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ENFPC 2012

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  • 1. Outline Motivation Stochastic Warm Inflation model Results and conclusionsPower spectrum for inflation models with thermaland quantum noisesLeandro A. da Silva, Rudnei O. RamosRio de Janeiro State UniversityDepartment of Theoretical PhysicsXXXIII Encontro Nacional de F´ısica de Part´ıculas e Campos28/08/2012
  • 2. Outline Motivation Stochastic Warm Inflation model Results and conclusions1 Motivation2 Stochastic Warm Inflation model3 Results and conclusions
  • 3. Outline Motivation Stochastic Warm Inflation model Results and conclusionsContributions to the Power SpectrumGreat achievement of Inflationary models: fluctuations generatedduring an early phase of inflation provide a source of nearly scaleinvariant density perturbations, what nicely agrees withobservations.Cold inflation: quantum fluctuations of the inflaton field makethe sole contributionWarm inflation: the dominant contribution comes fromthermal fluctuations⇓These models look like extreme cases... Questions:What about the intermediate regime?How can we explicitly account for both quantum and thermalcontributions under the same framework?In what conditions do thermal fluctuations overcome thequantum ones?
  • 4. Outline Motivation Stochastic Warm Inflation model Results and conclusionsThe stochastic approach to warm inflationPossible answer: stochastic approach to warm inflationExtension of the original stochastic inflation model(Starobinsky ∼ 1987)Explicitly accounts for both quantum and thermal fluctuationsin a very transparent wayRecovers cold inflation and warm inflation in appropriate limitsStart point:ds2= dt2− e2Htd2x∂2∂t2+ (3H + Υ)∂∂t−1a22Φ +∂Veff,r(Φ)∂Φ= ξT ,Effective equation of motion ⇒ Langevin-like equation with localdissipation (Υ) and white thermal noise (ξT ).(Berera, A., Moss, I. G., Ramos, R. O. (2009).Reports on Progress inPhysics, 72(2))ξT (x, t)ξT (x , t ) = 2ΥTa−3δ(x − x )δ(t − t ) ,
  • 5. Outline Motivation Stochastic Warm Inflation model Results and conclusionsThe stochastic approach to warm inflationIntroducing the field splitΦ(x, t) = ϕ(t) + δϕ(x, t) + φq(x, t) ,whereϕ(t) =1Ω Ωd3xΦ(x, t) ,andφq(x, t) =d3k(2π)3/2W(k, t) φk(t)e−ik·xˆak + φ∗k(t)eik·xˆa†kWe get:∂2ϕ∂t2+ [3H + Υ(ϕ)]∂ϕ∂t+ V,ϕ(ϕ) = 0 ,∂2∂t2+ [3H + Υ(ϕ)]∂∂t−1a22+ Υ,ϕ(ϕ) ˙ϕ + V,ϕϕ(ϕ) δϕ = ˜ξq + ξT ,
  • 6. Outline Motivation Stochastic Warm Inflation model Results and conclusionsThe stochastic approach to warm inflation˜ξq = −∂2∂t2+ [3H + Υ(ϕ)]∂∂t−1a22+ Υ,ϕ(ϕ) ˙ϕ + V,ϕϕ(ϕ) φq ,˜ξq → quantum noise term; ˜ξq(x, t), ˜ξq(x , t ) = 0 → ClassicalbehaviourIn terms of z =kaHcoordinate, the fluctuation EoM becomes:δϕ (k, z) −1z(3Q + 2)δϕ (k, z) + 1 + 3η − βQ/(1 + Q)z2δϕ(k, z) =1H2z2ξT (k, z) + ˜ξq(k, z) .β and η → Slow-roll parameters, Q =Υ3H
  • 7. Outline Motivation Stochastic Warm Inflation model Results and conclusionsThe stochastic approach to warm inflationFrom the solution of EoM, we write the inflaton power spectrum:Pδϕ =k32π2d3k(2π)3δϕ(k, z)δϕ(k , z) = P(th)δϕ (z) + P(qu)δϕ (z) .P(th)δϕ (z) =ΥTπ2∞zdz z 2−4νG(z, z )2, P(qu)δϕ = [2n(k) + 1]k32π2|Fk(z)|2Fk(z) =π3/2zνH4k3/2∞zdz (z )3/2−ν[Jα(z)Yα(z ) − Jα(z )Yα(z)]×βQ1 + QW,z (z )H(1)µ (z ) + z W,z (z )H(1)µ−1(z )−3QzβQ1 + Q+ η W(z )H(1)µ (z ) − 3QW(z )H(1)µ−1(z )−π3/2zνH4k3/2∞zdz (z )5/2−ν[Jα(z)Yα−1(z ) − Jα−1(z )Yα(z)] W,z (z )H(1)µ (z )
  • 8. Outline Motivation Stochastic Warm Inflation model Results and conclusionsThe stochastic approach to warm inflationWhich filter function? Step filter case: W(k, t) ≡ θ(k − aH)Pδϕ(z) ≈HT4π23Q2√π22αz2ν−2α Γ (α)2Γ (ν − 1) Γ (α − ν + 3/2)Γ ν − 12 Γ (α + ν − 1/2)+HT2eH/T − 1+ 1 z2η,whereν = 3(1 + Q)/2 , α = ν2 +3βQ1 + Q− 3ηRestrict model parameters → Curvature perturbations, spectral index andtensor to scalar ratio:∆2R =H2˙φ2Pδϕ , ns = 1 +d ln ∆2Rd ln k, r =2 H2(1 + Q)2PδϕWMAP 7 years: ns = 0.967 ± 0.014 (at 68% CL), r < 0.24(WMAP+BAO+SN) Martin, J., Ringeval, C. (2010). Physical Review D,82(2), 1-17.
  • 9. Outline Motivation Stochastic Warm Inflation model Results and conclusionsResults and conclusionsV (φ) =10−14m4PpφmPpFigure: Full line: p = 2, dashed line: p = 4 and the dotted line: p = 6.
  • 10. Outline Motivation Stochastic Warm Inflation model Results and conclusionsResults and conclusionsFigure: (a) p = 2, (b) p = 4, (c) p = 6
  • 11. Outline Motivation Stochastic Warm Inflation model Results and conclusionsThe stochastic approach to warm inflationRecovering Cold Inflation and Warm Inflation:Q = 0 limit:ns = 1 + 2η − 6 .Q 1 limit:ns = 1 + 2η − 6 + (8 − 2η)Q + O(Q2) .Large Q, T limit:ns = 1 +1Q−94−34β +32η + O(Q−3/2) + O(1/T) ,
  • 12. Outline Motivation Stochastic Warm Inflation model Results and conclusionsResults and conclusionsStochastic warm inflation is able to describe in a natural wayboth cold and thermal fluctuations.It recovers Cold Inflation in Q = 0 limit and Warm Inflation inQ 1 limit.Thermal fluctuation and dissipation make V ∼ φp (p > 3)again compatible with observations.Perspectives: check the results using a more “physical” filter,like a gaussian one (work in progress)Impose others observational constraints, like running ns (workin progress)