After inflation: preheating and non-gaussianities

After inflation: preheating and non-gaussianities

Kari Enqvist

University of HelsinkiBielefeld 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

de Sitter universe: cosmological constant

tM PeRR23/8

0 inflation

with a scale invariant spectrum of perturbations: n=1

t→ t + t makes no difference

inflation = superluminal expansion of the universe

HOW TO REHEAT THE UNIVERSEWITH SM and CDM DOFs?

WMAP: n = 0.948 ± 0.018 inflaton

H = H(t)

V

slow roll: , << 1

Hm

slow roll ends

classical field

0)(3 2...

mtH

0)( 22212

21 mp t

tH 2/3Equation of state: effectively pressureless matter

average over 1 oscillation period:

V=½m22

[1/m]

gassume Yukawa:

One-loop corrections to EOM: J /

00

Im m = Im = /2

Abbott, Farhi, Wise, ’82

condensate decays with a single particle rate

8

2mg

when H i.e. at )3/(2 rt

4*

22

)(8

3)( rr

Pr TTg

Mt

with

instant thermalization

GeV102.0 9 Pr MT GeV10,10 146 mg

0])(3[ 2...

mtH

• 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

(k=0)

(k=0)

k

-k

when V() = 0, → non-adiabatic excitation of quanta (field fluctuations)

effective mass2 = g22

- initial 2 body PS distribution → subsequent thermalization

22221( gVV

… but does not yet tell how to get SM dofs

2222122

21 gmV example:

ikxkk

ikxkk etaetakdxt )()(),( *3

0)(

3 222

22

ktt g

ta

kH

if expansion ignored: H=0, a=1

k” + [ Ak - 2q sin(2z) ] k = 0

- inflaton oscillations start when m ~ H → many oscillations in one Hubble time

- amplitude = (H)

Mathieu equation:

z = mtAk = 2q + k2/m

2 q = g22/4m

2

Instability bands on (k,q) plane: k grows ↔ nk() grows exponentially (within 1 Hubble time)

’parametric resonance’

initial conditions

HO with time-dependentfrequency k

expansion

q << 1 ↔ m >> gnarrow resonance

q >> 1 ↔ m << gbroad resonance

inflaton decays into -particlesall the time – but resonancemay 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 driftadiabatic

q

k

narrow resonance

q ~ 0.1

=2/m

=

broad resonance

q ~ 2 102

growth of nk() → backreaction → end of preheatinggrowth of nk() → backreaction → end of preheating

(k=0)

(k=0)

k

-k

k ~ exp(mt) tend ~ ln(m/g)/m

highly non-perturbative

’Floquet index’

PREHEATING AND

CURVATURE PERTURBATION

field perturbations → metric perturbations

2)1(2)1(2

2222

)21()21()(

)()(

dxdta

dxdtads

x

t

horizon

eHt

t1/2

1/H

t

H2 ~

local Minkowski

frozen

inflation ends

almost scaleinvariantspectrum

At lowest order, perturbations are gaussian:

kllklk kP )()1()1(

etcmlk 0)1()1()1(

metric perturbations density perturbations photon temperature perturbations

dominantly gaussian

Statistics?

… but non-gaussianities are generated at second order

2)1()1( )( NLf

2)1()1()1()1()1()1( NLNL ff

Gaussian 2 : non-Gaussian

~ (10-5)1/2 ~ 10-5

~ 10-10

small effect if fNL << 105

gauge invariant curvature perturbations

-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

technical problem: non-gaussianities require 2nd order formalism

- single field inflation: fNL ~ slow roll parameters , << 1

-multifield inflation: max(fNL) ~ O(1)

WMAP3 limits:

-54 < fNL < 134 95% CL

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 fNL → large 2 / (1)2

need 2nd field

Preheating and non-gaussianities

REQUIRE:

- interactions violating slow roll

- isocurvature fluctuations that can source adiabatic perturbations after inflation

g2

-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:

Example: enhancing non-gaussianitywith NARROW RESONANCE

inflation ends when ~ MP

resonance narrow if or

g < H/MP << 1

H < m → can neglect expansion

KE, Jokinen, Mazumdar,Multamäki, Väihkönen

2222122

21 gmV

14 2

22

m

gq

Narrow resonance

5101 PM

Hgq

mass of during inflation

HMgm P 510

effectively massless

subject to inflationary fluctuations

= 0 + 1 + ½2 = 1 + ½2

<> = 0

field perturbations:

metric perturbations:

g00 = -a2 (1 +21 + 2) etc

1 ~ 11st order from inflaton alone:

<> = 0: 1 isocurvature fluctuation

(helps with analytic approximations)

Evolution of 2

sources: , 1,

D2 = J() + J(rest)D2 = J() + J(rest)

J() ~ (1)2 + (1’)2 → <1k * 1k-q> + < * >

source is convolution inFourier space

(D(H) + g202) 1 = 0(D(H) + g20

2) 1 = 0

EOM for the 1st order perturbation:

narrow resonance, manyoscillations in 1 Hubbletime → ignore expansion

ESTIMATE: 1 ~ A exp (2qeff m t) 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 amplitudeqeff = ½qmax ↔ width of the resonance

[ k-, k+ ]

slowly changing A(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)

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 = 55fNL() ~ e4 = 55

nonexp]16

1[2

2 2/

2

2'2'

0

''0''

2 tqm

Pkk Be

q

M

qm

2nd order metric perturbation:

approximation: '0

''0

m

exponentially growing solution

23

3

23

228

aH

k

k

HmqB

220

2 /4/9 Hg

V = ¼4 + ½ g222

massless case:

exactly solvable, expansion can betransformed 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 fNL numerically

3 inflaton oscillations

y = g2/

y=1.2

y=1.5

y=1.875

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)