1st.mar, 2017 cosmological constraints on lorentz symmetry ...grav-wakate/arai.pdfcosmological...

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

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


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


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


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)


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


EFT picture of the primordial scalar perturbations

mK

!/a

mK � H

Hp

(non-projectable version)

12/21



decoupledR =

1

ap2!R

e�iRd⌘!R

' =1

ap

2!'e�i

Rd⌘!'

mK

!/a

mK � H

Horizon crossing Hp

'R,

coupled


12/21



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


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


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.


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


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


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


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


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.


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

1st.mar, 2017 cosmological constraints on lorentz symmetry ...grav-wakate/arai.pdfcosmological...

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