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. -- 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

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


Nonlinear curvature perturbations in two field hybrid inflation

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


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.

tatemae

(建前)

hon-ne

(本音)

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

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


Simple examples

simple examples

product:

sum:


Nonlinear curvature perturbations in two field hybrid inflation

trajectories are radial in space

qa

na

q

N=0

qb

N=const.


Nonlinear curvature perturbations in two field hybrid inflation

  • 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


Nonlinear curvature perturbations in two field hybrid 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

V

f

inflation

4. Two (plus one)-field hybrid inflation

∙ can be easily generalized to n fields

“n-brid inflation”

  • slow-roll eom:

  • transformation of variables:


Nonlinear curvature perturbations in two field hybrid inflation

Assume that inflation ends at

and the universe is thermalized instantaneously.

realized by

Parametrize orbits by an angle at the end of inflation


Nonlinear curvature perturbations in two field hybrid 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


Nonlinear curvature perturbations in two field hybrid inflation

  • 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


Nonlinear curvature perturbations in two field hybrid inflation

  • dN to 2nd order in df:

  • comoving curvature perturbation spectrum

spectral index:

tensor/scalar:

  • single-field case

No non-Gaussianity if dfis Gaussian


Nonlinear curvature perturbations in two field hybrid inflation

Let

“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:

outputs:


Nonlinear curvature perturbations in two field hybrid inflation

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

hope you like it...


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)


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