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)