Lectures on Inflation 1   (given at Summer School on Cosmology, ICTP, July 2012)

Summer School on Cosmology

M. Sasaki 16 - 27 July 2012

Yukawa Institute, Kyoto

Yukawa Institute for Theoretical Physics Kyoto University

contents

contents

• horizon & flatness problems • slow-roll inflation

• reheating scenario

1. Inflationary Universe

• curvature (scalar-type) perturbation

• gravitational wave (tensor-type) perturbation

2. Cosmological Perturbations from Inflation

• origin of non-Gaussianity

• δN formalism: NG generation on superhorizon scales • other sources of NGs

3. Non-Gaussian Curvature Perturbation

1. Inflationary Universe

1. Inflationary Universe

now last scattering surface η x η = 0 • horizon problem 4 3 0 3 ( ) for 3 G aɺɺ = − π ρ + P a < P > − ρ 2 2 2 2 3 ( ) ( )( ) ds = a η −dη +dσ : conformal time ( ) dt d a t η = conformal time is

bounded from below

1 if a ∝ tn, n < _{gravity=attractive} 0 0 finite ( ) t t dt a t → =

∫

2 2 2 2 3 ( ) ( ) ds = −dt + a t dσ particle horizon E

now

last scattering surface η

η − 8

• solution to the horizon problem 4

3 0

3 ( )

G

aɺɺ = − π ρ + P a >

for a sufficient lapse of time in the early universe

0 0 0 ₀ ( ) t t t dt a t

η η

− =

∫

→_→ ∞ or large enough to

cover the present horizon size

NB: horizon problem≠

2 2 8 3 ; G K H K a

π

ρ

= − − ∞ < < +∞

• flatness problem (= entropy problem)

4

2

if a , |K | in the early universe.

a

ρ

_∝ −

ρ

_≫

2

0 0

conversely if at an epoch in the early universe, the universe must have either

or become completely collapsed (if ) empty (if ) by now. | |/ K K K a

ρ

> < ≈

alternatively, the problem is the existence of huge entropy within the curvature radius of the universe

3 3 3 3 0 3 3 87 0 0 0 10 | | | | a a S T T T H K K −     = _{ } ≈ _{ } > ≈     (# of states = exp[S])

solution to horizon & flatness problems

solution to horizon & flatness problems

spatially homogeneous scalar field:

2 2 1 1 2 V ( ), P 2 V ( ) ρ = φɺ + φ = φɺ − φ

(

2

)

2 3P 2 V ( ) 0 if V ( ) ρ + = φɺ − φ < φɺ < φ potential dominated 2 if ( ) ( ) P V V ρ ≈ − ≈ φ φɺ ≪ φ

V ~ cosmological const./vacuum energy

2 decreases rapidly . K const a

ρ

≈ inflation

“vacuum energy” converted to radiation after sufficient lapse of time

solves horizon & flatness problems simultaneously

2 8

3 .

G

slow

slow

-

-

roll inflation

roll inflation

• single-field slow-roll inflation

V( V(φφ)) φ φ 2 2 2

3

=0

8

1

3

2

( )

( )

H

V

a

G

H

V

a

φ

φ

φ

π

_φ

_φ

′

+

+

 

_≡

₌



₊



 

_

_

 





ɺɺ

ɺ

ɺ

ɺ

2 2 2 ( ) _ij i j ds = −dt + a t δ dx dx 2 2 2 2

3

3

2

₁

1

₂

2

H

H

_V

V

φ

_φ

φ

−

=

≈

+

ɺ

ɺ

ɺ

≪

ɺ

∙∙∙ slow variation of H

inflation!

metric field eq. Linde ’82, ...

3

( )

V

H

φ

φ

ɺ

= −

′

~

Ht

a e

slow

slow

-

_-

roll conditions

_{roll conditions}

2

1

2

;

P

M V

H

V

H

φ

ε

δ

ε

ε

δ

ε

φ

′′

≡

ɺɺ

= +

ɺ

≈ −

≪

ɺ

2 2 2 2 2 2 2

3

3

2

₁

1

₂

₂

2

P

H

M V

H

_V

V

V

φ

_φ

ε

φ

′

≡ −

=

≈

=

+

ɺ

ɺ

ɺ

≪

ɺ

condition for quasi-de Sitter (inflationary) expansion

condition for friction-dominated (over-damped) evolution

sufficient condition on potential:

2 2 2 v v 2

1

1

2

,

;

P P V

M V

M V

V

V

ε

≡

′

≪

η

≡

′′

η

≪

reheating

reheating

• standard scenario 2 3 2

3

H

m

_φ

0

a

/

cos(

m t

_φ

)

φ

_ɺɺ

₊

φ

_ɺ

₊

φ

₌

_⇒

φ

_∝

−

₊

α

int

~

Y

L

g

φψψ

ψ

ψ

φ

Y

g

e.g. decay rate:

Γ

~

g m

_Y2 _φ

;

m

_φ

≫

m

_ψ

effective equation of motion:

2 2

1

3

2

(

)

d

V

H

dt

φ

φ





⇔

_

+

_

= −

+ Γ





ɺ

ɺ

when damped oscillation:

φ

3

H

φ

V

′

( )

φ

φ

⇒

ɺɺ

+

ɺ

+

= −Γ

ɺ

,

m

_φ

≫

H

> Γ

2 2

1

2

( )

V

φ

=

m

_φ

φ

+⋯

effect of Γ

ρ

_φ

3

H

ρ

_φ

ρ

_φ

⇒

ɺ

+

= −Γ

energy conservation eqns

3

H

φ φ φ

ρ

ɺ

+

ρ

= −Γ

ρ

4

r

H

r φ

ρ

ɺ

+

ρ

= Γ

ρ

ρ

r : produced radiation • Γ<H

~

t-1 4 5 2

2

1

5

/ f r f f f

a

a

H

φ

a

a

ρ

ρ

−

_

_









Γ

_

_

=

_

_

_

_

_

_

_

_

−









_





_

3

,

f f

a

a

φ φ

ρ

ρ

−





=

_

_

_

_





2 5

8

1 48

3

/

.

r f

a

a

ρ

=

=

 

_{ }

≈

 

max at

• Γ>H ~ t-1

( )

4

0

,

_{r r R} R

a

t

a

φ

ρ

ρ

ρ

−





=

=

_

_





t

R

:def by

H t

( )

R

= Γ

reheating temperature & max temperature

reheating temperature & max temperature

1 4 1 4 2 / / max

~

~

f f P R P

H

H

T

M

T

M

Γ















_Γ











log(a) log(ρ) a_f a_max a_R ρ_r,max ρ_f ρ_R

( )

2 4

30

:

R r R eff R

T

ρ

t

=

π

N T

(

)

1 4 _{1 2} 2 2 2 1 4

36

/ _/ /

~

P P R eff eff

M

M

T

N

N

π



_Γ



_Γ

= 

_



_





indep. of ρ_f dep. on ρ_f

T_max is important for thermal history

T_R is important for horizon problem ρ_r

comoving scale

comoving scale

vs

vs

Hubble horizon radius

Hubble horizon radius

log log aa((tt)) log log LL t t==tt_endend a L k = k: comoving wavenumber 1 L = H− inflation subhorizon superhorizon subhorizon hot bigbang k H a = k H a > k H a < k H a > k H a =

e

e

-

-

folding number: N

folding number: N

[

]

end 1 ( ) ( ) t ~ ln ( ) t N N Hdt z φ φ φ = =

∫

+ end end ( ) exp[ ( )] ( ) a t N t t a t = → ⇒ redshift # of e-folds from

φ

=

φ

(t)

until the end of inflation

log log aa((tt)) log log LL L L==HH--1 1 _{~ ~ tt} t t==tt((φφ)) _tt==tt end end N N==NN((φφ)) L L==HH--11_{~ const~ const}

condition on e

condition on e

-

_-

folding number

_{folding number}

ignore variation of H during inflation.

entropy generated within present Hubble volume:

1 0 h H

φ

φ

− ≡ value of at which comoving scale of left horizon log log aa((tt)) log log LL a a((φφ_hh)) _{aaff aaRR} H H₀₀--11 H H_ff--11 3 3 3 N( _h) _R 3 f R f a S H e T a φ −   = _{ }   1 3 2/ H− ∝ a 1 2 H− ∝ a L ∝ a

3 2 3 3 3 3 3 2 4 3 3 3 ₀ 87 1 2 0 10 / ( ) ( ) ( ) / ~ ~ h h h f f N _R N f R R f P R N P R f a S H e T e T a M T T M e T H φ φ φ

ρ

ρ

ρ

− −       = _{     }         _{ } ≈ _{ } > _{ }     1 4 15 10 2 1 53 3 10 3 10 / ( )_h ln f ln TR N

φ



ρ

   ⇒ > + _{ } + _{ }       GeV GeV

• changing T_R by one order (by 10) changes N by 1

Q1. Show that conformal time η_hat φ=φ_h satisfies |η_h|>η₀ , where η₀ is the conformal time today.

preheating

preheating

Kofman, Linde & Starobinsky ‘94 If φ couples to other light scalar (bose) fields

2 2 2 2

int

~

,

L

g

φ χ

m

_χ

≪

m

_φ

e.g.

catastrophic χ- particle creation can occur

(

2 2

)

3 ( / ) 0 k H k k a g k

χ

ɺɺ +

χ

ɺ + +

φ χ

= 2 2 3 2 ( / ) sin f af a m tφ

φ

=

φ

for m t_φ∆ ≫_1~> H t∆ _{oscillating potential}

possible parametric amplification of χ_k

(

2 2 2

)

0 ( / ) sin k k a g m tφ k

χ

ɺɺ + +

φ

χ

=

(

2 cos2

)

0 k a b m tφ k

χ

χ

⇔ ɺɺ + − = _{Mathiew eqn}

2 2 2 2 4 k a b am g b m φ φ

φ

  = _{ } +     = _ _  

if b>1 initially, evolutionary path passes

through unstable region

b

a 2 , k m a b a ≪ φ = For instantaneous reheating

instability bands

(

2 cos2

)

0 k a b m tφ k

χ

ɺɺ + −

χ

=

2. Cosmological Perturbations from Inflation

2. Cosmological Perturbations from Inflation

curvature perturbation: intuitive derivation

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 λ ω   + + = = _{≡  }   ɺɺ ɺ ( ) ik x k k t e δφ =

∑

δφ i

harmonic oscillator with friction term and time-dependent

ω

k

δφ

→ const.

··· frozen when

λ

> c H-1

(on superhorizon scales)

δφ_k

gravitational wave modes also satisfy the same eq. physical wavelength _λ

• fluctuation amplitude (vacuum fluctuations=Gaussian) 2 2 3 2 1 2 / , ~ iw tk ; k k k k k k e w H a a w φ  ₌ ϕ ϕ − ₌ ≫ 2 2 3 ₂ 2 / ~ ( / ) ( / ) k k k k a H H H a k H a k H k ϕ ϕ δφ π   ≈ = ≈ ⇒ _{=  }   ≫ frozen at a =k/H

In the above, metric perturbations

δg

are ignored ~ a gauge in which

δg

is minimized

= hypersurface on which δR(3)_{=0: “flat” slice}

2 2 3 k K δ = R

R: called curvature perturbation 2 3 3 2 2 4 6 ( ) ( ) , K k R R a δ = a R ⇒ =

t xi 0 0

δφ

=   ≠  R 0 0

δφ

≠   =  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

generation of

“

“

comoving

comoving

”

”

curvature perturbation

curvature perturbation

φ

=const. 3-surface is called “comoving” slice.

• curvature perturbation on comoving slices:

c H _δφ φ = −

R

ɺ evaluated on flat slice gauge transf.

conservation of comoving curvature perturbation

conservation of comoving curvature perturbation

2 2 2 2 2 2 2 P; a z a M H φ ε ≡ ɺ = 2 3 1 2( ), H w H ε = − ɺ = + w P ρ = • eom 2 2 2 0 ( ) ; z k z ′ ′′ + ′ + = C C C R R R ' d a d dη dt = = 2 2 0 (z ) z ′ ′′ + ′ = C C R R 2 0 k → (k aH) (k aH) H δφ (k aH) φ   ≈ = = −_{ } =   ≪ ɺ C C R R

if R _C becomes const., “adiabatic” limit is reached

ε

: slow-roll parameter Kodama & MS ‘84 2 1 : z ′ ∝ C R _{decaying mode} . : const = C R _{“growing” mode}

Curvature perturbation spectrum

Curvature perturbation spectrum

2 1 2 1 2 2 _P / _{k aH} H M π ε ₌   =     • spectrum Mukhanov (‘85), MS (’86)

spectrum derived by 1st _{principle calculation}

more elegantly derived a la Faddeev-Jackiw method

Garriga, Montes, MS & Tanaka (’98)

generalized to k-inflation:

Garriga & Mukhanov (’99)

( , ); L = P X φ X = −gµν∂ ∂_µφ φ_ν 1 ( ) nS ; P kR = Ak − • spectral index 2 2 2 ( ) k aH H P k πφ ₌   =     R ɺ 2 2 2 1 2 3 2 6 S P V V V V n M V V η ε ′′ ′   − = _ − _ = −  

• generalized action for R

_C 2 3 2 2 2 2 2 2 _s C s C ; z S d d x c k c

η

 ′  =

∫

_R − R _ s c = sound velocity 2 3 2 1 3

4

1

2

| |

3

1

2

( )

| |

(

)

s

(

)

s k c k s P _{c k aH}

k

H

P k

r

c

w

M

η

π

π

=

π

₌





=

=

_

_

+

_

_

R 2 2 2 3 1( ) _P z = + w a M

(=1 for canonical case) canonical quantization: 2 2 R s S z c δ π δ − _′ = = ′ C C R R RC,πR  = iℏ † * ( ) ( ) ; k k k k a r η a r₋ η =  +  C R 1 2 s ic k s k s c r e z c k η _η − → ( → −∞) positive freq fcn

Garriga & Mukhanov (’99)

Q2. Derive the above spectrum by performing canonical quantization as outlined above.

•

δ

N - formula

( )

( )

tend end t

H

N

Hdt

φ

d

φ φ

φ

φ

φ

=

∫

=

∫

_ɺ

( )

_c k aH k aH

N

H

N

δ

φ

δφ

δφ

φ

₌

φ

₌







∂



=

_

_

= −

_

_

=

∂









R

ɺ

2 2 2 2 1 2 | | ( ) _{k k} ; k aH H N P k ϕ _η φ πφ = =    ∂  = _{ } = _{ } ∂     R ɺ 2 2 ₂ 2 k k k aH H ϕ δφ π ₌   = _{=  }   MS & Stewart (’96) A A A N N δ δφ φ ∂ = ∂

∑

geometrical justification

NL generalization Lyth, Malik & MS (’04)

Starobinsky (’85)

Tensor Perturbation

Tensor Perturbation

0

i TT ij TT ij ij

h

δ

h

∂

=

=

, ( ; ) ( ) ( ) . . ij k t a P kk ij k t h c σ σ σ φ ϕ =+ × =

∑

+ ( ) : k t

ϕ same as massless scalar

• canonically normalized tensor field

1 1 2 32 ; 8 TT _P TT ij ij ij P M h h M G G φ π π ≡ = ≡ : transverse-traceless 2 2 2 2 2 2 8 4 8 2 , , TT k ij ij P P P H h k k M M M σ σ ϕ σ φ σ π   = = _{=  }  

∑



∑

 • tensor spectrum 2 4 1 2 ~ ij S d x g t φ ∂   − _{ } + ∂  

∫

i i i Starobinsky (’79)

Tensor

Tensor

-

-

to

to

-

-

scalar ratio

scalar ratio

•

• scalar spectrum:scalar spectrum:

( )

( )

2

( )

2 3 2 2 3 2 4 s k H P k π N π = π ∇ •

• tensor spectrum:tensor spectrum:

( )

( )

4 8 2 2 3 2 3 2 2 ( ) g P k H P k M π π = π •

• tensor spectral index:tensor spectral index: 2

2 2 2 2 2 2 2 g P P H n H M H _{M N} φ φ φ − = − = = ⋅ ∇ ɺ ɺ ɺ ɺ a a dN H N dt φ = − = − ∇ i 8 g g s P r n P

≡ ≤ ··· valid for all slowvalid for all slow--roll modelsroll models with canonical kinetic term

with canonical kinetic term

1 s n k − ∝ g n k ∝ 1 8 2 2 g s P P P M N ≥ = ∇ ( ) _{a b} N H φ φ φ ∇ ≡ ∂ ∂

Comparison with observation

Comparison with observation

Standard (single

Standard (single--field, slowroll) inflation predicts scalefield, slowroll) inflation predicts scale- -invariant

invariant GaussianGaussian curvature perturbations.curvature perturbations.

CMB (WMAP) is consistent with the prediction.

Linear perturbation theory seems to be valid.

CMB constraints on inflation

CMB constraints on inflation

Komatsu et al. ‘10

scalar spectral index: n_s = 0.95 ~ 0.98

However,….

Inflation may be non-standard

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

Quantifying NL/NG effects is important

2 gauss gauss 3 5 5? C = + fNL + ; fNL ~> R R R _⋯

PLANCK, … may detect Non-Gaussianity

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

energy scale of inflation H2 >< 10-10M_Planck2 ?

modified (quantum) gravity? NG signature?