Particle Production from Coherent Oscillation

1. Particle ProductionfromCoherent Oscillation Hiroaki Nagao Graduate School of Science and Technology, Niigata University, Japan DESY Theory Workshop, October, 1st , 2009 In collaborationwith TakehikoAsaka (Niigata Univ.)

2. Introduction [ex:A.D.Linde (‘82,‘83)] • Inflation ・Solve the problems of Standard Big Bang Cosmology ・Provide the origin of density fluctuation ・Supported by CMBR observation • Reheating ?? ・Coherent oscillation of scalar field ・Energy transfer into elementary particles [ex:WMAP 5yr. (‘08)] Our focus! ?? SM , SUSY(?)…

3. Framework • Particle production from coherent oscillation (Neglect expansionof our univ.) How arethey produced?!

4. ・So far,…. [ex: M.S.Turner (‘83)] : φ decay occurs When is this approximation valid?

5. Our analysis [ex:N.N.Bogolyubov(‘58)] ◎Use the method based on Bogolyubov transformation ・Solve E.O.M for mode function ・Estimate distribution function Find the behavior of [ex:L.Kofman et al(‘94) M.Peloso et al(‘00)] e.g.) e.g.) In weak coupling limit to avoid the preheating effect

6. Perturbative expansion in coupling [ex:Y.Shtanov et al(‘94) A.D.Dolgov(‘01) ] ◎ Solution of E.O.M starts at starts at

7. Growth for mode k* Phase cancellation ・The mode k* is ensured to grow!

8. Analytical results ◎Distribution function of scalar ◎Growing mode ◎Number density

9. Evolution of occupation number for

10. Yield of produced scalar

11. Number density Provide Good Approximation！ Is this treatment valid forever ? 11

12. Non-perturbative effect ‘Bose condensation’ ・Effect of higher order corrections of couplinggS ・Reflect the statistical property of χ

13. Q.How to estimate thisexponent?? Much longer time scale than period of coherent oscillation Average over the oscillation period of φ “Averaging method”!! [ex:A.H.Nayfeh et.al (‘79)]

14. Analytical results ◎Distribution function Correspond to the energy conservation condition in non-rela. φdecay. where ◎Number density

15. Evolution of occupation number for

16. Yield of produced fermion

17. Non-perturbative effect ‘Pauli blocking’ Effect of higher order corrections of coupling gF Reflect the statistical property of ψ

18. How to estimate this frequency?? Averaging method! Long periodic oscillation around 1/2

19. Decay process of non-rela. φ Fermion Scalar Decay processes are forbidden for

20. Abundance of heavy particles Heavy particles can be produced are induced at

21. Summary • Particle production from coherent oscillation Neglect expansion Weak coupling limit • Obtain the exact distribution function up to by using Bogolyubov transformation →・Applicable in the beginnings of production ・Imply the production of heavy particles • Higher-order correction is crucial in the later time ・Provide the difference between χ andψ ・Can be estimated by the averaging method

22. Thank you for your attention.Dankeschön.

23. BACKUP SLIDE

24. Number density of coherent oscillation Same dilution rate Approximation Treat coherent oscillation as non-relativistic particles ・Estimate by decay of non-relativistic φ

25. Particle picture [ex:M.G.Schmidt et.al(‘04)] ・Field operator ・Hamiltonian densityunder the time dependent background Off-diagonal element! Disable the particle picture Eigenstate of Hamiltonian Diagonalization of Hamiltonian

26. Particle picture [ex: M.Peloso et al(‘00)] ・Field operator ・Hamiltoniandensity under the time dependent background Eigenstate of Hamiltonian Diagonalization of Hamiltonian

27. Diagonalization ◎Bogoliubov transformation ・Commutation relation(Equal time) ◎Diagonalized Hamiltonian Eigenstate of Hamiltonian where

28. Particle number ・Number operator ◎Number density of produced ψ ・Distribution function in k space Pauli exclusion principle

29. Solution for mode function ◎Solution for starts at

30. Leading contribution for β ・Leading order contribution Superposition of oscillation only contain oscillating behavior??

31. Growth ofβ Cause the phase cancellation at Growing mode = Energy conservation in decay process Grow! Growth of Growth of occupation number

32. Growth of occupation number starts at ・By taking Growth of occupation number ＠

33. Number density for scalar ◎ contribution ・Definition of number density ・Exchange the order of integration

34. ・Integration in momentum space ( General hypergyometric function ) ・Expandin terms of and perform integration in time

35. Averaging method ◎Variation of parameters where w/ [ex:A.H.Nayfeh et.al (‘79)] ◎Averaging ・Remove the short-periodic oscillation ・Only contain the long periodic terms

36. Later time behavior ◎Averaged solution for scalar Exponential growth! ◎Later time behavior ofoccupation number Its exponent is consistent with the result of parametric resonance [ex:M.Yoshimura(‘95)]

37. Averaging method [ex:A.H.Nayfeh et.al (‘79)] ◎Variation of parameters ◎Averaging Originate from Dirac eq.

38. Averaged solution ◎Averaged solution for fermion Long periodic oscillation around 1/2

39. Consistency ◎We obtain following results by the method of averaging

40. Evolution of number density ・Growth of number density would be stopped because of the absence of phase cancellation

41. Distribution function in k space