(RESCEU&IPMU)横山順一

Inflaton 　 φ

slow rollover

Reheating

[ ] 3 0H V

V[φ]

BEGINNING??

END?? Λ

But little is known about the beginning and end of inflation.

Slow-roll phase is now probed by astronomical observations.

2 2

2 2

1[ ]

3 2G

a KV

a a M

Klein Gordon Equation

Einstein Equation

Cosmic Scale Factor

aH

a

Hubble parameter

182.4 10 GeV8

PlG

MM

Fluctuation is generated continuously in each Hubbletime with initial wavelength and amplitudewhich is stretched by exponential expansion.

H 1 H 2

2

2

HH t

R depends on the potential and its derivative.

Since the right-hand-side evolves very slowly,we find a nearly scale-invariant spectrum.

32 1

3

4: 1

2nk

t k n

kR

time

scale

H-1

during inflation

The modes which left the Hubble horizonearlier are stretched more to constitute longer wavelength modes.

2r k

： polarization tensor with

satisfies the same eqn as a minimally coupled massless scalar field.

Long-wave quantum fluctuation is generated during inflation.

A

ij ij ijh h e h e A

ije A ijA AAije e

2 22 22 2

1 1 1 1

16 8 2h h h h

G G a a

RL =

( )Ak t

22

3( )

2Ak

Ht

k for ( )k a t H

Square amplitude of tensor perturbation

2 polarization modes of the graviton = 2 independent massless fields

Square amplitude per logarithmic frequency interval

Tensor perturbation has a nearly scale-invariant spectrumat formation.

2

2 22 2

[ ( )] 1 [ ( )] 1( ) 2 4

2 3 2ij

F ij hG G

H f V fh f h h

M M

1 2[ ( )]

[ ( )]3 G

V fH f

M

21 6 2 , 16 24 2ln

ss

dnn

d k

curvature perturbation 2 22 32

2 4

3 [ ]

2 2 [ ] 24RG

H H V

V M

tensor/scalar ratio 2

216h

R

r

spectral index and its running (scale dependence)

22 [ ]

2 [ ]GM V

V

2 [ ]

[ ]G

VM

V

42

[ ] [ ]

[ ]G

V VM

V

182.4 10 GeV8

PlG

MM

-200 T(μK) +200

Low frequency components of tensor perturbations may be observed by

B-mode polarization of CMB anisotropy

• Polarization is generated by quadrupole temperature anisotropy.• E-mode from both scalar (density) and tensor perturbations.• B-mode only from tensor perturbations.

E mode

B mode

Ongoing/Planned projects and their target sensitivity

PLANCK 　ｒ ～ 0.1 BICEP 　ｒ ～ 0.05 PolarBear ｒ ～ 0.01 QUIET ｒ ～ 0.01CLOVER 　　ｒ　～ 0.01 　　 EPIC 　ｒ ～ 0.001 　　　　 B-POL ｒ ～ 0.001

High frequency components may be observed by

future space-based laser interferometers. Deci-hertz Interferometer Gravitational Wave Observatory

N. Seto, S. Kawamura, & T. Nakamura, PRL 87(2001)221103

amplitude of GW to achieve S/N>1 after ten years of correlation analysis

DECIGO©cooray.org

In terms of B-mode polarization, we can measure around the wavenumber corresponding to .

2 22 ( )h Fh f

:h inflaton field value when comoving scale corresponding to large-scaleCMB observation ( ) left the Hubble radius during inflation

( ) :f inflaton field value when comoving scale corresponding to frequency f today left the Hubble radius during inflation

10.002Mpck

17ln ln 38 for 1Hz

2 10 Hzh

f fN f

f

# of e-folds between the two epochs.

We can extrapolate to using slow-roll parameters at .

2 ( )h hf 2 ( )h f10.002Mpck

172 10 Hzhf 10.002Mpck

This gives the initial amplitude of gravitational radiation when the mode with comoving frequency f reentered the Hubble radius at .( )int t f

2

2 ( ( ))( ) 8 ( )

2hG

H ff V f

M

なので、 CMB スケールに対応する　　　　　　から以下のように展開できる。

hV f

2 3

2 32 3

1 1 11

2 6h

dV d V d VV f V f N N N

V dN V dN V dN

N Hdt

2 22

22

3 G

dV dV d dt V V VM V

dN d dt dN H H V

2 2 2

2 3 2 24 ( )

d V V VVV V

dN H H H

22 [ ]

2 [ ]GM V

V

2 [ ]

[ ]G

VM

V

などにより、

2 2

2 2

3

2 2

( ( ))( ) 8 ( ) 1 2 ln 2 ln

2

1 (12 16 4 2 ) ln

3

h h hG h h

h

H f f ff f

M f f

f

f

17ln ln 38 for 1Hz

2 10 Hzh

f fN f

f

と求まる。

Amplitude of GW is constantwhen its wavelength is longerthan the Hubble radius between and .

scale

time

H-1

a(t)λ inflation

Hubble horizon

After entering the Hubbleradius, the wavelengthdecreases as and the energy density as

.

H-1

Radiationdominant

Matter dominant

4 ( )a t

1( )a t

1 32

3 12(1 )

0( , ) ( ) , 2 ( )1 ( ) ( )

pp

pp

p kh f a a t J k fa t

p a t H t

( ) ( 1)pa t t p When , the tensor perturbation evolves as

( )int f( )outt f

( )outt f ( )int f

Density parameter in GW per logarithmic frequency interval

222 2 2inf inf

( ) 1( , ( )) ( ) ( ) ( ( )) ( )

16 16 12in

gw in cr in h

H t ff t f h f h f a f f

G G

21( , ( )) ( )

12GW in hf t f f

( , )GW f t

When the mode reentered the Hubble horizon at ,the angular frequency is equal to , so we find

( )int t f ( )inH t f

( , )( , )

( )GW

GWcr

f af a

a

4 ( )a t

ハッブルホライズンに入ったあとには、

3(1 ) ( )wa t totw p

: equation of state

放射優勢期には、これは一定である。

放射優勢期にホライズンに入ったモードは初期のスペクトルの形状をそのまま留めている。

標準宇宙論では、今日周波数 f のモードがホライズンに入った ときの宇宙の温度は、

6( ) 3.4 10 GeV0.1Hz

fT f

インフレーション後の再加熱温度がこれより高ければ、生成時のスペクトルがそのまま DECIGO によって観測されるであろう。

16r

0.2

(5 year WMAP)

r

UC

U

C

大スケールのC

MB

偏光観測で重力波振幅を予言可

能

2 2h R

( , )( , )

( )GW

GWcr

f af a

a

4 ( )a t

After entering the Hubble horizon,

3(1 ) ( )wa t totw p

: equation of state

During radiation domination, it is constant.

We can probe the change of the equation of state.

Evolution of the density parameter

(Seto & JY 03)

can be determined bylarge-scale observations

can be observedby DECIGO/BBO

We can determine the equation ofstate in the early Universe.

We can determine thermal history of the early Universe.

RT reheating temperatureafter inflation

再加熱温度が低い場合はスペクトルが変形する。

水色：重力波を検出できる領域　　

空色：さらに再加熱温度も決定　　　　できる領域

CMB の B モード偏光が検出できると、 DECIGO 帯の重力波を予言できる

DECIGO によって再加熱時期が観測できる

Ongoing/Planned projects and their target sensitivity

PLANCK 　 ｒ ～ 0.1 BICEP 　ｒ ～ 0.05 PolarBear ｒ ～ 0.01 QUIET ｒ ～ 0.01CLOVER 　 ｒ　～ 0.01 　　 EPIC 　 ｒ ～ 0.001　　　　B-POL ｒ ～ 0.001