Fluctuation-dissipation dynamics in the early Universe

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1 Fluctuation-dissipation dynamics in the early Universe Arjun Berera The University of Edinburgh Miami 2014, Florida, USA, December 2014 T H E U N I V E R S I T Y O H F E D I N B U R G

2 Review warm inflation Overview Density fluctuations at finite temperature Quantum field theory calculation - dissipation, radiative corrections, thermalization λφ 4 model warm inflation predictions compared to Planck results Other applications of fluctuation-dissipation dynamics in cosmology

3 Scalar field ( inflaton ) dynamics V ρ = φ 2 +V(φ)+ ( φ)2 2R 2 φ Cold inflation: p = φ 2 Just Choose V(φ) V(φ) ( φ)2 6R 2 φ + 3H φ + V (φ) = 0 Potential energy dominated 3H φ φ, slow-roll Warm Inflation: Υ dominates φ + 3H φ + Υ φ + V (φ) = 0 Slow-roll now means Υ φ 3H φ, φ, overdamped

4 Two basic inflation pictures Cold Inflation V(φ) Inflation Reheating φ Warm Inflation V( φ) Inflation + continuous radiation production φ φ φ φ ρ ρ v ρ ρ r t ρ r ρ v t

5 Energetics Consider GUT scale inflation: M GeV ρ v M GeV 4 H GeV T > H requires (influences structure formation) T > 1GeV requires (makes reheating unnecessary) > 1 part in of ρ v ρ r > 1 part in of ρ v ρ r Inflation pictures 1. Cold inflation: basic assumption is no dissipation 2. Warm inflation: radiation production inherent [AB, (1995)] Theoretical consideration Equipartition Hypothesis of Statistical Mechanics: Scalar field should distribute its energy evenly amongst all degrees of freedom. Dynamical Question: Will relevant time scales during inflation prohibit the minute radiation production given above?

6 Nature of the fluctuations Thermal/quantum fluctuations leave ripples in early universe - imprinted on CMB at Last Scattering - amplified by gravity to make structure Cold inflation: Quantum (Guth, Pi 82; Hawking 82; t 0 γ γ 1 2 Last Scattering Surface Warm Inflation: Thermal Starobinsky 82; δφ = k1/2 F T1/2 2π Bardeen et. al. 83) Υ < 3H : k F = H (Moss 85, AB,Fang 95) δφ = H 2π δρ ρ = Hδφ φ HORIZON EXIT Υ > 3H : k F = ΥH (AB 99)

7 Worked example - no η-problem Model: V = m2 2 ϕ2 = N e 2 2 ϕ 0 m m Υ, T m3/4 m 1/4 P ϕ1/4 0 P Υ 1/4, δρ ρ (ϕ 0 m )3/8 ( m Υ ) 9/8 P For N e = 60, δρ ρ = 10 5 = ϕ 0 m 6 108, Υ m P e.g., m = 10 9 GeV = H m 0.17, ϕ 0 m P , T 10 4 m No η-problem: m > H No ϕ amplitude problem: ϕ < m P No graceful exit problem: inflation RD automatic No quantum-to-classical trans. problem: δϕ classical

8 Scalar field (Φ) QFT dynamics L Φ = 1 2 ( µφ) 2 V (Φ) N ψ L I = Φ h j ψ j ψ j, 1 j=1 2 N χ j=1 g 2 j Φ2 χ 2 j + Decompose into background ϕ and fluctuations φ: Φ = ϕ + φ Equation of motion for ϕ(t) from quantum effective action: δγ V = ϕ(t) + 3H ϕ(t) + δϕ ϕ + d 4 x Σ R (x x ) ϕ = 0 Adiabatic evolution: ϕ = ϕ(t t ) + Yielding a friction term at leading order

9 Closed Time Path approach - Goal Compute Observable Thermal initial state at T < : Ô(t) Tr(ˆρ(t)Ô) Tr(ˆρ(t)) ρ(t < ) = exp( βh) = U(T < iβ, T < ) ( U(t, t ) exp[ ih(t t )], time evolution operator) Thus Ô(t) = Tr[U(T < iβ, T < ) U(T <, t) Ô U(t, T < )] Tr[U(T < iβ, T < )] Also add large positive time T > Ô(t) = Tr[U(T < iβ, T < ) U(T <, T > ) U(T >, t) Ô U(t, T < )] Tr[U(T < iβ, T < )]

10 Closed Time Path approach - Method + O^ T < t T > T < iβ Can express as a path integral recall U(t, t ) DΦexp ( t ) i t d4 xl[φ] = Z[J +, J, J β ] = Tr[U(T < iβ, T < ; J β )U(T <, T > ; J )U(T >, T < ; J + )] DΦ + DΦ DΦ β exp ( i T > d 4 x[l J+ [Φ + ] L J [Φ ]] + i T < T < iβ T < d 4 xl Jβ [Φ β ] ) e.g. scalar field theory: L J [Φ] = 1 2 [ µφ µ Φ m 2 Φ 2 ] λ 4! Φ4 + JΦ

11 Ñ ¾ Đ³ ³ ¾ ³ ½ ¾ Đ³ Ñ ¾ ³ ½ ¾ ³ ½ ³ ½ ¾ ¾ ½ Å ³ ½ Å ³ ½ Å ³ ½ Å ³ Example:S = d 4 x How to obtain the ϕ-effective EOM Tadpole Method: demand φ = 0 and compute [ 1 2 µ Φ µ Φ m2 2 Φ2 λ 4! Φ4 ], Φ = ϕ + φ = S = d 4 x[ ϕ 1 2 [ + m2 ]ϕ φ 1 2 [ + m2 ]φ φ[ + m 2 ]ϕ λ 4! (ϕ4 + 4ϕ 3 φ + 6ϕ 2 φ 2 + 4ϕφ 3 + φ 4 )] φ = 0 at lowest nontrivial order = ½ ¾ ³ ½ ³ ½ ³ ¾ Æ ³ ¾ ¾ ³ ¾ Æ µ x i x G 0ij φ (x,x ) j i,j = ± ½ ¾ ³ ½ ³ ½ ³ ¹ ¾ ³ ¾ Æ ³ ¾ ³ ¾ ¾ Æ µ ¼

12 ¼ Ü ¼ Ü Ü µ ¼ Ê ¾ º º ¾ Æ ¾ ¾ ¾ Æ ¾ Tadpole Method (cont.) - Explicit EOM ( ) =0 = Æ ¾ = [ ] = 0, now convert to analytic expression ½ ¾ ³ Ø ¼ µ Ê Ý Æ ¼ ݵ Ü ¼ ݵ Ü Ýµ ܼ ݵ ³ ¾ Ø Ý µ ܼ ¾ ³ Ø ¼ µ Ê ÝÁÑ Ü ¼ ݵ Ü ¼ ݵ ³ ¾ Ø Ý µ Ø Final Expression: ϕ + 3H ϕ + m 2 ϕ + λ 6 ϕ3 + local 1 loop V eff terms λ2 ϕ(t ) λ2 d 4 yim[g ++ φ (x, y)g ++ φ (x, y)g ++ φ (x, y)]ϕ(t y ) = 0 d 4 yim[g ++ φ (x, y)g ++ φ (x, y)]ϕ 2 (t y )

13 Green s functions relations (AB, Moss and Ramos, in preparation 2007) One-loop effective equation of motion: [ ] + M 2 φ ϕc (x) + λ 3! ϕ3 c(x) ϕ c(x) +ϕ c (x) d 3 x t = ϕ c (x)ξ 1 (x) + ξ 2 (x) 2 d 3 x ϕ 2 c(x, t)d 1 (x x,0) t d 3 x dt ϕ c (x, t ) ϕ c (x, t )D 1 (x x, t t ) + where defining C i (x x, t t ) = t D i (x x, t t ) C 1 (x x, t t ) = λ 2 Im d 3 x ϕ c (x, t)d 2 (x x,0) [ ] 2 G ++ φ (x, x ) sgn(t t ) + 4g 4 Im [ G ++ χ j (x, x ) ] 2 sgn(t t ) dt ϕ c (x, t )D 2 (x x, t t ) [ C 2 (x x, t t ) = λ2 3 [ ] 3 Im G ++ φ (x, x )] sgn(t t ) + 4g 4 Im G ++ χ (x, x )G ++ φ (x, x )G ++ χ (x, x ) sgn(t [ ] ξ 1 (x)ξ 1 (x ) N 1 (x x, t t ) = λ2 2 2 Re G ++ φ (x, x ) + 2g 4 Re [ G ++ χ (x, x ) ] 2 ξ 2 (x)ξ 2 (x ) N 2 (x x, t t ) = λ2 6 Re [ G ++ φ (x, x )] 3 + 2g 4 Re [ ] G ++ χ (x, x ) 2 G ++ φ (x, x )

14 Fluctuation dissipation theorem - relation Relation between real and imaginary parts of Green s function leads to generalized fluctuation-dissipation theorem (in Fourier space): Ñ 1 (p, ω) = 2ω [ n(ω) ] Γ 1 (p, ω) where D i (p, ω) = 2 Γ i (p, ω) Relates dissipation and noise In local form, i.e. when N(t t ) N 0 δ(t t ), get familiar expression: N 0 = Ñ(0) = 2T Γ(0)

15 Warm inflation [AB PRL 75, 1995] Stochastic Evolution equation includes dissipation and noise (AB and Fang, PRL74, 1912 (1995)): φ 1 a 2 2 φ + 3H φ + Υ φ + V (φ) = ξ Dissipation term leads to radiation production during inflation, Density perturbations: R(k) = Ḣ φ Strong dissipative regime: Weak dissipative regime ρ r = 4Hρ r + Υ φ 2 δφ, δφ = k1/2 F T1/2 2π Υ > 3H, T > H, k F = ΥH Υ < 3H, T > H, k F = ΥH(FD), H(thermal)

16 Inflaton mass Coupling of inflaton induces χ mass term: m χ = gϕ Thermal corrections to inflaton mass: { m 2 g 2 φ = T 2, g 2 e mχ/t, m χ T m χ > T Inflaton can not be directly coupled to radiation [AB, Gleiser, Ramos (1998); Yoyoyama,Linde (1998)]

17 Two-stage dissipation SUSY protects inflaton potential from radiative corrections Superpotential: W = F(Φ) + g 2 ΦX2 + h 2 XY 2 Heavy fields decay into light degrees of freedom [AB,Ramos (2003)] L = g 2 φ 2 χ 2 + hmχσ 2 Dissipation produces radiation bath (σ) Heavy (χ)-fields protect inflaton potential from radiative corrections

18 Inflaton effective potential SUSY broken at finite energy density and temperature [Hall,Moss (2005), Bastero-Gil,AB,Ramos,Rosa (2013)] σ i ψ σi ψ χ ψ σi ψ χ χ ψ σi χ m 2 χ = g 2 ϕ 2 ± g 2 V (ϕ) + h2 N y 8 T2 σ i χ χ χ σ i χ m 2 ψ χ = g 2 ϕ 2 + h2 N y 8 T2 σ i Flatness of potential is protected: V 1 (ϕ, T) = g2 N χ 32π2V (ϕ)log m2 χ µ 2 small radiative corrections for g 2 N χ < 10 and h 2 N y < 1

19 Dissipation coefficient One-loop correction to inflaton self-energy: [Moss,Xiong (2008); Bastero-Gil,AB,Ramos (2011)] Υ = 4 T g4 ϕ 2 d 4 p (2π) 4ρ2 χn B (1 + n B ) Spectral function (Briet-Wigner): where ω p = p 2 + m 2 χ ρ χ = 4ω p Γ χ (p 2 0 ω2 p )2 + 4ω 2 p Γ2 χ χ g 2 ϕ 2 2 g 2 ϕ 2 2 χ

20 Dissipative coefficient g 2 T h N Y h N Y 0.01 h N Y 0.1 h N Y m Χ T Υ = 0.02h 2 N y N χ T 3 ϕ (g 2 N χ < 10, h 2 N y < 1) g 2 N χ 2π h 2 N y m χ Te m χ/t Virtual modes (low-momentum) + on-shell modes (pole)

21 Thermalization [Bastero-Gil,AB,Ramos,Rosa (2013)] Decay rates Γ i > H, ϕ ϕ : χ σ + σ, ψ σ + ψ σ, σ + σ + φ χ R χ I + φ I, ψ χ ψ φ Scattering rates nσ i v > H: σ σ σ χ σ σ (a) σ σ (b) σ σ ψ χ σ ψ σ χ σ ψ σ χ ψ σ ψ σ ψ σ ψ σ σ (c) (d) (e) ψ σ ψ σ σ χ σ σ χ φ φ σ φ σ χ φ (f) (g)

22 Warm inflation dynamics Superpotential: W = W(Φ) + gφx 2 + hxy 2 Dissipative coefficient: Υ C φ T 3 φ 2 C φ 0.02h 2 N χ N y Slow-roll: φ V φ [3H(1+Q)] Q Υ/(3H) 4ρ R 3Q φ 2 ρ R = π 2 g T 4 /30 ǫ φ, η φ < 1 + Q Q < (>)1 - weak (strong) dissipative regime

23 Observational consequences Adiabatic scalar perturbations from thermal fluctuations [Bartrum et al. (2013)]: P R = ( ) 2 (H ) ( H 2 ϕ 2π 1 + 2n + 2 3πQ 3 + 4πQ T H ). Tensor-scalar ratio [Bartrum,AB,Rosa (2013)]: r = 16ǫ φ (1 + Q) 5/2 H T < 8 n t Presence of radiation during inflation lowers tensor-scalar ratio [Bastero-Gil + AB (2009)]

24 λφ 4 potential - weak dissipation [Bartrum, et al (2013)] λ 10 14, < g 4 N χ 10 6 Planck (Dec. 2014, Twitter) n s = ± , r < 0.09 (dust-cleaned WMAP low l polarization)

25 ( T T Isocurvature mode and r eff [Bastero-Gil et al (2014)] ) 2 P ζ (1 + 4Bm 2 + 4B m + 56 ) ( r P ζ ) 6 r eff ζ = Hδρ/ ρ and B m S m /ζ, with matter isocurvature perturbations S m = Ω c Ω m S c + Ω b Ω m S b where S b = δρ b ρ b 3 4 δρ γ ρ γ,

26 Bispectrum: Non-gaussianity [Bastero-Gil, et al (2014)] B(k 1, k 2, k 3 )δ(σk) = cyc Φ 1 (k 1, t f )Φ 1 (k 2, t f )Φ 2 (k 3, t f ) Φ i (k, t) gauge invariant variable of order i. Non-linearity parameter: f NL = 18B(k, k, k)/5p(k) 2 Planck bounds: f local NL 10 3 = 2.7 ± 5.8 fequi NL = 42 ± 75 fs NL = 4 ± K r =/ 0, Γ=/ 1 K r =/ 0, Γ=1 K r =0, Γ=/ 1 K r =0, Γ=1 T/H=30 f NL 10 1 T/H=20 T/H=10 f NL 10 1 T/H= Q Q

27 Template shapes of bispectrum Discriminate models by determining the shape of the bispectrum Template bispectra: P Local, BL = cyc k1 3k2 3 P Equilateral, BE = cyc 3k1 3k2 3 2k1 2k2 2k k1 1k2 2k cyc k1 k2 (k1 + k2 )k1 k2 Strong warm inflation BS = P Weak warm inflation BW = P 3 3 cyc k1 k2 k1 k2

28 Bispectrum shape Matching numerical bispectrum to given template, distance function: B 1 B 2 = B dk 1 dk 2 dk 1 (k 1, k 2, k 3 )B 2 (k 1, k 2, k 3 ) 3 δ(k 1 + k 2 + k 3 );. P(k 1 )P(k 2 )P(k 3 ) Q 1 B W Q 1 B S

29 Coupled cosmological stochastic equations [Bastero-Gil,AB,Moss,Ramos (in progress)] Scalar field + radiation equations with dissipation, viscosity and associated stochastic forces: 2 φ(x, t) + Υ φ(x, t) + Ω,φ = (2ΥT) 1/2 ξ (φ) (x, t) δ ρ (f) + (ρ (f) + p (f) ) δu (f) + s,φ δq = δq (φ) {(ρ (f) + p (f) ) δu (f) } + δp (f) η 2 δu (f) where δu (f) = δv (f) ( ζ + 1 ) η δu (f) = δq (φ) 3 + Σ δq (φ) = δj (φ) ξ (φ) (x, t)ξ (φ) (x, t ) = δ (3) (x x )δ(t t ) δq (φ) = δυ φ 2 2Υ φ δ φ + (2ΥT) 1/2 φ ξ (φ) + P δj (φ) = Υ φ δφ + 2 Ṗ P = C P (2ΥT) 1/2 φ 2 ξ (φ) Σ ij (x, t)σ kl (x, t ) = 2T (ηδ ik δ jl + ηδ il δ jk + (ζ 23 ) η)δ ijδ kl δ (3) (x x )δ(t t )

30 Fluctuation-dissipation in cosmology [AB (1996), Bastero-Gil et al (2014), Bartrum et al (2014)] Treatment of early universe cosmological problems usually classical evolution with some quantum effects superposed, i.e. inflation, reheating, phase transitions etc... This is an approximation for a multi-particle problem Include effective dynamics from all these degrees of freedom using dissipation terms which by fluctuation-dissipation relations will be accompanied by stochastic forces Learn from warm inflation that dissipation and stochastic effects do affect observable predictions. These effects can explain the Planck data within a well motivated model. An alternative to developing elaborate models which now even involve quantum gravity. Lesson here could apply to other cosmological physics. Dissipation and noise applied to phase transitions, reheating, Axion inflation, thermal fluctuations models, trapped inflation, string cosmology...

31 Warm inflation summary Dissipative effects and temperature during inflation alter tensor-scalar ratio and tilt QFT solutions are perturbative but have large number of fields N χ 10 6 Models exist with large GUTs, branes, extra-dimensions Large number of fields is a tuning but in strong dissipative regime there is no η-problem, thus removing a tuning from cold inflation models Expect fewer fields in pole-dominated dissipative regime Bulk and shear viscosity affect the radiation bath and density perturbations [Bastero-Gil et al. (2014)] Control gravition over-abundance [Sanchez et al. (2011); Bartrum et al (2013)] Baryogenesis [Bastero-Gil,AB,Ramos,Rosa (2012)]

32 Conclusion Excellent consistency with Planck/BICEP2 data is still possible for one of the simplest inflation models by including some thermodynamic considerations All the good properties of the φ 4 inflation model remain in tact: Renormalizable - familiar quantum field theory interactions Initial conditions - eternal inflation still occurs in the warm regime [AB, Rangarajan (2013)] General warm inflation picture of interactions, particle production and dissipation during inflation could have relevance in explaining CMB data. Other dynamical realizations and models of warm inflation should be explored

33 Two stage dissipative mechanism (AB and R. Ramos, PRD 63, (2001); I. G. Moss and C. Xiong, hep-ph/ ; Bastero-Gil, AB, Ramos, [hep-ph]) Basic Lagrangian - inflaton field coupled to heavy field (> T) which in turn coupled to light fields (< T) Examples: φ χ ψ with m χ > 2m ψ > m φ L I = g 2 1 φ φχ χ h 2 [χ ψ σ P R ψ σ + χ ψ σ P L ψ σ ], Υ = 0.11g1 4 h4 T7 2ϕ2 m 2 ψ χ φ χ y with m χ > 2m y > m φ L I = g 2 1 φ φχ χ h 1 M[χ σ 2 + χ(σ ) 2 ], Υ = 0.026g 4 1 h4 1 ϕ2t3 M 4 m 8 χ φ ψ χ ψ d, y with m χ > m ψy + m y > m φ L I = 1 2 g 2 ϕ ψ χ ψ χ h 3 [σ ψ χ P R ψ σ + σ ψ χ P L ψ σ ], Υ = g 2 2 h4 3 T 5 m 4 ψ χ

34 Physical picture of two stage dissipative mechanism (Moss, Graham, PRD78, (2008); AB, Ramos, PLB607, 1 (2005)) Particle production rate of radiation bath particles due to interactions: ṅ = Im 2 t dt e iω(p)(t t ) 2ω(p) Σ 21 (p, t, t ) Σ 21 = Σ σ,21 + Σ ψχ,21 is the sum of the σ and ψ σ self-energies Noting ρ r = d 3 p (2π) 3ω(p)n implies: Υ = ρ r ϕ 2 = 1 ϕ 2 d 3 p (2π) 3 [ ωσ (p)ṅ σ + ω ψσ (p)ṅ ψσ ]

35 Interaction generic in inflation models g 2 φ 2 χ 2 generic to inflation models (for reheating) for g > 10 3 require SUSY for flat potential Minimal SUSY model with this interaction W = λφ 3 + g ΦX 2 + h XY (Φ = φ + θψ + θ 2 F, X = χ + θψ χ + θ 2 F χ, Y = y + θψ y + θ 2 F y ) = L int 1 4 g2 φ 2 χ gφψ χψ + g 2 φ ψ χ ψ χ hχ ψ y ψ y + For φ ϕ 0 SUSY is broken with m χ1 m ψχ m χ2 (V 1 loop gλϕ 4 < V tree λϕ 4, so flatness preserved) = dissipative mechanism through φ χ ψ y + ψ y,2y,..., just like our toy model, so all results follow

36 Local limit of noise and dissipation [AB, Moss and Ramos, (2007)] f(t) slowly varying on timescale τ if Fourier transform f(ω) satisfies f(ω) = 0 for ω > 2π/τ kernel function K(t) described as localized on a timescale τ with accuracy 1 ǫ if Fourier transform K(ω) satisfies K(ω) K(0) K(0) < ǫ for ω < 2π/τ f(t) slowly varying on a timescale τ and K(t) localized on a timescale τ, K(t t )f(t )dt = K(0)f(t) + O(ǫ) K(t) K(0)δ(t t ) (Note: if K(ω) analytic in ω kernel can be localised and kernel admits a local derivative expansion. In general, derivative expansion might not exist, even when the kernel is localized in above sense.)

37 Non-gaussianity Power spectrum: R(k 1 )R(k 2 ) = (2π) 3 δ(k 1 + k 2 )P R (k 1 ) Bispectrum: R(k 1 )R(k 2 )R(k 3 ) = (2π) 3 δ(k 1 + k 2 + k 3 )B R (k 1, k 2, k 3 ) e.g. local model: B R (k 1, k 2, k 3 ) = 6 5 f NL(P R (k 1 )P R (k 2 ) + perm.) f NL - Nonlinearity parameter WMAP bounds: 10 < fnl local < 74 (k 3 k 1 k 2 ); 214 < f equil NL < 266 (k 3 k 1 k 2 ) Planck (f NL O(10)) standard inflation models f NL 0(ǫ, η) - Single field inflation canonical kinetic terms slow-roll initial vacuum state (Maldacena 03, Acquaviva, et al., 03)

38 Non-gaussianity in warm inflation (Gupta, et al., PRD66, (2002); Moss and Xiong, JCAP 0704, 007 (2007)) Φ(x) + (3H + Υ) Φ(x) + Υa 2 v α α Φ(x) a 2 2 Φ(x) + V Φ = ξ(x) 2nd order term V - scalar velocity perturbation Bispectrum B v r(k 1, k 2, k 3 ) 18L(Q) cyclic P R (k 1 )P R (k 2 ) ( 1 k k 2 2 ) k 1 k 2 L(Q) ln(1 + Q/14) Relation for f NL : Q 100 f NL O(30) 15L(Q) < f v NL < 33 2 L(Q)

39 Observational tests of inflation Spectra of energy density fluctuations (scalar spectra) Spectra of gravitational waves (tensor spectra) Non-gaussian deviations Isocurvature fluctuations Present day cosmological constant Particle Spectra EXPERIMENTS CMB - COBE, MAP, Planck, Boomerang, Maxima Redshift surveys - Sloan, 2df... Big Bang Nucleosynthesis Supernovae 1A Hubble Space Telescope

40 Summary of WMAP data WMAP 7 year data and angular power spectra: Scalar amplitude: A S = (2.43 ± 0.11) 9 Scalar spectral index: n S = ± Running of spectral index: dn S /dln k = ± Tensor to scalar ratio: r < 0.36(95%CL) Nongaussianty: fnl loc = 32 ± 21 (68% CL) Consistent with Gaussian within 95% CL Yadav and Wandelt, 2008 found in WMAP 3 year data 27 < fnl loc < 147 (95% CL) with rejection of floc NL = 0 at 2.8σ

41 What Planck can achieve ESA Planck Bluebook, 2005 CMB power spectrum for concordance ΛCMB model (red line) compare WMAP to (projected) Planck data

42 Distinguishing models ESA Planck Bluebook, 2005 Solid red lines are concordance ΛCDM model with spectral index n S = 0.95 and 1 WMAP has difficulty distinguishing between the models vs. Planck can distinguish very well

43 Parameter forecast for Planck ESA Planck Bluebook, 2005 Bartolo, et al., 2005 Error on scalar index reduced to half a percent Tensor-scalar ratio detectable down to 0.05 (optimistic) If no detection of r in Planck low energy scale V 1/4 of inflation, V 1/4 = r 1/4 GeV

44 Standard inflation models - predictions Slow roll parameters: ǫ m2 ( pl V ) 16π V 1 η m2 pl 8π ( V V ) 1 f NL < 1 in all models p = 4 ruled out by WMAP data Kinney, et al., 2001 If r < 0.1 is found by Planck, monomial cold inflation models ruled out If f NL > 5 is found by Planck all these simplest of inflation models would be ruled out

45 Superpotential: Warm inflation models (AB and Ramos, PLB 607, 1 (2005)) W = W(Φ) + gφx 2 + hxy 2 Dissipative coefficient: Υ C φ T 3 φ 2 C φ 0.64h 4 N χ N 2 decay Slow-roll: φ V φ [3H(1+Q)] Q Υ/(3H) 4ρ R 3Q φ 2 ρ R = π 2 g T 4 /30

46 Warm inflation models - monomial potential (Bastero-Gil and AB, Int. J. Mod. Phys A24, 2207 (2009)) V = V 0 [ φ m p ] p d(t/h) dn e > 0 Weak DR Strong DR Solves eta - problem, m φ > H Solves large φ amplitude problem - φ < m P C φ 0.16N χ N 2 decay T/H > 1

47 Bulk viscosity and background evolution (Mimoso,Nunes,Pavon PRD73, 06; Del Campo, Herrera, Pavon PRD75 07; Del Campo et al., ) Viscous pressure arising from interactions of particles and decay within fluid: Π 3ζ b H From calculation of ζ b : ζ b ρ s R > 0 Π < 0 Modified equation of state: p R = (γ 1)ρ R, 1 γ 2 Evolution of radiation: ρ R + 3H(ρ R + p R + Π) = Υ φ 2 Π < 0 increases the duration of inflation Presence of bulk pressure also modifies primordial spectrum

48 Stringy warm inflation (AB, Kephart, PRL 83, 1084 (1999); Bastero-Gil, et al., [astro-ph.co]) Recurrent problem in embedding inflation in string models is the eta - problem, i.e. quantum corrections and SUGRA contributions to inflaton potential ruin required flatness Warm inflation solution to the eta - problem: large dissipation Υ H V H 2, i.e. much bigger than scale for SUGRA corrections. Necessary ingredient for warm inflation is large number of fields - naturally available in string theory, i.e. moduli fields, branes, Kaluza-Klein modes come in the hundreds of thousands. Example: trapped warm inflation - scalar inflaton field trapped in decaying oscillation about ESP, coupling to other fields leads to warm inflation.

49 Refining the theory Establishing fluctuation-dissipation theorem Origin of derivative expansion Finite temperature effective potential in SUSY models Origin of dissipation from more general non-equilibrium derivation Higher loops, resummations etc... Apply to more models

50 Summary of warm inflation Treats dynamical effects of inflaton interacting with other fields during inflation Model Building Can have inflation models with m φ > H Particle physics during inflation phase, eg. magnetic fields baryogenesis... Observational implications Running spectral index in simple models Blue and red spectra are possible Non-gaussianity at strong dissipation f NL O(10) low tensor-scalar ratio r 0.1

51 Green s function G ij φ (x, x ) - Basic Properties G φ (x, x ) = G++ φ (x, x ) G + (x, x ) G G + φ φ (x, x ) φ (x, x ) = ( i T+ φ(x)φ(x ) i φ(x )φ(x) i φ(x)φ(x ) i T φ(x)φ(x ) Fourier space: G φ (x, x ) = i d 3 q (2π) 3eiq.(x x ) G φ (q, t t ) G ++ φ (q, t t ) = G > φ (q, t t )θ(t t ) + G < φ (q, t t )θ(t t), G φ (q, t t ) = G > φ (q, t t )θ(t t) + G < φ (q, t t )θ(t t ), G + φ (q, t t ) = G < φ (q, t t ), G + φ (q, t t ) = G > φ (q, t t ). Hermiticity: G > φ (q, t t ) = G > φ (q, t t) Continuity: d dt [ G > (q, t t ) G > φ (q, t t) ] t=t = iδ(t t )

52 Green s Function - equilibrium approximation Gives lower bound estimate of dissipative effects (Moss and Xiong, hep-ph/ ) Low temperature regime (m χ < T): G equil (k, t) = where ω k = k 2 + m 2 χ, f(ω, k, t) = exp(iωt) 4πω i 2(ω k iγ χ ) exp[ i(ω k iγ χ)t] + f(ω k iγ χ, t) f(ω k + iγ χ, t) At small time, i.e. t τ χ = Γ 1 χ exponential decay approximation. E 1(i( k + ω)t) exp( iωt) E 1 (i( k ω)t) 4πω log m2 χ Γ 2 χ At larger time, power-law decay behavior. Leads to dissipative coefficient: Υ equil (ϕ, T) = g 2 h 4 ( gφ m χ, the behavior same as the ) 4 T 3 m 2 χ

53 Green s Function - equil. approx. (cont) (Hosoya and Sakagami, PRD 29, 2228 (1984); Berera, Ramos, Gleiser, PRD 58, (1998)) High Temperature limit: Υ 132 ( ) 2T πt ϕ2 N χ ln 2µ(T) where thermal mass µ(t) = gt 12

54 Local limit of one-loop kernels Homogeneous field ϕ c (x) = ϕ c (t) Ñ 1 (ω) g 4 ( e βω + 1 ) d 3 k (2π) 3 x ω/t g 4 h 4M4 T 4 F 1 (βω) m 8 χ dω 2π n(ω )n(ω ω ) ρ χ (k, ω ) ρ χ (k, ω ω )

55 Warm inflation on a computer [AB, G. Lacagnina (lattice gauge), C. Verdozzi (condensed matter)] Purpose: Study of overdamped motion and its universal features study of how equipartition is achieved Feasible goals: Numerical simulations of classical/quantum models from condensed matter: Fermi-Paste-Ulam Caldeira-Leggett Simulations of lattice quantum field theory models: Caldeira-Leggett, φ 4... Example of overdamping in the classical regime

56 Improving parameter estimation 1 and 2 σ contour regions for cosmological parameters blue contours WMAP 4 red contours Planck projected after 1 year of observation ESA Planck Bluebook, 2005