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cifar08/Lindefest 5-9 March, 2008. Nonlinear curvature perturbations in two-field hybrid inflation. -- d N in exactly soluble models --. Yukawa Institute (YITP) Kyoto University. Misao Sasaki. Happy Kanreki, Andrei!. kanreki = one cycle of chinese zodiacal calendar. 祝還暦. 60 yrs =

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nonlinear curvature perturbations in two field hybrid inflation


5-9 March, 2008

Nonlinear curvature perturbations in two-field hybrid inflation

-- dN in exactly soluble models --

Yukawa Institute (YITP)

Kyoto University

Misao Sasaki


Happy Kanreki, Andrei!

kanreki = one cycle of

chinese zodiacal calendar


60 yrs =

12 animals

x 5 elements



this (Andrei’s) year

Red signifies

a baby

mouse x earth

also, Red is a color

of happiness



A person is regarded as reborn on the 60th birthday

1 success of inflation
1. Success of Inflation

(slow-roll) inflation

Linde ’82, ...

explains the origin of cosmological perturbations.


finally observed by COBE & WMAP!

Inflation from string theory

KKLMMT ’03, ...

brane/DBI inflation, moduli inflation, ...

may lead to large non-Gaussianity.

−9 < fNLlocal < 111 (WMAP 5yr)

may be detected in the very near future

why care about soluble models
Why care about soluble models?

In this talk, I analyze full nonlinear curvature perturbations

in exactly soluble models of slow-roll inflation

Using dN formalism,

(dN··· e-folding number perturbation)

  • We can explicitly see how and when curvature
  • perturbations are generated.
  • Non-Gaussianity can be explicitly evaluated.

We may deepen our understanding of

cosmological perturbations.





Because it’s fun!

2 dn for curvature perturbations
2. dN for curvature perturbations

Starobinsky ‘85, MS & Stewart ‘96, MS & Tanaka \'98,

Lyth, Malik & MS ‘04,....

Separate universe (gradient expansion) approach

Belinski et al. ’70, Tomita ’72, Salopek & Bond ’90, …

  • each Hubble region evolves independently
  • perturbations on superhorizon scales
  • ≈ difference between different FLRW universes
  • calculate dN for different FLRW universes

٠٠٠ curvature perturbation

on comoving slice

defined by an integral

=non-local quantity

→ gauge invariance

d n for slowroll type inflation
dN for ‘slowroll-type’ inflation

MS & Tanaka ’98, Lyth & Rodriguez ’05, ...

  • In slowroll inflation, all decaying mode solutions of the
  • (multi-component) scalar field f die out.

→ N=indep. of df/dt

where df =dfF (on flat slice) at horizon-crossing.

  • The above formula is valid for any model in which
  • N is only a function of f but not of df/dt.

(e.g., power-law inflation)

3 exactly soluble slow roll models
3. Exactly soluble slow-roll models

(a,b= 1,2,...,n)

Slow-roll equations of motion

e-folding number from the end of inflation

an exactly soluble class
An exactly soluble class


for each a

(a= 1,2,...,n)

this is sufficient

(slowroll unnecessary)



new field space


··· na is conserved

solvability  (n-1) constants of integration

simple examples
simple examples




e-folding number

  • Nonlinear dN is
  • Linear curvature perturbation is given by

where (q, na) and (dq, dna) are the values at horizon

crossing during inflation


curvature perturbation generated from adiabatic & “entropy” perturbations

perturbations orthogonal to dqa/dN

adiabatic perturbation

entropy perturbation during inflation

Polarski & Starobinsky ’94,

Mukhanov & Steinhardt ’97, MS & Tanaka ’98,...

entropy perturbation at the end of inflation

Bernardeau, Kofman & Uzan ‘04, Lyth ‘05, ...

4 two plus one field hybrid inflation




4. Two (plus one)-field hybrid inflation

∙ can be easily generalized to n fields

“n-brid inflation”

  • slow-roll eom:
  • transformation of variables:

Assume that inflation ends at

and the universe is thermalized instantaneously.

realized by

Parametrize orbits by an angle at the end of inflation


(∙∙∙ const of motion)

This determines g in terms of f1 & f2 .


  • dN valid to full nonlinear order is simply given by

To be precise, one has to add a correction term to adjust

  • the energy density difference at the end of inflation


(assuming instantaneous thermalization)

However, this correction is negligible



dN to 2nd order in df:

  • comoving curvature perturbation spectrum

spectral index:


  • single-field case

No non-Gaussianity if dfis Gaussian



“true” entropy perturbation

linear entropy perturbation

contributes at 2nd order

practically any non-Gaussianity is possible

numbers just in case
numbers just in case...

(respecting WMAP5yr)

model parameters:


5 summary
5. Summary
  • Exactly soluble models are useful in understanding
  • generation of (nonlinear) curvature perturbations
  • Curvature perturbation may be generated from both
  • adiabatic and entropy perturbations during inflation
  • In multi-field models, final amplitude of curvature
  • perturbation depends crucially on how inflation ends.
  • n-brid inflation looks like an almighty model!

(classically. negligible quantum corrections assumed)