super-horizon scales⇓Inflaton fluctuations behave effectivelly as classical fluctuationmodes with random amplitudes⇓Emulates the growth of vacuum fluctuations by an effectivestochastic noise field which drives the dynamics of thevolume-smoothed inflaton → effective dynamics forcoarse-grained field φ
  • 4. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic Approach to InflationHow?Usual approach → decomposition of φ in a classical,coarse-grained component and in a quantum fluctuation part:Φ(x, t) → φ(x, t) + q(x, t) .φ → coarse-grained scalar field averaged over approximatellyall de Sitter horizon size 1/χq(x, t) → summarizes high frequency (k kh ≈ χ) quantumfluctuations.q(x, t) aproximated as a free, massless scalar field.
  • 5. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic Approach to Inflationde Sitter metric:ds2= dxµdxνgµν = −dt2+ e2χtdx2,Lagrangian density:L =12√−g [gµν∂µΦ∂νφ − 2V (Φ)]Equation of motion(EoM):−3χ∂∂t−∂2∂t2+ e−2χt 2Φ(x, t) −∂V (Φ)∂Φ= 0
  • 6. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic Approach to InflationMaking the field decomposition (slow-roll: ¨φ(x, t) ≈ 0):3χ∂∂t− e−2χt 2[φ(x, t) + q(x, t)] +∂V (φ)∂φ= 03χ∂∂t− e−2χt 2φ(x, t) +∂V (φ)∂φ= 3χη(x, t) ,Noise term:η(x, t) ≡ −∂∂t+e−2χt3χ2q(x, t)Fourier mode expansion in de Sitter background:q(x, t) ≡ d3kWχ(k) σk(t)e−ik·xˆak+ σ∗k(t)eik·xˆa†k.Wχ(k) → filter or window function. Sharp momentum cutoffimplementation: Wχ(k) ≡ θ(k − χeχt).
  • 7. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic Approach to Inflationσk(t) ≡12k(2π)3χτ − iχke−ikτ.Commutator:[η(x, τ), η(y, τ)] = 0⇓Classical behaviour of quantum noise!
  • 8. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic Approach to Inflationσk(t) ≡12k(2π)3χτ − iχke−ikτ.Commutator:[η(x, τ), η(y, τ)] = 0⇓Classical behaviour of quantum noise!Propagator:0 | η(x, t)η(y, t ) | 0 =χ34π2δ(t − t )sin τ | x − y |τ | x − y |.
  • 9. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationWarm InflationSame basic ideas of cold inflationInflaton interacts with other fields → radiation productionA reheating process is no more necessarySmooth transition to the radiation dominated regimeThermal origin for the density perturbationsA. Berera and L. Z. Fang,Phys. Rev. Lett. 74, 1912 (1995):Inflaton dynamics → Langevin-like equation
  • 10. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationMicroscopic motivation to Warm InflationExample:S[φ, χ, σ] = d4x12(∂µφ)2−12m2φφ2−λ4!φ4+12(∂µχ)2−12m2χχ2+12(∂µσ)2−12m2σσ2−g22φ2χ2− f χσ2.φ → classical field in which dynamics we are interested inχ → intermediate field that couples to σ and φσ → Thermally equilibrated field at temperature T
  • 11. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationMicroscopic motivation to Warm InflationDetailed calculation: Berera and Ramos, PRD63, 103509(2001),Gleiser and Ramos, PRD50, 2441 (1994):General Effective Equation of Motion (Homogeneousapproximation)d2φ(t)dt2= −dVeff(φ)dφ− φn(t)t−∞dt φn(t ) ˙φ(t )Kχ(t − t )+ φn(t) ξ (t) ,n = 0: additive noise→ φχ2 interactionn = 1: multiplicative noise → φ2χ2 interaction
  • 12. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationMarkovian ApproximationNon-Markovian Equation of Motion:∂2t + m2φ +λ3!φ(t)2φ(t) + φn(t)tt0dt K(t − t )φn(t ) ˙φ(t )= φn(t)ξ(t) .Markovian Approximation:φn(t)tt0dt K(t − t )φn(t ) ˙φ(t ) φ2n(t) ˙φ(t)tt0→−∞dt K(t − t )→ Υ φ2n(t) ˙φ(t) .Markovian Equation of Motion:¨φ + Υ φ2n ˙φ + m2φφ +λ6φ3= φn(t) ξ(t)
  • 13. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationMarkovian ApproximationFigure: (a) mχ = 50H(0), (b) mχ = 150H(0) and (c) mχ = 250H(0)
  • 14. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationMarkovian ApproximationMore details about Markovian dynamics reliability:R. L. S. Farias, R. O. Ramos and L. A. da Silva, Nonlineareffects in the dynamics governed by non-Markovian stochasticLangevin-like equations, JPCS, in pressR. L. S. Farias, R. O. Ramos and L. A. da Silva, NumericalSolutions for non-Markovian Stochastic Equations of MotionComp. Phys. Comm. 180, 574 (2009).R. L. S. Farias, L. A. da Silva and R. O. Ramos,Non-Markovian stochastic Langevin equations: Markovian andnon-Markovian dynamics, Phys. Rev. E 80, 031143 (2009)R. L. S. Farias, R. O. Ramos and L. A. da Silva,LangevinSimulations with Colored Noise and Non-MarkovianDissipation Brazilian Journal of Physics, vol. 38 , no. 3B,(2008)
  • 15. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationMarkovian ApproximationConsidering the additive case:¨φ + [3H + Υ] ˙φ + V (φ) = ξ ,¨a = −8π3m2plρr + ˙φ2− V (φ) a ,˙ρφ = −3˙aa˙φ2− Υ ˙φ2+ ν ˙φ , ˙ρr = −4˙aaρr + Υ ˙φ2− ξ ˙φ .
  • 16. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic approach to Warm InflationBack to stochastic approach: implement Warm InflationModification of EoM:∂φ(x, t)∂t=13χ + Υe−2χt 2φ(x, t) −∂V (φ)∂φ+η(x, t)+ξ(x, t) .
  • 17. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic approach to Warm InflationBack to stochastic approach: implement Warm InflationModification of EoM:∂φ(x, t)∂t=13χ + Υe−2χt 2φ(x, t) −∂V (φ)∂φ+η(x, t)+ξ(x, t) .Global analysis:∂φ(t)∂t= −13χ + Υ∂V (φ)∂φ+ η(t) + ξ(t) ,η(t)η(t ) =χ34π2δ(t − t )ξ(t)ξ(t ) = βδ(t − t ) .
  • 18. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic approach to Warm InflationConsidering a general SDE of the form˙φ = h(φ, t) + g(φ, t)η(t) + f (φ, t)ξ(t) ,we obtain a Fokker-Planck equation for η(t)ξ(t ) = 0 of the form∂P(φ, t)∂t= −∂∂φD(1)+∂2∂φ2D(2)P(φ, t) .where the Kramers-Moyal coefficients are:D(1)(φ, t) = h(φ, t) +α2∂g(φ, t)∂φg(φ, t) +β2∂f (φ, t)∂φf (φ, t)D(2)(φ, t) =α2g(φ, t)2+β2f (φ, t)2.(1)
  • 19. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic approach to Warm InflationIn our particular case:∂P(φ, t)∂t= −∂∂φ−13χ + Υ∂V (φ)∂φ+∂2∂φ2χ34π2+β2P(φ, t) .A trick to obtain φn(t) :∂∂tφn(t) ≡∞−∞dφφn ∂∂tP(φ, t) .∂∂tφn(t) =χ38π2+β2n(n−1) φn−2(t) −n3χ + Υφn−1(t)∂V (φ)∂t.φ2(t) =3χ + Υ2m2φχ34π2+ β 1 − exp −2m2φ3χ + Υt .
  • 20. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic approach to Warm InflationThe Fokker-Planck operatorLFP ≡ −∂∂φ−13χ + Υ∂V (φ)∂φ+∂2∂φ2χ34π2+β2,is so that LFP = L†FP.⇒ Transformation of variables:t → (3χ + Υ)tφ → (3χ + Υ)χ34π2+βψ
  • 21. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic approach to Warm InflationThe Fokker-Planck becomes:∂P(ψ, t )∂t=2∂2P(ψ, t )∂ψ2+∂∂ψ∂ ˜V (ψ)∂ψP(ψ, t ) ,where˜V (ψ) ≡ (3χ + Υ)χ34π2+β−1V φ = (3χ + Υ)χ34π2+βψOne more transformationP(ψ, t ) = exp1 ˜V (0) − ˜V (ψ) F(ψ, t ) ,and then the Fokker-Planck equation becomes...
  • 22. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic approach to Warm Inflation...a Schr¨odinger-like equation with imaginary time:−2∂2∂ψ2+ U(ψ) F(ψ, t ) = −∂F(ψ, t )∂t, (2)withU(ψ) ≡12 ∂ ˜V (ψ)∂ψ2−∂2 ˜V (ψ)∂ψ2 . (3)Analogy with quantum mechanics:ψ | α, t =λψ | λ λ | α, t → F(ψ, t ) =λcλe− iEλtϕλ(ψ) ,withHϕλ(ψ) = Eλϕλ(ψ) , H ≡ −2∂2∂ψ2+ U(ψ) , t →it
  • 23. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationStochastic approach to Warm InflationThen using standard operator techniques and recovering theoriginal variables φ and t, we get:P(φ, t) =λcλe−m2φσ−1λt×Λ12π14√2λλ!m2φ(λ2 − 34 )m2φΛ−1φ −∂∂φλexpm2φΛ−1φ2,And the propagator K(φ2, t2; φ1, t1):K(φ2, t2; φ1, t1) = 1 − e−2m2φσ−1(t2−t1)− 12×exp−m2φΛ−1φ22 + φ21 − 2φ2φ1e−m2φσ−1(t2−t1)1 − e−2m2φσ−1(t2−t1)with Λ ≡ (3χ + Υ) χ34π2 + βand σ ≡ 3χ + Υ
  • 24. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationDiscussions and perspectivesPlanck energy scale at V (φ = φmax ) → breakdown ofsemiclassical picture of spacetime⇓P(φmax ) = 0⇓Bounded from above eigenvaluesH eigenvalues ∈ N → LFP eingenvalues ∈ Z−⇓Highest eigenvalue (˜λmax ) of volume-weighted FP equation(LFP → LFP + 3H) can be negative or positive: warm inflation→ ˜λmax > 0 → number of inflating domains increases withoutlimit. (to appear in JCAP)
  • 25. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationDiscussions and perspectivesWork out the local (φ(x, t)) analysisStudy the pathway to classicalization: when the thermalfluctuations overcome the quantum ones?Better quantify eternal inflation based on Warm Inflation
  • 26. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationThanks for your attention!!!