slide1 n.
Download
Skip this Video
Download Presentation
Yukawa Institute for Theoretical Physics Kyoto University

Loading in 2 Seconds...

play fullscreen
1 / 32

Yukawa Institute for Theoretical Physics Kyoto University - PowerPoint PPT Presentation


  • 95 Views
  • Uploaded on

Relativity & Gravitation 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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Yukawa Institute for Theoretical Physics Kyoto University' - donar


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

Relativity & Gravitation

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
0. Horizon & flatness problem
  • horizon problem

gravity=attractive

conformal time is

bounded from below

E

particle horizon

slide3

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

slide4

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
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
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
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

fdetermines comoving scale k

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
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
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)

slide11

Tensor Perturbation

Starobinsky (’79)

: transverse-traceless

  • canonically normalized tensor field
  • tensor spectrum
slide12

Tensor-to-scalar ratio

  • scalar spectrum:
  • tensor spectrum:
  • tensor spectral index:

··· valid for all slow-roll models

with canonical kinetic term

comparison with observation
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
CMB constraints on inflation

Komatsu et al. ‘10

  • scalar spectral index: ns = 0.95 ~ 0.98
  • tensor-to-scalar ratio: r < 0.15
slide15

However,….

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
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
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

origin self interaction non trivial vacuum
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
1a non canonical kinetic term dbi inflation

=

0

g 3+2

g 3

1a. Non-canonical kinetic term: DBI inflation

Silverstein & Tong (2004)

kinetic term:

~ (Lorenz factor)-1

perturbation expansion

large NG for large g

1b non trivial vacuum
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

rtot

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
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 d n formalism
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
3 types of d n

f2

f1

3 types of dN

end of/after

inflation

originally adiabatic

entropy/isocurvature → adiabatic

slide26

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

slide27

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
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

slide29

df

  • 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
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
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.

slide32

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?) ...