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Nonlinear curvature perturbations in two-field hybrid inflation

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cifar08/Lindefest

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

祝還暦

60 yrs =

12 animals

x 5 elements

chan-chanko

clothing

this (Andrei’s) year

Red signifies

a baby

mouse x earth

also, Red is a color

of happiness

reliability

productivity

A person is regarded as reborn on the 60th birthday

(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

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.

tatemae

(建前)

hon-ne

(本音)

Because it’s fun!

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

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)

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

Slow-roll equations of motion

e-folding number from the end of inflation

If

for each a

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

this is sufficient

(slowroll unnecessary)

Then

...

new field space

coordinates

··· na is conserved

solvability (n-1) constants of integration

product:

sum:

trajectories are radial in space

qa

na

q

N=0

qb

N=const.

- 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, ...

V

f

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 .

whereg=g(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

where

(assuming instantaneous thermalization)

However, this correction is negligible

if

- dN to 2nd order in df:

- comoving curvature perturbation spectrum

spectral index:

tensor/scalar:

- single-field case

No non-Gaussianity if dfis Gaussian

Let

“true” entropy perturbation

linear entropy perturbation

contributes at 2nd order

practically any non-Gaussianity is possible

(respecting WMAP5yr)

model parameters:

outputs:

So, here is a birthday present for you, Andrei.

hope you like it...

- 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)