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Yukawa Institute for Theoretical Physics Kyoto University

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100 yrs after Einstein in Prague

26 June 2012

Inflation and Birth of

Cosmological Perturbations

Misao Sasaki

Yukawa Institute for Theoretical Physics

Kyoto University

0. Horizon & flatness problem

- horizon problem

gravity=attractive

conformal time is

bounded from below

E

particle horizon

solution to the horizon problem

for a sufficient lapse of time in the early universe

or large enough to

cover the present

horizon size

NB: horizon problem≠

homogeneity & isotropy problem

flatness problem (= entropy problem)

alternatively, the problem is the existence of huge

entropy within the curvature radius of the universe

(# of states = exp[S])

solution to horizon & flatness problems

spatially homogeneous scalar field:

potential dominated

V ~ cosmological const./vacuum energy

inflation

“vacuum energy” converted to radiation

after sufficient lapse of time

solves horizon & flatness problems simultaneously

1．Slow-roll inflation and vacuum fluctuations

- single-field slow-roll inflation

Linde ’82, ...

metric：

V(f)

field eq.：

f

may be realized for various potentials

∙∙∙ slow variation of H

inflation!

comoving scale vs Hubble horizon radius

k: comoving wavenumber

log L

superhorizon

subhorizon

subhorizon

log a(t)=N

t=tend

inflation

hot bigbang

e-folding number: N

# of e-folds from f=f(t)

until the end of inflation

redshift

log L

N=N(f)

L=H-1 ~ t

L=H-1~ const

log a(t)

t=t(f)

t=tend

Curvature (scalar-type) Perturbation

- intuitive (non-rigorous) derivation

- inflaton fluctuation (vacuum fluctuations=Gaussian)

rapid expansion renders oscillations frozen at k/a < H

(fluctuations become “classical” on superhorizon scales)

- curvature perturbation on comoving slices

∙∙∙ conserved on superhorizon scale,

for purely adiabatic pertns.

evaluated on ‘flat’ slice (vol of 3-metric unperturbed)

Curvature perturbation spectrum

- spectrum

~ almost scale-invariant

(rigorous/1st principle derivation by Mukhanov ’85, MS ’86)

- dN - formula

Starobinsky (’85)

geometrical justification

MS & Stewart (’96)

non-linear extension

Lyth, Marik & MS (’04)

Starobinsky (’79)

: transverse-traceless

- canonically normalized tensor field

- tensor spectrum

- scalar spectrum:

- tensor spectrum:

- tensor spectral index:

··· valid for all slow-roll models

with canonical kinetic term

Comparison with observation

- Standard (single-field, slowroll) inflation predicts scale-invariant Gaussian curvature perturbations.

WMAP 7yr

- CMB (WMAP) is consistent withthe prediction.
- Linear perturbation theory seems to be valid.

CMB constraints on inflation

Komatsu et al. ‘10

- scalar spectral index: ns = 0.95 ~ 0.98
- tensor-to-scalar ratio: r < 0.15

Inflation may be non-standard

multi-field, non-slowroll, DBI, extra-dim’s, …

PLANCK, …may detect non-Gaussianity

(comoving) curvature perturbation:

B-mode (tensor)may or may not be detected.

energy scale of inflation

modified (quantum) gravity? NG signature?

Quantifying NL/NG effects is important

2. Origin of non-Gaussianity

- self-interactions of inflaton/non-trivial “vacuum”

quantum physics, subhorizon scaleduring inflation

- multi-field

classical physics, nonlinear coupling to gravity

superhorizon scaleduring and after inflation

- nonlinearity in gravity

classical general relativistic effect,

subhorizon scaleafter inflation

Origin of NG and cosmic scales

k: comoving wavenumber

log L

classical/local

effect

classical gravity

quantum effect

log a(t)

t=tend

inflation

hot bigbang

Origin１：self-interaction/non-trivial vacuum

Non-Gaussianity generated on subhorizon scales

(quantum field theoretical)

- conventional self-interaction by potential is ineffective

Maldacena (’03)

ex. chaotic inflation

∙∙∙ free field!

(grav. interaction is Planck-suppressed)

extremely small!

- need unconventional self-interaction
- → non-canonical kinetic termcangenerate large NG

∝

∝

0

g 3+2

g 3

1a. Non-canonical kinetic term: DBI inflationSilverstein & Tong (2004)

kinetic term：

~ (Lorenz factor)-1

perturbation expansion

large NG for large g

fNLlarge for equilateral configuration

Bi-spectrum (3pt function) in DBI inflationAlishahiha et al. (’04)

WMAP 7yr：

1b. Non-trivial vacuum

- de Sitter spacetime = maximally symmetric

(same degrees of sym as Poincare (Minkowski) sym)

gravitational interaction (G-int) is negligible in vacuum

(except for graviton/tensor-mode loops)

- slow-roll inflation : dS symmetry is slightly broken

G-int induces NG but suppressed by

But large NG is possible if the initial state (or state at

horizon crossing) does NOT respect dS symmetry

(eg, initial state ≠ Bunch-Davies vacuum)

various types of NG :

scale-dependent, oscillating, featured, folded ...

Chen et al. (’08), Flauger et al. (’10), ...

Origin 2：superhorizon generation

- NG may appear if T mndepends nonlinearly ondf,
- even if df itself is Gaussian.

This effect is smallinsingle-field slow-roll model

(⇔ linear approximation is valid to high accuracy)

Salopek & Bond (’90)

- For multi-field models, contribution to T mnfrom each field
- can be highly nonlinear.

NG is always of local type:

x

WMAP 7yr：

dN formalism for this type of NG

Origin 3：nonlinearity in gravity

ex. post-Newtonian metric in asymptotically flat space

Newton

potential

NL (post-Newton) terms

(in both local and nonlocal forms)

- important when scales have re-entered Hubble horizon

distinguishable from NL matter dynamics?

- effect on CMB bispectrum may not be negligible

?

Pitrou et al. (2010)

(for both squeezed and equilateral types)

3. dN formalism

What is dN?

- dN is the perturbation in # of e-folds counted
- backward in time from a fixed final time tf

therefore it is nonlocal in time by definition

- tf should be chosen such that the evolution of the
- universe has become unique by that time.

=adiabatic limit

- dN is equal to conserved NL comoving curvature
- perturbation on superhorizon scales at t>tf

- dN is valid independent of gravity theory

Chooseflat slice at t = t1 [ SF (t1) ] and

comoving (=uniform density) at t = t2 [ SC (t1) ] :

NonlineardN - formula

( ‘flat’ slice: S(t) on which )

SC(t2) : comoving

r (t2)=const.

R(t2)=0

SF(t2) : flat

SF (t1) : flat

R(t1)=0

Nonlinear dNfor multi-component inflation :

wheredf =dfFis fluctuation on initial flat slice at or after horizon-crossing.

dfF may contain non-Gaussianity from

subhorizon (quantum) interactions

eg, in DBI inflation

4．NG generation on superhorizon scales

two efficient mechanisms to convert

isocurvature to curvature perturbations:

curvaton-type & multi-brid type

- curvaton-type

Lyth & Wands (’01), Moroi & Takahashi (‘01),...

rcurv(f) << rtot during inflation

highly nonlinear dep of drcurv on df possible

rcurv(f) can dominate after inflation

curvature perturbation can be highly NG

- multi-brid inflation

MS (’08), Naruko & MS (’08),...

sudden change/transition in the trajectory

curvature of this surface

determines sign of fNL

tensor-scalar ratio r may be large in multi-brid models,

while it is always small in curvaton-type if NG is large.

5. Summary

- inflation explains observed structure of the universe

flatness:W0=1 to good accuracy

curvature perturbation spectrum

almost scale-invariant

almost Gaussian

- inflation also predicts scale-invariant tensor spectrum

will be detected soon if tensor-scalar ratio r>0.1

any new/additional features?

non-Gaussianities

- 3 origins of NG in curvature perturbation

1. subhorizon ∙∙∙ quantum origin

NG from

inflation

2. superhorizon ∙∙∙ classical (local) origin

3. NL gravity ∙∙∙ late time classical dynamics

- DBI-type model: origin 1.

need to be quantified

may be large

- non BD vacuum: origin 1.

any type ofmay be large

- multi-field model: origin 2.

may be large:

In curvaton-type modelsr≪1.

Multi-brid model may giver~0.1.

Identifying properties of non-Gaussianity

is extremely important for understanding

physics of the early universe

not only bispectrum(3-pt function) but also

trispectrum or higher order n-pt functions

may become important.

Confirmation of primordial NG?

PLANCK (February 2013?) ...

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