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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

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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


conformal time is

bounded from below


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
solution to horizon & flatness problems

spatially homogeneous scalar field:

potential dominated

V ~ cosmological const./vacuum energy


“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, ...



field eq.:


may be realized for various potentials

∙∙∙ slow variation of H


comoving scale vs hubble horizon radius
comoving scale vs Hubble horizon radius

k: comoving wavenumber

log L




log a(t)=N



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


log L


L=H-1 ~ t

L=H-1~ const

log a(t)



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)


Tensor Perturbation

Starobinsky (’79)

: transverse-traceless

  • canonically normalized tensor field
  • tensor spectrum

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


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 gravity

quantum effect

log a(t)



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



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


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:


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



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



3 types of dN

end of/after


originally adiabatic

entropy/isocurvature → adiabatic


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.


SF(t2) : flat

SF (t1) : flat



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



  • 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
  • 3 origins of NG in curvature perturbation

1. subhorizon ∙∙∙ quantum origin

NG from


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?) ...