1. After inflation: preheating and non-gaussianities Kari Enqvist University of Helsinki Bielefeld 16.5.06

2. End of inflation  hot universe –But thermalization dynamics leave no signature Preheating: ”non-perturbative reheating” –Certain types (”narrow resonance”) may give rise to observable non-gaussianity in CMB

3. de Sitter universe: cosmological constant  inflation with a scale invariant spectrum of perturbations: n=1 t→ t +  t makes no difference inflation = superluminal expansion of the universe

4. HOW TO REHEAT THE UNIVERSE WITH SM and CDM DOFs? WMAP: n = 0.948 ± 0.018 inflaton H = H(t) V  slow roll: ,  << 1 slow roll ends

5. classical field Equation of state: effectively pressureless matter average over 1 oscillation period:

6. V=½m 2  2 [1/m]

7. assume Yukawa: One-loop corrections to EOM: 00 00  Im m =Im =  /2 Abbott, Farhi, Wise, ’82

8. condensate decays with a single particle rate wheni.e. at with instant thermalization

9. weak Yukawas → low reheat temperature ’inefficient reheating’ decay to scalars:  →  –large  density → backreaction –potentially explosive particle production preheating Kofman, Linde, Starobinsky

10. PREHEATING - oscillating inflaton condensate → source for quantum field   (k=0) kk  -k when V(  ) = 0, → non-adiabatic excitation of quanta (field fluctuations) effective mass 2 = g 2  2 - initial 2 body PS distribution → subsequent thermalization … but does not yet tell how to get SM dofs

11. example: if expansion ignored: H=0, a=1

12.  k ” + [ A k - 2q sin(2z) ]  k = 0 - inflaton oscillations start when m  ~ H → many oscillations in one Hubble time - amplitude  =  (H) Mathieu equation: z = m  t A k = 2q + k 2 /m  2 q = g 2  2 /4m  2 Instability bands on (k,q) plane:  k grows ↔ n k (  ) grows exponentially (within 1 Hubble time) ’parametric resonance’ initial conditions HO with time-dependent frequency  k

13. expansion q > g  narrow resonance q >> 1 ↔ m  << g  broad resonance inflaton decays into  -particles all the time – but resonance may be washed out by expansion bursts of  -production as k-modes drift through the instability bands fixed k Expansion of universe: Mathieu eq OK if drift adiabatic q k

14. narrow resonance q ~ 0.1 =2  /m ==

15. broad resonance q ~ 2  10 2

16. growth of n k (  ) → backreaction → end of preheating  (k=0) kk  -k  k ~ exp(  m  t)t end ~ ln(m  /g)/  m  highly non-perturbative ’Floquet index’

17. PREHEATING AND CURVATURE PERTURBATION field perturbations → metric perturbations

18. x t horizon e Ht t 1/2 1/H t H 2 ~  local Minkowski frozen inflation ends almost scale invariant spectrum

19. At lowest order, perturbations are gaussian: metric perturbations  density perturbations  photon temperature perturbations dominantly gaussian Statistics?

20. … but non-gaussianities are generated at second order Gaussian  2 : non-Gaussian ~ (10 -5 ) 1/2 ~ 10 -5 ~ 10 -10 small effect if f NL << 10 5

21. gauge invariant curvature perturbations -comoving curvature perturbation R -uniform density curvature perturbation  1st order: agree at large scales 2nd order:  2 (LR) =  2 (MW) + 2  1 2 R 2 has spurious time evolution ~  ’,  ’ Vernizzi preheating: look for large non-gaussianities → O(1) differences irrelevant technical problem: non-gaussianities require 2nd order formalism

22. - single field inflation: f NL ~ slow roll parameters ,  << 1 -multifield inflation: max(f NL ) ~ O(1) WMAP3 limits: -54 < f NL < 134 95% CL

23. How to get large non-gaussianities? need: 1st order curvature perturbation does not grow 2nd order curvature perturbation grows curvature perturbation  ’   ’non-adiabatic pressure’ = isocurvature Langlois, Vernizzi large f NL → large  2 / (  1 ) 2 need 2nd field

24. Preheating and non-gaussianities REQUIRE: - interactions violating slow roll - isocurvature fluctuations that can source adiabatic perturbations after inflation  g2g2 -2nd order effects become significant (backreaction) -small scales couple to large scales (initial conditions extend over 1/H) -enhancement of pre-existing perturbations can be large:

25. Example: enhancing non-gaussianity with NARROW RESONANCE inflation ends when  ~ M P resonance narrow if or g < H/M P << 1 H < m  → can neglect expansion KE, Jokinen, Mazumdar, Multamäki, Väihkönen

26. Narrow resonance mass of  during inflation  effectively massless subject to inflationary fluctuations

27.  =  0 +  1  + ½  2   =  1  + ½  2  = 0 field perturbations: metric perturbations: g 00 = -a 2 (1 +2  1 +  2 ) etc 1 ~ 11 ~ 1 1st order from inflaton alone: = 0:  1  isocurvature fluctuation (helps with analytic approximations)

28. Evolution of  2 sources: ,  1,  D  2 = J(  ) + J(rest) J(  ) ~ (  1  ) 2 + (  1  ’) 2 → + source is convolution in Fourier space (D(H) + g 2  0 2 )  1  = 0 EOM for the 1st order perturbation: narrow resonance, many oscillations in 1 Hubble time → ignore expansion

29. ESTIMATE:  1  ~ A exp (2q eff m  t) in the resonance, = 0 elsewhere A(k) = amplitude at the end of inflation = H/(2k 3 ) ½  … q ~  2 < 1 given by the inflaton amplitude q eff = ½q max ↔ width of the resonance [ k -, k + ] slowly changing A(k) → k ± = ½m  (1 ± q/2) J k→0 ~ ~  dk k 2 (  1  k ) 2 + … ~ amplitude   dk k 2 = stuff  exp(qm  t/2) source for  2 generated by 1st order local perturbations in the interval [ k -, k + ]

30. Back to the metric perturbation: D  2 = stuff  exp(…t) + rest →  2 ~ exp(qm  t/2) D  2 = stuff  exp(…t) + rest →  2 ~ exp(qm  t/2) f NL (  ) ~  2k / k ~ exp(Nq/2) N = # oscillations during preheating Example: chaotic inflation backreaction kicks in after N=10-30 osc → take N =10, q = 0.8 f NL (  ) ~ e 4 = 55

31. 2nd order metric perturbation: approximation: exponentially growing solution

32. V = ¼  4 + ½ g 2  2  2 massless case: exactly solvable, expansion can be transformed away Jokinen, Mazumdar EOM: X” + f(  ) X = 0 X = scaled  pert. Jacobian elliptic function Lame eq. -J & M average over oscillations -non-local terms vanish at large scales (spatial gradients neglected) → follow the evolution of  2 and f NL numerically

33. 3 inflaton oscillations y = g 2 / y=1.2 y=1.5 y=1.875 y=1.875: f NL = -1380 WMAP: massless preheating ruled out for 1 < y < 3

34. SUMMARY preheating: large fluctuations → large 2nd order effect f NL ~ O(1000) possible future limits f NL ~ O(1) →potentially significant constraints model-dependent; e.g. instant preheating not constrained backreaction suppresses? (e.g. Nambu, Araki)