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Particle Production from Coherent Oscillation. Hiroaki Nagao Graduate School of Science and Technology, Niigata University, Japan. DESY Theory Workshop, October, 1 st , 2009 . In collaboration with Takehiko Asaka. ( Niigata Univ.). Introduction. [ e x:A.D.Linde (‘82,‘83)].
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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.)
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(?)…
Framework • Particle production from coherent oscillation (Neglect expansionof our univ.) How arethey produced?!
・So far,…. [ex: M.S.Turner (‘83)] : φ decay occurs When is this approximation valid?
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
Perturbative expansion in coupling [ex:Y.Shtanov et al(‘94) A.D.Dolgov(‘01) ] ◎ Solution of E.O.M starts at starts at
Growth for mode k* Phase cancellation ・The mode k* is ensured to grow!
Analytical results ◎Distribution function of scalar ◎Growing mode ◎Number density
Number density Provide Good Approximation! Is this treatment valid forever ? 11
Non-perturbative effect ‘Bose condensation’ ・Effect of higher order corrections of couplinggS ・Reflect the statistical property of χ
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)]
Analytical results ◎Distribution function Correspond to the energy conservation condition in non-rela. φdecay. where ◎Number density
Non-perturbative effect ‘Pauli blocking’ Effect of higher order corrections of coupling gF Reflect the statistical property of ψ
How to estimate this frequency?? Averaging method! Long periodic oscillation around 1/2
Decay process of non-rela. φ Fermion Scalar Decay processes are forbidden for
Abundance of heavy particles Heavy particles can be produced are induced at
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
Number density of coherent oscillation Same dilution rate Approximation Treat coherent oscillation as non-relativistic particles ・Estimate by decay of non-relativistic φ
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
Particle picture [ex: M.Peloso et al(‘00)] ・Field operator ・Hamiltoniandensity under the time dependent background Eigenstate of Hamiltonian Diagonalization of Hamiltonian
Diagonalization ◎Bogoliubov transformation ・Commutation relation(Equal time) ◎Diagonalized Hamiltonian Eigenstate of Hamiltonian where
Particle number ・Number operator ◎Number density of produced ψ ・Distribution function in k space Pauli exclusion principle
Solution for mode function ◎Solution for starts at
Leading contribution for β ・Leading order contribution Superposition of oscillation only contain oscillating behavior??
Growth ofβ Cause the phase cancellation at Growing mode = Energy conservation in decay process Grow! Growth of Growth of occupation number
Growth of occupation number starts at ・By taking Growth of occupation number @
Number density for scalar ◎ contribution ・Definition of number density ・Exchange the order of integration
・Integration in momentum space ( General hypergyometric function ) ・Expandin terms of and perform integration in time
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
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)]
Averaging method [ex:A.H.Nayfeh et.al (‘79)] ◎Variation of parameters ◎Averaging Originate from Dirac eq.
Averaged solution ◎Averaged solution for fermion Long periodic oscillation around 1/2
Consistency ◎We obtain following results by the method of averaging
Evolution of number density ・Growth of number density would be stopped because of the absence of phase cancellation