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After inflation: preheating and non-gaussianities Kari Enqvist University of Helsinki Bielefeld 16.5.06 • 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 inflation = superluminal expansion of the universe de Sitter universe: cosmological constant inflation R R0 e 8 / 3 M P2 t with a scale invariant spectrum of perturbations: n=1 t→ t + t makes no difference V WMAP: n = 0.948 ± 0.018 inflaton slow roll ends H = H(t) slow roll: , << 1 m H HOW TO REHEAT THE UNIVERSE WITH SM and CDM DOFs? classical field .. . 3H (t ) m 0 2 average over 1 oscillation period: p (t) m 0 1 2 Equation of state: effectively pressureless matter 2 1 2 2 2 H 3 / 2t V=½m22 [1/m] assume Yukawa: g One-loop corrections to EOM: / J Abbott, Farhi, Wise, ’82 Im m = Im 0 0 = /2 condensate decays with a single particle rate 2 g m 8 when H .. [3H (t ) ] m 0 i.e. at 2 tr 2 /(3) 3 M (t r ) g* (Tr )Tr4 8 2 with . 2 P instant thermalization Tr 0.2 M P 10 GeV 9 g 106 , m 1014 GeV • weak Yukawas → low reheat temperature • ’inefficient reheating’ • decay to scalars: → – large density → backreaction – potentially explosive particle production preheating Kofman, Linde, Starobinsky PREHEATING - oscillating inflaton condensate → source for quantum field V V ( g 1 2 (k=0) (k=0) 2 2 2 k when V() = 0, → non-adiabatic excitation of quanta -k (field fluctuations) effective mass2 = g22 - initial 2 body PS distribution → subsequent thermalization … but does not yet tell how to get SM dofs V m g 2 1 2 example: (t , x) d k ak k (t )e 3 ikx 2 1 2 a (t )e k * k 2 ikx 2 2 2 2 k 2 2 t 3H t 2 g k 0 a (t ) if expansion ignored: H=0, a=1 - inflaton oscillations start when m ~ H → many oscillations in one Hubble time initial conditions - amplitude = (H) Mathieu equation: k” + [ Ak - 2q sin(2z) ] k = 0 z = mt Ak = 2q + k2/m2 q = g22/4m2 HO with time-dependent frequency k Instability bands on (k,q) plane: k grows ↔ nk() grows exponentially (within 1 Hubble time) ’parametric resonance’ q q << 1 ↔ m >> g narrow resonance inflaton decays into -particles all the time – but resonance may be washed out by expansion q >> 1 ↔ m << g broad resonance bursts of -production as k-modes drift through the instability bands expansion fixed k k Expansion of universe: Mathieu eq OK if drift adiabatic narrow resonance = q ~ 0.1 =2/m broad resonance q ~ 2 102 growth of nk() → backreaction → end of preheating k (k=0) highly non-perturbative -k (k=0) k ~ exp(mt) ’Floquet index’ tend ~ ln(m/g)/m PREHEATING AND CURVATURE PERTURBATION field perturbations → metric perturbations ds a( ) (dt dx ) 2 2 2 2 a( ) (1 2 )dt (1 2 )dx 2 (1) 2 (1) 2 x inflation ends t1/2 frozen horizon almost scale invariant spectrum eHt t 1/H H2 ~ local Minkowski t Statistics? At lowest order, perturbations are gaussian: kl (1) (1) k l (1) (1) (1) k l m P (k ) kl 0 etc metric perturbations density perturbations photon temperature perturbations dominantly gaussian … but non-gaussianities are generated at second order f NL ( ) (1) (1) 2 2 : non-Gaussian Gaussian ~ (10-5)1/2 ~ 10-5 f NL (1) (1) (1) (1) f NL (1) (1) 2 ~ 10-10 small effect if fNL << 105 gauge invariant curvature perturbations technical problem: non-gaussianities require 2nd order formalism -comoving curvature perturbation R -uniform density curvature perturbation 1st order: agree at large scales 2nd order: 2(LR) = 2(MW) + 212 R2 has spurious time evolution ~ ’, ’ Vernizzi preheating: look for large non-gaussianities → O(1) differences irrelevant WMAP3 limits: -54 < fNL < 134 95% CL - single field inflation: fNL ~ slow roll parameters , << 1 -multifield inflation: max(fNL) ~ O(1) How to get large non-gaussianities? curvature perturbation ’ Langlois, Vernizzi ’non-adiabatic pressure’ = isocurvature need 2nd field large fNL → large 2 / (1)2 need: 1st order curvature perturbation does not grow 2nd order curvature perturbation grows Preheating and non-gaussianities g2 REQUIRE: - interactions violating slow roll - isocurvature fluctuations that can source adiabatic perturbations after inflation can be large: -2nd order effects become significant (backreaction) -small scales couple to large scales (initial conditions extend over 1/H) -enhancement of pre-existing perturbations Example: enhancing non-gaussianity with NARROW RESONANCE KE, Jokinen, Mazumdar, Multamäki, Väihkönen V m g 1 2 2 2 1 2 2 2 2 inflation ends when ~ MP g 2 2 1 resonance narrow ifq 2 4m or g < H/MP << 1 H < m → can neglect expansion Narrow resonance H 5 q 1 g 10 MP mass of during inflation m g 10 M P H 5 effectively massless subject to inflationary fluctuations field perturbations: = 0 + 1 + ½2 = 1 + ½2 <> = 0 (helps with analytic approximations) metric perturbations: g00 = -a2 (1 +21 + 2) etc 1st order from inflaton alone: 1 ~ 1 <> = 0: 1 isocurvature fluctuation Evolution of 2 sources: , 1, D2 = J() + J(rest) J() ~ (1)2 + (1’)2 → <1k * 1k-q> + < * > source is convolution in Fourier space EOM for the 1st order perturbation: (D(H) + g 2 02) 1 = 0 narrow resonance, many oscillations in 1 Hubble time → ignore expansion ESTIMATE: 1 ~ A exp (2qeff m t) in the resonance, = 0 elsewhere A(k) = amplitude at the end of inflation = H/(2k3)½ … q ~ 2 < 1 given by the inflaton amplitude qeff = ½qmax ↔ width of the resonance slowly changing A(k) → [ k-, k+ ] k± = ½m(1 ± q/2) Jk→0 ~ < * > ~ dk k2 (1k)2 + … ~ amplitude dk k2 = stuff exp(qmt/2) source for 2 generated by 1st order local perturbations in the interval [ k-, k+ ] Back to the metric perturbation: D2 = stuff exp(…t) + rest → 2 ~ exp(qmt/2) fNL() ~ 2k / <1 * 1>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 fNL() ~ e4 = 55 2nd order metric perturbation: ' q qm t / 2 2 2k [1 ] Be nonexp 2 MP 16 '' 2k 2m2 q '' 0 ' 0 q m B 8 2 3 H 2 k 3 2k aH approximation: 3 2 9 / 4 g 202 / H 2 m '' 0 ' 0 exponentially growing solution massless case: V = ¼4 + ½ g222 exactly solvable, expansion can be transformed away EOM: Lame eq. X” + f() X = 0 Jokinen, Mazumdar X = scaled pert. Jacobian elliptic function -J & M average over oscillations -non-local terms vanish at large scales (spatial gradients neglected) → follow the evolution of 2 and fNL numerically y = g2/ y=1.875 y=1.5 y=1.2 3 inflaton oscillations y=1.875: fNL = -1380 WMAP: massless preheating ruled out for 1 < y < 3 SUMMARY • preheating: large fluctuations → large 2nd order effect • fNL ~ O(1000) possible • future limits fNL ~ O(1) →potentially significant constraints • model-dependent; e.g. instant preheating not constrained • backreaction suppresses? (e.g. Nambu, Araki)