Yukawa Institute for Theoretical PhysicsKyoto University

Misao Sasaki

Relativity & Gravitation100 yrs after Einstein in Prague

26 June 2012

2

0. Horizon & flatness problem0. Horizon & flatness problem• horizon problem

43 0

3 3 for ( )

Ga P a P

&&

2 2 2 23( )( )( )ds a d d conformal time:

( )dt

da t

conformal time isbounded from below

1if , na t n gravity=attractive

0 0

finite( )

t

t

dta t

2 2 2 23( )( )ds dt a t d

particle horizon

E

3

• solution to the horizon problem4

3 03

( )G

a P a &&

for a sufficient lapse of time in the early universe

0

00 0

( )

t

tt

dta t

or large enough tocover the present

horizon size

NB: horizon problem≠homogeneity & isotropy problem

4

22

8

3 ;

G KH K

a

• flatness problem (= entropy problem)

42

if in the early universe.| |

,K

aa

?

2

0

0

conversely if at an epoch in the early universe,

the universe must have either

or become completely

collapsed (if )

empty ( by noif w.)

| | /

K

K

K

a

alternatively, the problem is the existence of hugeentropy within the curvature radius of the universe

3 3

3 3 3 3 8700 0 0 10

| | | |

aaS T T T H

K K

(# of states = exp[S])

5solution to horizon & flatness solution to horizon & flatness problemsproblems

spatially homogeneous scalar field:

2 21 1

2 2 ( ), ( )V P V & &

2 23 2 0 if ( ) ( )P V V & &

potential dominated2 if ( ) ( )P V V &=

V ~ cosmological const./vacuum energy

2 decreases rapidly.

Kconst

a

inflation

“vacuum energy” converted to radiationafter sufficient lapse of time

solves horizon & flatness problems simultaneously

2 8

3.

GH const

6

11 ．． Slow-roll inflation and vacuum Slow-roll inflation and vacuum fluctuationsfluctuations

• single-field slow-roll inflation

V(V())

2

2 2

3 =0

8 1

3 2

( )

( )

H V

a GH V

a

&& &

& &

2 2 2( ) i jijds dt a t dx dx

22

22

332 1

1 22

HH VV

&& &=

&∙∙∙ slow variation of H

inflation!

metric：

field eq.：

Linde ’82, ...

3

( )VH

&

~ Hta e

may be realized for various potentials

7

comoving scale vs Hubble horizon radiuscomoving scale vs Hubble horizon radius

log log aa((tt)=N)=N

log log LL

tt==ttendend

aL

k

k: comoving wavenumber

1L H

inflation

subhorizon

superhorizon

subhorizon

hot bigbang

kH

a

kH

a

kH

a

kH

a

kH

a

8

e-folding number: Ne-folding number: N

log log aa((tt))

log log LL

LL==HH-1 -1 ~ ~ tt

tt==tt(()) tt==ttendend

end

1( )

( ) ~ln ( )t

tN N Hdt z

NN==NN(())

endend

( )exp[ ( )]

( )a t

N t ta t

LL==HH-1-1~ const~ const

redshift# of e-folds from =(t) until the end of inflation

determines comoving scale k

9

• inflaton fluctuation (vacuum fluctuations=Gaussian)2 2 1

2

, ~ ;kiw tk k k

k

kk e w H

aw

v?

rapid expansion renders oscillations frozen at k/a < H(fluctuations become “classical” on superhorizon scales)

22

3

22 / ~

~ ;k kk a H

H k HH

ak

=

Curvature (scalar-type) PerturbationCurvature (scalar-type) Perturbation intuitive (non-rigorous) derivation

• curvature perturbation on comoving slices

c

H

R & ∙∙∙ conserved on superhorizon scale, for purely adiabatic pertns.

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

10

Curvature perturbation spectrumCurvature perturbation spectrum

• N - formula

( )( )

end endt

t

HN Hdt d

&

( ) ck k

H Ha a

N HN

R&

2 222

2( ) ;kk Hk aH

a

H NP k

R &

2322

22 2k k

k H

~ almost scale-invariant22

2( )

kH

a

HP k

R &• spectrum

MS & Stewart (’96)AA

A

NN

geometrical justification

non-linear extension Lyth, Marik & MS (’04)

Starobinsky (’85)

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

11

Tensor PerturbationTensor Perturbation0i TT ij TT

ij ijh h

,

( ; ) ( ) ( ) . .ij k ij kk t a P k t h c

( ):k t same as massless scalar

• canonically normalized tensor field

1 1

232 8;TT TTP

ij ij ij P

Mh h M

G G

: transverse-traceless

2 22 2

2 2 2

4 82 4

2, ,

( )kTT

ij ijP P P

Hh k k

M M M

v v• tensor spectrum

2

4 1

2~ ijS d x g

t

ggg

Starobinsky (’79)

12

Tensor-to-scalar ratioTensor-to-scalar ratio

• scalar spectrum:scalar spectrum: 2

22

2s

HP k N

• tensor spectrum:tensor spectrum: 8

2

2

2 2( )g

P

HP k

M

• tensor spectral index:tensor spectral index:2

2 2

2 2 2 22g

P P

Hn

H M H M N

& &&

&

aa

dNH N

dt

g

8gg

s

Pr n

P ··· ··· valid for all slow-roll modelsvalid for all slow-roll models

with canonical kinetic termwith canonical kinetic term

1snk

gnk

1

822

g

sP

P

PM N

2( )ab

a b

N NN G

13

Comparison with observationComparison with observationStandard (single-field, slowroll) inflation predicts Standard (single-field, slowroll) inflation predicts scale-invariant scale-invariant GaussianGaussian curvature perturbations. curvature perturbations.

CMB (WMAP) is consistent with the prediction.

Linear perturbation theory seems to be valid.

WMAP 7yr

14

CMB constraints on inflationCMB constraints on inflationKomatsu et al. ‘10

scalar spectral index: ns = 0.95 ~ 0.98

tensor-to-scalar ratio: r < 0.15

15

However,….

Inflation may be non-standard

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

Quantifying NL/NG effects is important

2gauss gauss

3

55? C ~;NL NLff R R R L

PLANCK, … may detect non-Gaussianity

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

energy scale of inflation -10 2Planck10 ?2H M

modified (quantum) gravity? NG signature?

(comoving) curvature perturbation:

16

2. Origin of non-Gaussianity2. Origin of non-Gaussianity

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

• nonlinearity in gravity

• multi-field

classical physics, nonlinear coupling to gravity superhorizon scale during and after inflation

quantum physics, subhorizon scale during inflation

classical general relativistic effect, subhorizon scale after inflation

17

Origin of NG and cosmic scalesOrigin of NG and cosmic scales

log log aa((tt))

log log LL

tt==ttendend

aL

k

k: comoving wavenumber

1L H

inflation

classical gravity

classical/localeffect

quantum effect

hot bigbang

18

OriginOrigin １：１： self-interaction/non-trivial vacuumself-interaction/non-trivial vacuum

• conventional self-interaction by potential is ineffective

4 V

• need unconventional self-interaction → non-canonical kinetic term can generate large NG

Non-Gaussianity generated on subhorizon scales

(quantum field theoretical)

ex. chaotic inflation2 21

2

V m ∙∙∙ free field!

(grav. interaction is Planck-suppressed)

Maldacena (’03)

21~ ( / )PlO M1510~

extremely small!

19

1a. Non-canonical kinetic term: DBI inflation1a. Non-canonical kinetic term: DBI inflation

1 21~ ( ) ( )K ff &kinetic term：

Silverstein & Tong (2004)

1 1f

~ (Lorenz factor)-1

perturbation expansion

0 1 2 3K K K K K

0

= ∝ ∝

3 3+2

2 20 0~

large NG for large

3 21

2 ;X X f

&

20

Bi-spectrum (3pt function) in DBI inflationBi-spectrum (3pt function) in DBI inflation

fNLlarge for equilateral configuration

　　1 2 3~ ~p p pv v v

1 2 3 0p p p v v v

2pv

3pv

1pv

1 3

1 21

2

2 3

( ) ( ) ( )

~ ( ) ( ), (( ), )

C C

L

C

j C Cj

N

p p

f p p p

p

p p p R R R

R R cyclic

2~NLf

equil NL NLff

WMAP 7yr： 241 266 95equil CL( % )NLf

Alishahiha et al. (’04)

21

1b. Non-trivial vacuum1b. Non-trivial vacuum

• de Sitter spacetime = maximally symmetric(same degrees of sym as Poincare (Minkowski) sym)

gravitational interaction (G-int) is negligible in vacuum

• slow-roll inflation : dS symmetry is slightly brokenG-int induces NG but suppressed by

(except for graviton/tensor-mode loops)

2/ PlH M &

But large NG is possible if the initial state (or state athorizon 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), ...

3 1( , )SO

23

Origin 2Origin 2：： superhorizon generationsuperhorizon generation

• NG may appear if T depends nonlinearly on , even if itself is Gaussian.

This effect is small in single-field slow-roll model(⇔ linear approximation is valid to high accuracy)

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

NG is always of local type:

1 2 3local const.( , , )NL NLf p p p f

Salopek & Bond (’90)

N formalism for this type of NG

x

tot

=A tot

WMAP 7yr： 10 74 95local CL( % )NLf

24

Origin 3Origin 3：： nonlinearity in gravitynonlinearity in gravity

ex. post-Newtonian metric in asymptotically flat space

2 2 2 2 21 2 2 1 2 2ds dt dr

NL (post-Newton) termsNewtonpotential

• important when scales have re-entered Hubble horizon

5~ ( )NLf O

• effect on CMB bispectrum may not be negligible

Pitrou et al. (2010)

(for both squeezed and equilateral types)

(in both local and nonlocal forms)

distinguishable from NL matter dynamics?

?

25

3. 3. N formalismN formalism

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

• N is equal to conserved NL comoving curvature perturbation on superhorizon scales at t>tf

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

• N is valid independent of gravity theory

What is What is N?N?

therefore it is nonlocal in time by definition

=adiabatic limit

26

3 types of 3 types of NN

originally adiabatic

end of/afterinflation

entropy/isocurvature → adiabatic

ft t

1

2

27

NonlinearNonlinear N - formulaN - formulaChoose flat slice at t = t1 [ F (t1) ] andcomoving (=uniform density) at t = t2 [ C

(t1) ] :( ‘flat’ slice: (t) on which )

F (t1) : flat

C(t2) : comoving (t2)=const.

(t1)=0

F(t2) : flat

0 2 12 11 2, , , ; iFN t N t t N t t xt

C2 21, ,; i iF t t x t xN R

(t2)=0

(3) 3 3 ( , ) 3det ( ) ( )t xa t e a t R

28

• Nonlinear N for multi-component inflation :

1 2

1 2

1

!

n

n

A A A

nAA A

AA An

N N N

N

n

where =F is fluctuation on initial flat slice at or after horizon-crossing.

F may contain non-Gaussianity from subhorizon (quantum) interactions

eg, in DBI inflation

2944．． NG generation on superhorizon NG generation on superhorizon scalesscales

• curvaton-type Lyth & Wands (’01), Moroi & Takahashi (‘01),...

two efficient mechanisms to convertisocurvature to curvature perturbations:

curv() << tot during inflation

highly nonlinear dep of curv on possible

curv() can dominate after inflation

curvature perturbation can be highly NG

curvaton-type & multi-brid type

30

• multi-brid inflationMS (’08), Naruko & MS (’08),...

sudden change/transition in the trajectory

21

2a a b

a abN N N L

tensor-scalar ratio r may be large in multi-brid models,while it is always small in curvaton-type if NG is large.

curvature of this surface determines sign of fNL

31

5. Summary5. Summary

• inflation explains observed structure of the universe

flatness: 0=1 to good accuracy

curvature perturbation spectrum

almost scale-invariantalmost Gaussian

• inflation also predicts scale-invariant tensor spectrum

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

any new/additional features?

32

• 3 origins of NG in curvature perturbation

• multi-field model: origin 2.

• DBI-type model: origin 1.

1. subhorizon ∙∙∙ quantum origin

2. superhorizon ∙∙∙ classical (local) origin

3. NL gravity ∙∙∙ late time classical dynamics

equilNLf may be large

localNLf may be large: In curvaton-type models r≪1.

Multi-brid model may give r~0.1.

NG frominflation

need to be quantified

• non BD vacuum: origin 1.

NLfany type of may be large

non-Gaussianitiesnon-Gaussianities

33

Identifying properties of non-Gaussianityis extremely important for understanding

physics of the early universe

not only bispectrum(3-pt function) but alsotrispectrum or higher order n-pt functions

may become important.

Confirmation of primordial NG?

PLANCK (February 2013?) ...