NG from inflation

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2-4 November, 2009 11th Italian-Korean
Symposium Sogang University
Non-Gaussianity from Inflation
Misao Sasaki YITP KIAS
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contents
1. Inflation and curvature perturbations
dN formalism
2. Origin of non-Gaussianity
subhorizon or superhorizon scales
3. Non-Gaussianity from superhorizon scales
two typical models curvaton multi-brid
inflation
4. Multi-brid inflation
5. Summary
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1. Inflation and curvature perturbations

• single-field inflation, no other degree of
  freedom

Linde 82, ...
slow-roll evolution
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N Number of e-folds of inflation

• number of e-folds counted backward in time
• from the end of inflation log(redshift)

log L
NN(f)
LH-1 t
LH-1 const
log a(t)
inflation
tt(f)
ttend
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Cosmological curvature perturbations
- standard single-field slow-roll case -
Inflaton fluctuations (vacuum fluctuationsGaussia
n)
Oscillation freezes out at k/a lt H ( classical
Gaussian fluctuations on superhorizon scales)
Curvature perturbations
conserved on superhorizon scales
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dN formalism

• spectrum of Rc

scale-invariant

• curvature perturbation e-folding number
  perturbation (dN)

Starobinsky (85)
general slow-roll inflation
MS Stewart (96)
nonlinear generalization
Lyth, Malik MS (04)
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Theory vs Observation

• Prediction of the standard single-field slow-roll
  inflation
• almost scale-invariant Gaussian random
  fluctuations

• perfectly consistent with CMB experiments

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However, nature may not be so simple...

• Tensor (gravitational wave) perturbations?

tensor perturbations induce B-mode polarization
of CMB
rlt0.2 (95CL)
tensor-scalar ratio
WMAPBAOSN (08)
cf. chaotic inflation r0.15

• Non-Gaussianity?

Komatsu Spergel (01)
( - )gravitational pot.
minus
-9lt fNLlt111 (95CL)
WMAP (08)
(fNL5130 at 1s)
What would the presence of non-Gaussianity mean ?
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2. Origin of Non-Gaussianity

• three typical origins

• Self-interaction of inflaton field

quantum physics, subhorizon scale during inflation

• Multi-component field

classical physics, nonlinear coupling to gravity
superhorizon scale during and after inflation

• Nonlinearity of gravity

classical general relativistic effect,
subhorizon scale after inflation
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Origin of non-Gaussianity and cosmic scales
k comoving wavenumber
log L
classical/local NL effect
classical NL gravity
quantum NL effect
log a(t)
ttend
inflation
hot Friedmann
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Origin of NG (1) Self-interaction of inflaton
field
Non-Gaussianity from subhorizon scales
(QFT effect)

• interaction is very small for standard
  self-couplings

(potential-type)
Maldacena (03)
ex. chaotic inflation
free field!
(gravitational interaction is Planck-suppressed)

• non-canonical kinetic term
• ? strong self-interaction ? large
  non-Gaussianity

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example DBI inflation
Silverstein Tong (2004)
kinetic term
(Lorenz factor)-1
perturbation expansion

?
?
?
0
g -1
g 3
g 5
large non-Gaussianity for large g
Seery Lidsey (05), ... , Langlois et al.
(08), Arroja et al. (09)
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bispectrum (3-pt function) of curvature
perturbation from DBI inflation
Alishahiha et al. (04)
WMAP 5yr constraint
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Origin of NG (2) Nonlinearity of gravity
ex. post-Newton metric in harmonic coordinates
Newton potential
NL terms
may be important after the perturbation
scale re-enters the Hubble horizon
Effect on CMB bispectrum seems small (but
non-negligible?)
Bartolo et al. (2007)
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Origin of NG (3) Superhorizon scales
Even if df is Gaussian, dT mn may be
non-Gaussian due to its nonlinear dependence on
df
This effect is small for a single-field slow-roll
model (? linear approximation is extremely good)
Salopek Bond (90), ...
But it may become large for multi-field models
Lyth Rodriguez (05), ....
Non-Gaussianity in this case is local
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In the rest of this talk we focus on this case, ie
non-Gaussianity generated on superhorizon scales
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3. Non-Gaussianity from superhorizon scales
- two typical models for fNLlocal -

• curvaton model

Linde Mukhanov (1996), Lyth Wands (2001),
Moroi Takahashi (2001), ...

• multi-brid inflation model

MS (2008), Naruko MS (2008)
both may give large fNLlocal
but in the case of curvaton scenario tensor-scalar
ratio r will be very small.
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Curvaton model
Inflation is dominated by inflaton f
Curv. perts. are dominated by curvaton c
during inflation
f dominates curvature perturbations during
inflation
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After inflation, f thermalizes, c undergoes
damped oscillation.
Assume c dominates final amp of curvature
perturbations
fNL 1/q
Lyth Rodriguez (05) Malik Lyth (06) MS,
Valiviita Wands (06)
Large NG is possible if qltlt1
But tensor mode is strongly suppressed
inflaton
curvaton
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4. Multi-brid inflation

• hybrid inflation
• inflation ends by a sudden destabilization
  of vacuum

• multi-brid inflation multi-field hybrid
  inflation

Slow-roll eom
N as a time
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simple 2-brid example the case
of radial inflationary orbits on (f1,f2)
f2
q
f
N0
f1
Nconst.
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Three types of dN
( ) indicates field perturbations
f2
f1
end of inflation or phase transition
originally adiabatic
(standard scenario)
entropy ? adiabatic
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condition to end the inflation
f1, f2 inflaton fields (2-brid inflation)
c waterfall field
during inflation
V0
inflation ends when
c
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In general,
For 2-component case,
true entropy perturbation
these could be important sources of NG
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Simple analytically soluble model
MS (08)

• exponential potential

parametrize the end of inflation

• dN to 2nd order in df

• Spectrum of curvature perturbation

spectral index
tensor/scalar
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isocurvature perturbation
isocurvature contributes at 2nd order
possibility of large non-Gaussianity
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Just an example ...
input parameters
outputs
indep. of waterfall c
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WMAP 5yr
Komatsu et al. 08
WMAPBAOSN
WMAP
present example
tensor-scalar ratio r is not suppressed
AND fNLlocal 50 (positive and large)
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Another example O(2) SSB model
Inflation ends at
symmetry breakdown at the end of inflation
f2
Alabidi Lyth 06
f1
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• curvature perturbation spectrum

• spectral index

• tensor/scalar

• non-gaussianity

for
Again, fNLlocal may be large and positive.
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(No Transcript)
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It seems fNLgt0 always in multi-brid inflation
Any good reason for
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Perhaps yes!
inflaton trajectories
y
For hyperbolic end of inflation condition fNL
may become negative.
Huang 09
x
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simplest example
End of inflation condition
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The result is
If the end surface is concave (hyperbolic), fNL
can become negative.
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5. Summary

• 3 types of non-Gaussianity

1. subhorizon quantum origin
These are important
2. superhorizon classical (local) origin
3. gravitational dynamics classical origin

• DBI inflation --- type 1.

can be large

• curvaton scenario, multi-brid inflation --- type
  2.

can be large. sign is important, too.
but curvaton scenario predicts r1 if fNL is
large.
multi-brid inflation can give r0.1
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Non-Gaussianity plays an important role in
determining (constraining) models of inflation
4-pt function (trispectrum) may be detected in
addition to 3-pt function (bispectrum)
Arroja Koyama (08), Huang (09),
PLANCK started to take data!
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Planck first light survey
WMAP 5yr
NG may be detected in the very near future!