Preheating : Density Perturbations from the Shock-in-Time Connecting Inflation to the Hot Big Bang

Transcription

1 Preheating : Density Perturbations from the Shock-in-Time Connecting Inflation to the Hot Big Bang Jonathan Braden University College London LCDM, January 29, 2015 w/ J. Richard Bond, Andrei Frolov, and Zhiqi Huang

2

3 Why Is This Regime Interesting Theoretical Consistency Inflationary cosmology is incomplete without this transition Understand nonequilibrium quantum field theory More practical concerns N ln(a 0 /a end ) needed to match observations to inflationary models Production of nongaussian density perturbations [Bond,Frolov,Huang,Kofman],[Rajantie,Chambers] tensors [Easter,Giblin,Lim],[Figueroa,Garcia-Bellido],[Dufaux,Felder,Kofman,Huang] Linear structure growth depends on background expansion Nonequilibrium - baryogenesis?, nonthermal DM production?

4 Outline Review of linear preheating Nonlinear stage of preheating Entropy Production Production of adiabatic perturbations from isocurvature modes

5 Starting the Big Bang Inflation Hot Big Bang Cold (T 0), S V 0 Few active d.o.f. Hot (T > MeV ), S V g eff (T )T 3 Many active d.o.f. Huge entropy production But how does it happen?

6 Preheating : Linear Regime [Kofman, Linde, Starobinski] QFT = inhomogeneity [δφ i, δ φ i ] 0 = δφ k 2, δ φ i 2 0 Generic Equations for L mat = 1 2 µφ I µ φ I V ( φ) ( ) k tt (a 3/2 2 δφ) i + a 2 + F ij( φ(t), a) (a 3/2 δφ j ) = 0 Harmonic oscillator with ( periodic) time-dependent frequency Exponential Growth of Inhomogeneities Narrow Parametric Resonance Broad Parametric Resonance Tachyonic Resonance Spinodal (Tachyonic) Instability

7

8 Preheating : Nonlinear Evolution of ln ρ/ ρ φ i + 3H φ i a 2 φ 2 i + i V = 0 3H 2 = ρ

9 Entropy and Coarse Graining Shannon S = Entropy Dϕf [ϕ] ln f [ϕ] ϕ(x) = (ϕ 1 (x),..., ϕ Ns (x)) (ϕ 1 (x 1 ), ϕ 1 (x 2 ),..., ϕ 1 (x N ), ϕ 2 (x 1 ),... )

10 Entropy and Coarse Graining Shannon (or von Neumann) Entropy S = Dϕf [ϕ] ln f [ϕ]= Trˆρ( ˆϕ) ln ˆρ( ˆϕ) ϕ(x) = (ϕ 1 (x),..., ϕ Ns (x)) (ϕ 1 (x 1 ), ϕ 1 (x 2 ),..., ϕ 1 (x N ), ϕ 2 (x 1 ),... )

11 Entropy and Coarse Graining Shannon (or von Neumann) Entropy S = Dϕf [ϕ] ln f [ϕ]= Trˆρ( ˆϕ) ln ˆρ( ˆϕ) Maximize S Subject to Measured C(x, y) ϕ(x)ϕ (y) S ME N lat ln 2π 2 N lat 2 = 1 ln detc 2

12 Entropy and Coarse Graining Shannon (or von Neumann) Entropy S = Dϕf [ϕ] ln f [ϕ]= Trˆρ( ˆϕ) ln ˆρ( ˆϕ) Maximize S Subject to Measured C(x, y) ϕ(x)ϕ (y) S ME N lat ln 2π 2 N lat 2 = 1 ln (k i ) 2 k i Statistically Homogeneous : (k i ) = det ϕ(k i ) ϕ(k i )

13 Entropy and Coarse Graining Shannon (or von Neumann) Entropy S = Dϕf [ϕ] ln f [ϕ]= Trˆρ( ˆϕ) ln ˆρ( ˆϕ) Maximize S Subject to Measured C(x, y) ϕ(x)ϕ (y) S ME N lat ln 2π 2 N lat 2 = 1 2 ln ( Πki (k i ) J 2 ) +... Statistically Homogeneous : (k i ) = det ϕ(k i ) ϕ(k i ) Noncanonical : ΠP(k) Vfluc 2, J 2 = ϕ 2 ϕ can V 2 quantum Linear Evolution of Field Variables ds ME dt = 0 and Gaussianity preserved

14 S Preheating : The Shock-in-Time 1 L = µφ µ φ 2 µχ µ χ 2 m2 φ 2 2 g 2 φ 2 χ 2 2 N eff ln(ρ/ ρ) 3(1 +w)ln(a) +ln(ρ) Phonons (ln ρ, t ln ρ) ln(a/a end) N 1 eff ds/dt Fields (φ i,π φi ) m 1 k cut = ds nφ /dt ds nχ /dt ds φ /dt ds χ /dt ds tot /dt mt

15 Low-Point Statistics P/Pmax mt = log(ρ/ ρ) t log(ρ/ ρ) Gaussian fit Gaussian fit P/Pmax mt = a 3/2 δφ a 3/2 δχ a 3/2 δ φ a 3/2 δ χ φ Fit χ Fit κ4 k 3 P (k) mt = ln(ρ/ ρ) t ln(ρ/ ρ) κ4 k 3 P (k) mt = φ χ k/m k/m During Shock During Shock

16 Low-Point Statistics : Gaussian ln ρ Post-Shock P/Pmax mt = log(ρ/ ρ) t log(ρ/ ρ) Gaussian fit Gaussian fit P/Pmax mt = a 3/2 δφ a 3/2 δχ a 3/2 δ φ a 3/2 δ χ φ Fit χ Fit κ4 k 3 P (k) mt = ln(ρ/ ρ) t ln(ρ/ ρ) κ4 k 3 P (k) mt = φ χ k/m k/m Post-Shock Post-Shock

17 Multiscale View of Isocurvature Fluctuations χ To decay, inflaton must couple to other fields (e.g. g 2 φ 2 χ 2 ) Can χ i influence preheating?

18 Preheating Density Perturbations [Chambers, Rajantie],[Bond,Frolov,Huang,Kofman] ζ tot = ζ inflaton + F NL (χ)

19 Preheating Density Perturbations [Chambers, Rajantie],[Bond,Frolov,Huang,Kofman] ζ tot = ζ inflaton + F NL (χ)

20 Mechanism I : Modulation of the Shock-in-Time [Bond,JB] F NL = F NL (g 2 ( χ)) V (φ) = 1 2 m2 φ 2 + g 2 2 φ2 σ 2 Fixed g 2 ln (a/aend) gm P /m (ds/dt)normalized

21 Mechanism I : Modulation of the Shock-in-Time [Bond,JB] F NL = F NL (g 2 ( χ)) V (φ) = 1 2 m2 φ 2 + α 2 χ2 φ 2 σ 2 Fixed g 2 Dynamical g 2 eff = α χ2 ln (a/aend) gm P /m (ds/dt)normalized ln (a/aend) g eff M P /m (ds/dt)normalized

22 Preheating Density Perturbations [Chambers, Rajantie],[Bond,Frolov,Huang,Kofman] ζ tot = ζ inflaton + F NL (χ)

23 Mechanism II : Density Perturbations from Ballistic Motion Caustics [Bond,JB,Frolov,Huang] Jordan Frame L g = M2 P 2 (1 + ξφ2 )R 1 2 µφ µ φ 1 2 µχ µ χ λ 4 φ4 g 2 2 φ2 χ V/M 4 P ξ =0.001 ξ =0.01 ξ =0.1 ξ =1.0 ξ =10.0 ξ =100.0 ξ = ξ = φ canonical /M P

24 Mechanism II : Density Perturbations from Ballistic Motion Caustics [Bond,JB,Frolov,Huang] Einstein Frame L g = M2 P 2 R 1 2 λ ξ(1 + 6ξ)φ 2 (1 + ξφ 2 ) 2 µ φ µ φ 1 µ χ µ χ ξφ 2 φ 4 (1 + ξφ 2 ) 2 g 2 φ 2 χ 2 2 (1 + ξφ 2 ) V/M 4 P ξ =0.001 ξ =0.01 ξ =0.1 ξ =1.0 ξ =10.0 ξ =100.0 ξ = ξ = φ canonical /M P

25 Mechanism II : Density Perturbations from Ballistic Motion Caustics [Bond,JB,Frolov,Huang] Einstein Frame L g = M2 P 2 R 1 2 λ ξ(1 + 6ξ)φ 2 (1 + ξφ 2 ) 2 µ φ µ φ 1 µ χ µ χ ξφ 2 φ 4 (1 + ξφ 2 ) 2 g 2 φ 2 χ 2 2 (1 + ξφ 2 ) 2 For certain choices of g 2 λ, χ k=0 mode is unstable χ χ i has superhorizon fluctuations from inflation Chaotic billiards in a potential caustics

26 Ballistic Motion of Trajectories

27 Caustics Lead to Curvature Spikes

28 Density Perturbations from Caustics τ α ln(χ 0 /φ 0 )/µ 0 T Decoupled Trajectories

29 Density Perturbations from Caustics Lattice Simulations

30 ζ = ζ inf + F (χ i )

31 Back up the Hierarchy - A CMB Pixel 1.5e-05 F nl (χ 0 ) σ F nl (χ 0 ) σ 2 F nl (χ 0 ) 1e-05 5e e-06-1e e e-06 4e-06 6e-06 8e-06 1e-05 ζ = ζ inf + F (χ i )

32 Sample CMB Maps - Vary χh0

33 Sample CMB Maps - Vary χh0

34 Sample CMB Maps - Vary χh0

35 Sample CMB Maps - Vary χh0

36 Conclusions Post-inflation instabilities are common Sharp transition from ballistic to strongly coupled evolution shock-in-time Can generate density perturbations from preheating 1. Caustic formation from pre-shock billiards 2. Modulation of shock surface Generic given isocurvature mode coupled to inflaton