ENFPC 2010

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ENFPC 2010

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 26. Outline Stochastic Approach to Inflation Warm Inflation Stochastic approach to Warm InflationThanks for your attention!!!

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