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Workshop on gravity and cosmology for young researchers@YITP
Shun Arai (Nagoya University C-lab, M2)
1st.Mar, 2017
Cosmological constraints on Lorentz symmetry violation宇宙論的観測を用いたLorentz対称性の破れの検証
Shun Arai, S.Sibiryakov and Y.Urakawa in prep.Shun Arai, D.Nitta and H.Tashiro Phys. Rev. D 94, 124048
Introduction
What’s the Lorentz symmetry?
A.Einstein(1879-1955)
“The whole Galileo-Newton system thus sank to the level of a first approximation, becoming progressively less exact as the velocities concerned approached that of light.”
By Hendrik Lorentz
1/21
まず、イントロです。Lorentz対称性というのはA.Einsteinにより提唱された特
GR QFT String theory
Lorentz symmetry
Electro-Magnetism Newton mechanics
Special Relativity
Introduction
Observational constraints for Lorentz symmetry ❶
宇宙論的な距離を飛んでくる間にshapiro 遅延効果は大きくなるので、宇宙論的距離にあるGRBを用いると一番強い制限になる。
今回の発表では、数式をほとんど載せずに説明してみる。
C.Will. 2014
2/21
Introduction
Constraints for high energy scale
・Interferometers
・ CPT theorem V. A. Kostelecky et al. 2011 J.Ellis et al.2013
・ GRB V.Vasileiou et al. 2013
・ Binary pulsars, GW K. Yagi et al. 2013 D. Bias et al. 2016
・ GZK Cut-off HiRes Collaboration et al. 2007
Michealson - Morley, 1887
The strongest constraint for photonELV > 1012GeV
Observational constraints for Lorentz symmetry ❷
3/21
宇宙論的な距離を飛んでくる間にshapiro 遅延効果は大きくなるので、宇宙論的距離にあるGRBを用いると一番強い制限になる。
今回の発表では、数式をほとんど載せずに説明してみる。
Constraints for gravity sector
Introduction
Theoretical predictions of Lorentz Violation (LV)
Phenomena induced by LV are diverse and sometimes non trivial.
・extension of NG theorem
・ Renormalization of gravity
・Renormalization & unitarity in LV
H.Watanabe and H. Murayama, 2012,2014
P. Horava, 2009 J. Barvinski et al. 2015
T. Fujimori et al. 2015
etc…
・Renormalization group in LV R. Iengo et al. 2009
4/21
Introduction
Modern Cosmology
redshift z
Age of the Universe
Quantum fluctuation
initial singularity
S I Z E
Baryongenesis
Dark matter
Neutrinos
Structure formations Accelerating expansion (Dark energy)
380,000yrs 13.8byrs10-36s
1 030 ~ 61100
Inflation
CMB
5/21
Introduction
Modern Cosmology
redshift z
Age of the Universe
Quantum fluctuation
initial singularity
S I Z E
Baryongenesis
Dark matter
Neutrinos
Structure formations Accelerating expansion (Dark energy)
380,000yrs 13.8byrs10-36s
1 030 ~ 61100
Inflation
CMB
5/21
2. Late-time constraints on LV with CMB distortions
Talk plan
3. Summary
1. Constraints on LV from primordial fluctuations in inflationary era.
SA, S.Sibiryakov and Y.Urakawa in prep.
これはAmazing factです。今まで知られていなかったことが明らかとなったため
SA, D.Nitta and H.Tashiro Phys. Rev. D 94, 124048
6/21
2. Late-time constraints on LV with CMB distortions
Talk plan
3. Summary
1. Constraints on LV from primordial fluctuations in inflationary era.
SA, S.Sibiryakov and Y.Urakawa in prep.
これはAmazing factです。今まで知られていなかったことが明らかとなったため
SA, D.Nitta and H.Tashiro Phys. Rev. D 94, 124048
6/21
1.Constraints on LV in inflationary era
Planck scale
physical scale
current particle horizon
1/MP
a/k
a2
a32
ainfd01/Hinf
1/H
a
InflationHorizon crossing
7/21
d0
1.Constraints on LV in inflationary era
Planck scale
physical scale
current particle horizon
1/MP
a/k
a2
a32
ainfd01/Hinf
1/H
a
InflationHorizon crossing
1/M⇤
7/21
d0
1.Constraints on LV in inflationary era
Horava Lifshitz Gravity (HL gravity)Renormalizable theory of gravity P. Horava, 2009 J. Barvinski et al. 2015
HL gravity + (single) scalar field inflationR, hij �(t) + '
t ! bzt, x ! bxanisotropic scaling t ! t = t(t), x ! x = x(t,x)time foliation diffeomorphism
new d.o.f ; scalar graviton R2 tensors + 1 scalar E > M⇤
LV energy scaleM⇤ =
p↵MP
M⇤
local Lorentz symmetry 2 time derivatives 2z spatial derivatives
no ghost & power counting renormalizable
今回は、くりこみ可能な重力理論という前提で話を進める。そしてBack Upにpower counting renormalizabilityについてのスライドを入れる。
8/21
Renormalizability of Horava gravity
If a scalar field with scaling , we can calculate s with invariance of the canonical kinetic term
� ! b�s�
The n-th order interaction term behaves as
If , power counting renormalizable.z � 3
From now on, we mainly focus on the case z = 3
s =D � z
2
the case spatial dimension is D
Remark
t ! bzt, x ! bx
s =3� z
2
Zdtd
3x
�
2
2
Zdtd
3x�
n / E
�(z+3�ns)/z
1.Constraints on LV in inflationary era 9/21
1.Constraints on LV in inflationary era
In addition to the power-counting renormalizability, there are 2 different types of spacetime in HL gravity:
Types of Horava Lifshitz gravity
・Non projectable version: N = N(t, x)
ai = @ilnN↵aiai
- No ghost and IR instability
- Additional terms are possible
・Projectable version: N = N(t)
- continuous connection to GR
IR instability strong coupling (D.Blas et al. 2010)!
10/21
Non projectable version is the only healthy LV theory of gravity so far (D.Blas et al. 2010)
1.Constraints on LV in inflationary era
Behavior of Scalar graviton
- Projectable version:
hij = e2R�ij・4d Diff theory Lagrangian is invariant under a dilatation and shift transformation
・HL gravity
simply the same as a system of 2 light Lifshitz scalars with their interactions
In the non projectable version, massive scalar graviton gets decoupled from inflaton before Hubble crossing.
- Non projectable version:
mass term appears because of non canonical kinetic term
mK = O✓
Hp↵
◆� H
X ⌘⇣ p
aH
⌘2✓
p
aM⇤
◆2(z�1)
mR = 0x ! esx,R ! R� s
LR ⇠ a2�XR02 � a2m2
KXR2�
11/21
1.Constraints on LV in inflationary era
EFT picture of the primordial scalar perturbations
mK
!/a
mK � H
Hp
(non-projectable version)
12/21
1.Constraints on LV in inflationary era
EFT picture of the primordial scalar perturbations
decoupledR =
1
ap2!R
e�iRd⌘!R
' =1
ap
2!'e�i
Rd⌘!'
mK
!/a
mK � H
Horizon crossing Hp
'R,
coupled
(non-projectable version)
12/21
1.Constraints on LV in inflationary era
EFT picture of the primordial scalar perturbations
Isocurvature(Khronon)
P⇣ / 1
"1
✓H
MP
◆3/z�1
decoupledR =
1
ap2!R
e�iRd⌘!R
' =1
ap
2!'e�i
Rd⌘!'
mK
!/a
mK � H
Horizon crossing Hp
⇣ = R� H
�'
Adiabatic⇣
conserved in time
coupled
'R, Khronon⇣,
coupled decoupled
(non-projectable version)
but purely gauge mode of time foliation
sub planckにとっていたら必ずm_KがHよりも大きくなる。
12/21
S�� =M
2P
8
Zd
4xa
3c
�2T
�
2ij � c
2T(@k�ij)2
a
2
�
gµ⌫ 7! gµ⌫ + (1� c2T (t))nµn⌫
gµ⌫ 7! c�1T (t)gµ⌫
- disformal transformation
- conformal transformation
Scalar to tensor ratio & consistency relationP. Creminelli et al. Phys.Rev.Lett 113, 231301(2014)
L = a3M2P |
˙H|c2T˙⇡2 � c�2
T
(@i⇡)2
a2� (1� c�2
T ) ˙⇡(@i⇡)2
a2
�
t =
Zdtc1/2T (t) a = c�1/2
T a(t)
cs = c�1T
- power spectrum of tensor fluctuation
h�p�p0i = (2⇡)3�(p+ p0)1
2p3
H
MP
!2
nt = � r
8cs
\varepsilon > 0 も仮定しているか。
1.Constraints on LV in inflationary era 13/21
1.Constraints on LV in inflationary era
Cosmological Observables of LV inflation
この話をするためには、インフレーションの導入スライドが必要
14/21
it is a direct evidence of LV in the early universe.
power spectrum of primordial gravitational wave
tensor to scalar ratior ⌘ Pt
Ps
Pt =
✓Hinf
2⇡
◆2
/✓
k
k⇤
◆�nt
nt = 0 (z = 3)
P. Creminelli et al. (2014)
4d Diffs and NEC
nt = � r
8cs< 0,
HL gravity (4d Diffs) + NEC
4 dimensional general covariance = 4d Diffs Null Energy Condition = NEC
2. Late-time constraints on LV with CMB distortions
Talk plan
3. Summary
1. Constraints on LV from primordial fluctuations in inflationary era.
SA, S.Sibiryakov and Y.Urakawa in prep.
これはAmazing factです。今まで知られていなかったことが明らかとなったため
SA, D.Nitta and H.Tashiro Phys. Rev. D 94, 124048
15/21
CMB
CMB distortionsE
Black body radiation
z = 1100
z = 0
hot plasmain a galaxy cluster
CMB spectral distortions
y distortionμ distortion
2. Constraints on LV with CMB distortions
y distortion
16/21
Science of CMB distortions
・primordial magnetic field
・primordial fluctuations at a small scale
・the nature of dark matter
・high energy physics
2. Constraints on LV with CMB distortions 17/21
Science of CMB distortions
・primordial magnetic field
・primordial fluctuations at a small scale
・the nature of dark matter
・Test of the General Relativity
・high energy physics
2. Constraints on LV with CMB distortions 17/21
Energy dependent metric
Spectral distortions after recombination
In general y distortion and distortionµ
g ' 1 + h(E), h(E) ⌧ 1
@nE
@t� a
aE
✓1� d log g
d logE
◆�1 @nE
@E= 0
Modification of a redshift evolution
・Boltzmann equation for a photon
※On a static fluid frame
CMBの温度揺らぎによる黒体放射を歪める効果は2次のオーダーにあるので、摂動の1次を今回考
usual redshift evolution
2. Constraints on LV with CMB distortions 18/21
ds
2 = �dt
2 +a
2
g
2(E)�ijdx
idx
j
CMB
CMB distortionsE
Black body radiation
z = 1100
z = 0
CMB spectral distortions w LV
μ distortiony distortion
photon 1
photon 2
2. Constraints on LV with CMB distortions 19/21
Constraints w COBE/FIRAS
−2.0
−4.0
4.0
2.0
0.0
0 5 10 15 2520
∆E×
104
E [K]青点はCOBE/FIRASの黒体放射からのずれ
�E ⌘ nE � nE,BB
nE,BBh(E) / E
Δz/z < 10-5
2. Constraints on LV with CMB distortions 20/21
3.Summary
If the Einstein Equivalence Principle is broken, redshift evolution of a photon propagating after recombination is modified with energy dependence of the photon. This modification make CMB distortions, which enables us to constrain local Lorentz symmetry from observation.
Summary
Primordial gravitational waves could provide us for a smoking gun of the Lorentz invariance in the context of Horava Lifshitz gravity.
これはAmazing factです。今まで知られていなかったことが明らかとなったため
Inflation in a spacetime described by Horava Lifshitz gravity, an extra d.o.f in the gravity sector, scalar graviton get interactive with inflaton fluctuation. However, scalar graviton become massive at sub-horizon scale and inflaton fluctuation get decoupled at horizon crossing time, and finally only the adiabatic perturbation is left as a primordial density fluctuation.
21/21
Thank you for your attention:-)
Space Shuttle Discovery re-entry