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(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