] r/H [ Log *2 10

Open Inflation

in the String Landscape

Misao Sasaki

(YITP, Kyoto University)

Chuo University 6 December, 2011

D. Yamauchi, A. Linde, A. Naruko, T. Tanaka & MS,

PRD 84, 043513 (2011) [arXiv:1105.2674 [hep-th]]

K. Sugimura, D. Yamauchi & MS, arXiv:1110.4773 [gr-qc]

1. Brief overview of Inflation

2

isotropic component isotropic component

dipole (motion of solar system) dipole (motion of solar system)

WMAP 7 years data WMAP 7 years data

10-3 371 km/s (δT T/ _CMB)_ℓ=1 ≈ ⇒ =v

-5

20 10 Large Scale Structure (δT T/ _CMB)_ℓ∼ ≈ ⇔

multipole components multipole components

=2.73 K TCMB

-5

2 700 10

(δT T/ _CMB) _{≤ ≤ℓ} ≈

COBE-DMR (1990) WMAP (2003~)

Observed CMB anisotropy Map

Why the detection of δT/T at θ >10º was so important?

Because in the standard Friedmann universe, the size of causal volume (horizon size) grows like ~ ct.

Horizon problem

Hubble horizon size

Expansion of the Universe

Horizon grows faster than the cosmic expansion in the standard Friedmann (Bigbang) Universe a(t

)

a(t

)

a(t

) cH

~ct

a(t) ∝ t

for hot bigbang universe

Origin of horizon problem

Last Scattering Surface (t=4x10

yr)

t=0

time we are here

t=1.4x10

yr

Horizon problem in Big Bang Universe

size of causal region

~1º Now

Last Scattering Surface (t=3x10

yr)

t=0 horizon

size

There are ~10

causally independent patches on LSS

(t~10

yr)

Why the detection of δT/T at θ >10º was so important?

Because in the standard Friedmann universe, the size of causal volume (horizon size) grows like ~ ct.

• Thus, any causal, physical process cannot produce correlation on scales θ >1º.

• The angle sustaining the horizon size at LSS is ~ 1º.

• But (δT/T)

≠ 0 means there exists non-zero correlation.

Horizon problem

Inflationary Universe Inflationary Universe

•Universe dominated by a scalar (inflaton) field

•For sufficiently flat potential:

2 8 1 2

3G ( ) 2 ( )

H π V V

φ ^ φ φ ^

≈ ⇐ 

 ɺ ≪ 

• φ slowly rolls down the potential: slow-roll (chaotic) inflation

2 2

3 1

2 ( ) H

H V

φ

⇒ = φ

ɺ ɺ

≪

• H is almost constant ~ exponential expansion = inflation

• Inflation ends when φ starts damped oscillation.

φ V(φ)

φ decays into thermal energy (radiation) Birth of Hot Bigbang Universe

Linde (1983)

→ solves the horizon problem.

c H

Universe expands exponentially,

while the Hubble horizon size remains almost constant.

a(t)~e

; H~const.

A small region of the universe

An initially tiny region can become much larger than the entire observable universe

Hubble horizon during inflation

log L

log a(t) L=c H

Size of the observable

universe L ∝ a(t)

Inflationary Universe Bigbang Universe

Length Scales of the Inflationary Universe

Size of our observable universe small universe

expands by a factor >10

Birth of a gigantic universe

looks perfectly flat

Flatness can be explained only by Inflation

Flatness of the Universe

Zero-point (vacuum) fluctuations of φ :

2 2 2

2

2 2

0

3 ; ( )

( ) (

( ) )

k H k k k c

t a t t

t π

δφ δφ δφ ω

ω ^ λ ^

+ + = = ≡  

 

ɺɺ ɺ

( ) ^{ik x}

k k

δφ

∑ δφ

harmonic oscillator with friction term and time-dependent ω δφ

→ const.

··· frozen when λ > c H

(on superhorizon scales)

δφ_k

gravitational wave modes also satisfy the same eq.

physical wavelength λ(t)∝a(t)

Seeds of cosmological perturbations

curvature perturbation R ≈ gravitational potential Ψ

t

x

0

δφ 0

 =

 ≠

 R

0

δφ 0

 ≠

 =

 R

• δφ is frozen on “flat” ( R =0) 3-surface (t=const. hypersurface)

• Inflation ends/damped osc starts on φ =const. 3-surface.

end of inflation

hot bigbang universe

0 T = const., R ≠

Generation of curvature perturbations

•

Photons climbing up from grav potential well are redshifted.

Ψ Ψ Ψ Ψ

E E

E E

emit

( ) T 1 ( )

T n x

T T

∆ ≡ − = Ψ

•

In an expanding universe, this is modified to

( ) 3 ( )

T n x

T

∆ = Ψ

Sachs-Wolfe effect

•

There is also the standard Doppler effect:

( ) ( _emit ) T n n v x

T

∆ = −

i

For Planck distribution,

c=1 units

d

0

CMB anisotropy from curvature perturbation

LSS LSS

1 minor corrections

( ) 3 ( ) ( ) (

T n x n v x T

∆ = Ψ − +

i ⋯ )

Last Scattering Surface (LSS)

Observer

Ψ

•v

WMAP 7 year data (2010)

CMB anisotropy spectrum

2

2 _{k a H/} H

πφ ₌

=

R ɺ

• Amplitude of curvature perturbation:

• Power spectrum index:

3 2

1 2

3 2

4 1 2 3

2 ( ) ;

( )

nS

S pl

k V V

P k Ak n M

V V

π π

−  ′′ ′ 

= − =  − 

 

R

1 18

2 4 10 GeV: Planck mass

8 ~ .

Mpl

πG

≡ ×

Mukhanov (1985), Sasaki (1986)

• Tensor (gravitational wave) spectrum:

3 2

3

4 1

3

2 8

( ) ; ( )

( ) ( )

nT

T T

T

k P k

P k Ak n

V P k

π φ

π ^{= = − = −}

ɺ R

Liddle-Lyth (1992)

WMAP 1 0 049 0 017 1 0 04

, . . ~ .

S S

n − = − ± ⇔ n − − for a typical model

5 1 4 16

COBE ~10 ⁻ ⇒ V ^/ ( ) ~φ 10 GeV R

to be observed by PLANCK!

Summary of inflationary cosmology

• inflation (accelerated expansion) is a mechanism to solve horizon and flatness problems.

• slow-roll inflation can explain the observed structure of the universe.

• but need to identify the “inflaton” in unified theory, perhaps in string theory.

• any hint from observation/experiments?

20 20

2. String theory landscape

There are ~ 10

vacua in string theory

• vacuum energy ρ

may be positive or negative

• some of them have ρ

<<Μ

• typical energy scale ~ Μ

Lerche, Lust & Schellekens (’87), Bousso & Pochinski (’00), Susskind, Douglas, KKLT (’03), ...

which

?

21

Is there any way to know what kind of landscape we live in?

Or at least to know what kind of

neighborhood we live in?

22

• dS space: ρ

>0, O(4,1) symmetry

3

2 2 2 2 1 2

( )

( )

: cosh ,

/ 3

ds = − dt + a t d Ω a = H

Ht H = ρ M

2 1

(8 ) :

MP =

π

Ht

a ∝ e for t → ∞

( )

3

2 2 2 2 2

( )

sin

sin

d Ω

= d χ + χ d θ + θ φ d

de Sitter (dS) space

3-sphere

3 3

~ a ∝ e

Volume

• AdS space: ρ

<0, O(3,2) symmetry

3

2 2 2 2 1 2

( )

( ) : cos , |

| / 3

ds = − dt + a t d Ω

a = H

Ht H = ρ M

( )

3

2 2 2 2 2

( )

sinh

sin

d Ω

= d χ + χ d θ + θ φ d hyperbolic space

Anti-de Sitter space

collapses within t~ 1/H

24 24

A universe jumps around in the landscape by quantum tunneling

• it can go up to a vacuum with larger ρ

• if it tunnels to a vacuum with negative ρ

, it collapses within t ~ M

/|ρ

|

.

• so we may focus on vacua with positive ρ

: dS vacua

0 ρ

Sato, MS, Kodama & Maeda (’81)

( dS space ~ thermal state with T =H/2π )

25 25

Quantum Tunneling

= motion through a classically forbidden region

= described by motion with imaginary time

V(x)

x

2 2

1 1 ( )

2( )

2 2

dx dx dx

V E V E V E

dt d d

τ

τ τ

  + = ⇒   + = ⇔ = −

  

−  

  

E

x

tunneling probability:

τ = it

0 0

2

2

2

2 2( ) 2 2

2 1 .

2

1 .

2

out out out

out

x x

x x

dx dx

S V E dx dx d

d d

dx V d const d

dx V d const d

τ

τ

τ τ τ

τ τ

τ τ

−∞

−∞

∞

−∞

 

= − = =  

 

   

=    +  +

 

 

 

   

=    +  +

 

 

 

∫ ∫ ∫

∫

∫

“Euclidean bounce” action

= instanton x

“Euclidean” “Lorentzian”

26 26

3. Anthropic landscape

Susskind (‘03)

Not all of dS vacua are habitable.

“anthropic” landscape

• ρ

must not be larger than this value in order to account for the formation of stars and galaxies.

A universe jumps around in the landscape and settles down to a final vacuum with ρ

~ M

H

~(10

eV)

.

Just before it has arrived the final vacuum (=present universe), it must have gone through an era of (slow-roll) inflation and reheating, to create “matter and radiation.”

ρ

→ ρ

~ T

: birth of Hot Bigbang Universe

27

false vacuum decay via O(4) symmetric (CDL) instanton

inside bubble is an open universe

Coleman & De Luccia (‘80)

Most plausible state of the universe before inflation is a dS vacuum with ρ

~ M

. dS = O(4,1) O(5) ~ S

O(4) O(3,1)

2 2 2

x R

τ + =

2 2 2

t x R

− + =

bubble wall

false vacuum

creation of open universe

28

analytic continuation

open (hyperbolic) space

2 2 2

x R

τ + =

2 2 2

t x R

− + =

bubble wall

2 2

x co n st .

τ + =

2 2

t − x = co n s t .

29 29

Anthropic principle suggests that # of e-folds of inflation inside the bubble (N=H∆t) should be ~ 50 – 60 : just

enough to make the universe habitable.

Garriga, Tanaka & Vilenkin (‘98), Freivogel et al. (‘04)

Nevertheless, the universe may be slightly open:

2 3

1 − Ω =

10 ~ 10

Natural outcome would be a universe with Ω

<<1.

• “empty” universe: no matter, no life

Observational data excluded open universe with Ω

<1.

may be confirmed by Planck+BAO

Colombo et al. (‘09)

30

revisit open inflation!

What if 1-Ω ₀ is actually confirmed to be non-zero:~10 ^-2 -10 ^-3 ?

see if we can say anything about

Landscape

31 31

4. Open inflation in the landscape

• tunneling to a potential maximum ~ stochastic inflation

Simplest polynomial potential = Hawking-Moss model

Hawking & Moss (‘82)

2 3 4

2

2 3 4

V = m φ − ν φ + λ φ

• φ

potential:

Starobinsky (‘84)

– constraints from scalar-type perturbations –

V ′′ < H

φ slow-roll inflation

HM transition

• too large fluctuations of φ unless # of e-folds >> 60

Linde (‘95)

32 32

• If inflation is short,

too large perturbations from supercurvature mode of φ

2 2

~ ^{F R}

sc

H H

δσ π

π

H

: Hubble at false vacuum H

: Hubble after fv decay

MS & Tanaka (‘96)

0

2 2 (3) 2

| |; | | ( , )

sc K p m

p = p ≈ − K ∆ + p + K Y ^ℓ r Ω =

Two- (multi-)field model: “quasi-open inflation”

• a “heavy” field σ undergoes false vacuum decay

• another “light” field φ starts rolling after fv decay

~ perhaps naturally/easily realized in the landscape

σ

( ) ( )

2

2

, m 2

V φ σ =V_σ σ + ^φ φ

φ

Linde, Linde & Mezhlumian (‘95)

4 60

2

~ N κφ _<

=

creation of open universe &

supercurvature mode

bubble wall

open universe

dS vacuum wavelength > curvature radius

“supercurvature” mode

34

Two-field model 2:

a slightly more complicated two-field model

( ) ( ) ( )

2 2

V φ σ V_σ σ m^φ φ β

σ φ φ

= + + ² −

2

2 2

2

, 0

makes φ heavy at false vacuum kills the supercurvature mode tunneling from

σ=σ

to σ=0

(2 more parameters)

after tunneling, φ becomes light and starts slow-rolling

δφ : non-Gaussianity due to interaction?

(need study)

Sugimura, Yamauchi & MS (’11)

35 35

To summarize:

( ) ( )

2

2

, m 2

V

φ σ

σ

φ The models of the tunneling in the landscape with the simplest potentials such as

2 3 4

2

2 3 4

V = m

φ

ν φ

λ φ or

are ruled out by observations, assuming that inflation after the tunneling is short, N ~ 60.

The same models are just fine if N >> 60 (if Ω

=1) This means that we are testing the models of the landscape in combination with the probability

measures, which may or may not predict that the last stage of inflation is short.

NB. a slightly more complicated two-field model

may work.

36

How about more general single field models?

• if ρ

~ M

, the universe will most likely tunnel to a point where the energy scale is still very high unless potential is fine-tuned.

2

HF

φF

FV decay rapid roll

slow roll inflation

rapid-roll stage will follow right after tunneling.

• perhaps no strong effect on scalar-type pert’s:

2

~ 2

C

H R

πφ

ɺ

suppressed by

at rapid-roll phase 1/ φ ^ɺ

need detailed analysis

∙∙∙ future issue

Linde, MS & Tanaka (’99)

37 37

but tensor perturbations may not be suppressed at all.

?

TT

~

P

h H

M

tensor perturbations and

their effect on CMB

Memory of H

(Hubble rate in the false vacuum) may

remain in the perturbation on the curvature scale

If H

~M

, we would see a huge tensor perturbation!?

38 38

5. Single field open inflation

- evolution after tunneling -

2

HF

φF

Right after tunneling, H is dominated by curvature:

( )

, 4

a t V φ t

φ ^′

≈ ɺ ≈ −

rapid-roll phase

1

2 *

*

V t H

φ ε

−

≈ ≈

ɺ

: “slow-roll” parameter

2

V 2

ε

κ

 ′ 

≡  

 

1 3

2

2

a

a a

κ ρ

  = +

   ɺ

curvature dominant phase

kinetic energy grows until

at

39 39

2

HF

φF

1 2

2

V ε V

κ

 ′

≡  

 

( )

3 2

2 2

ln

ln /

d

d a V

ρ φ

φ φ

 

 = − 

 + 

 

ɺ ɺ

( )

1 1

* / * *

t H~^{> −} ε < H⁻

rapid-roll phase

at

1

2 *

*

V t H

φ ε

−

≈ ≈

ɺ

ρ starts to decay at

1 2 1

3 2

2

2

a V

a a

κ φ

  =  + +

   

   

ɺ ɺ

* ~1,

ε

for

( )

1 1

* * / *

t ≈ H⁻ ≪ H⁻ ε

* 1,

ε

V dominates (curvature dominance ends) at

no rapid-roll phase.

slow-roll inflation starts at

rapid-roll continues until

- continued

tracking is realized during rapid-roll phase ( ^φ

) ^ε

40 40

exponential potential model

( )

exp 2

V ∝ κε φ

(

)

(

⁽

⁾ )

V = H − H

κε φ φ

* 1

ε = Log 10[r/H *2 ]

Log₁₀[a(t) H_*]

* 0.1

ε =

* 0.5

ε =

2

* 10

ε =

4

* 10

ε =

potential kinetic term curvature term

added to realize slow-roll inflation /

V′ V = const. ε = const.

(

)

41 41

( )

 − ∂ ∂ −

−

=

φ φ φ

κ

µν V

g R

g

L 2

1 2

1

6. Tensor perturbations

some technical details...

( )

( )

2 2 2 2 2 2

2 2 2 2 2

( ) sin

( ) sin

E E E E E

C E E E E

ds dt a t dr r d a

η

η

= + + Ω

= + + Ω

• CDL instanton

( _E)

φ φ η

η

−∞ < < ∞

bubble wall

• action

ηE = +∞

ηE = −∞

rE

E 2

r π

=

42 42

• analytic continuation to Lorentzian space (through r

=π/ 2 )

2 ,

C E C E

r i r π

η η

 

=  −  =

 

C-region: ~ outside the bubble

( )

2 2 2 2 2 2

( ) cosh

C C C C C

ds = a η dη −dr + r dΩ

η ηη

η =const.

r =const.

Bubble wall r _C =0

R-region: inside the bubble

2 , 2 ,

R C R C R C

r r π i π i a i a

η η

= + = − − =

( )

2 2 2 2 2 2

( ) sinh

R R R R R

ds = a η −dη + dr + r dΩ

Euclidean vacuum C-region R-region

η_C⁼⁺ 8 η_C=- 8

R C

time

time

43 43

2( ) ( ) ( , )

TT

ij C p

p m ij C C Y

X r

h = a η η ^ℓ Ω

( ) ( )

2 1 0; 1

2

2

2 ^C ( ) _p

C C C

d a d

p X K

d a d

η

η η η

 ′ 

 + + +  = = −

 

 

( )

2 p

p

C

a X d aw dη

≡

;

2

2

2 _T( _C) ^{p p}

C

d U w p w

d η

η

 

− + =

 

  2

2

2

2 2

( ) ( )

( ) ( )

T C C

C C C

U

a t t η κ φ η

κ φ

= ′

= ɺ

• tensor mode function Euclidean vacuum

Y

: regular at r

=0

new variable w

:

U

η

bubble wall

ip C

e^{− η}

ip C

e ^η ρ ⁺

ip C

e ^η σ ⁻

2 2

ρ + σ =1

44 44

analytic continuation from C-region to open univ. (=R-region)

^{/2 /2}

C C R R

ip ip ip ip

p p p p

C R

w = ρe^{+ η} +e^{− η} → w = e ^π ρe^{− η} +e^{− π} e ^η

C R C π2 i

η →η = −η −

• there will be time evolution of w

in R-region:

effect of wall

( )

2 1 0 ;

( )

2

2

2 ( ) _p ^R

R R R

d d a

p X

d d a

η

η η η

  ′

+ + + = ≡

 

 

H H

η

epoch of bubble nucleation:

0 ;

2

2

2 2

2 _T( _R) ^p _T( _R)

R

d U p w U

d

η η κ φ

η

 

− + = = ′

 

 

or

final amplitude of X

depends both

on the effect of wall & on the evolution after tunneling

( )

2

1 _{p TT}

p

X d aw h

a dη ^{ }

=   ∝

45 45

high freq continuum + low freq resonance

wall fluctuation mode

p >1 p ~ 0

( )

2

2 ^{C C}

s κ dη φ η^′

∆ ≡

∫

Effect of tunneling/bubble wall on

∆s ~ ∞

∆s ~0.1

∆s ~0.02

∆s ~1

∆s ~0.7

; ²

1

~ 1 _C

P

s S S dt

M V φ

 

∆ =

 

 



∫

scale-invariant

(

)

T( ) p

P p ∝ X

wall tension

46 46

rapid-roll phase ( ε

-)dependence of P

(p)

ε <<1: usual slow roll

ε~1: small p modes remember H at false vacuum

ε >>1: No memory of H at false vacuum

47 47

7. CMB anisotropy

ℓ

ε~1: small ℓ modes remember initial Hubble

(1− Ω0)^ℓ

• scales as at small ℓ, scale-invariant at large ℓ

ε <<1: the same as usual slow roll inflation

ε >>1: No memory of initial Hubble

*

~ 1

small ℓ modes enhanced for ε

48 48

• CMB anisotropy due to wall fluctuation (W-)mode

0 1

(

( ) )

( )

W ; W T

W

C P P dp

C C C P p

s

= ∞ ∝

= ^ℓ + ^ℓ

∫

ℓ

ɶ

( 1

)

(W)

C ɶ

∝ − Ω

scale-invariant part

Α = Β =10⁵ Α =10⁵, Β =1 Α = Β =10⁴ Α = Β =10³ Α = Β =10²

0.2 0.5 1.0 2.0

10^-19 10^-16 10^-13 10^-10 10^-7 10^-4

Ε_*

C2HTLDs 2ΚH*2

ℓ =2

2

2 *

2

/ C

H s κ

=

∆

ℓ

ε*

∆s = 10

∆s = 10

∆s = 10

∆s = 10

dominates ℓ=2

MS, Tanaka & Yakushige (’97)

49 49

8. Summary

Open inflation has attracted renewed interest in the context of string theory landscape

Landscape is already constrained by observations

anthropic principle + landscape 1-Ω

~ 10

– 10

• simple polynomial potentials

φ

φ

φ

lead to HM-transition, and are ruled out

• simple 2-field models, naturally realized in string theory, are ruled out

due to large scalar-type perturbations on curvature scale

If inflation after tunneling is short (N ~ 60):

50 50

Tensor perturbations may also constrain the landscape

“single-field models”

• it seems difficult to implement models with short slow-roll inflation right after tunneling in the string landscape.

• there will be a rapid-roll phase after tunneling.

1 2

2 V ~1

ε V

κ

 ′ >

=  

 

right after tunneling

• unless ε>>1, the memory of pre-tunneling stage persists in the IR part of the tensor spectrum

due to either wall fluctuation mode or evolution during rapid-roll phase

We are already testing the landscape!

(1

)

∝ − Ω

large CMB anisotropy at small ℓ

if ε <<1, energy scale must have been already very low.