RDO · 2018. 4. 3. · rdo :h lqyhvwljdwh wkh sursdjdwlrq ri judylwdwlrqdo zdyhv *: lq d xqlyhuvh...

$: *RDO · 2018. 4. 3. · *rdo :h lqyhvwljdwh wkh sursdjdwlrq ri judylwdwlrqdo zdyhv *: lq d xqlyhuvh jryhuqhg e\ wruvlrqdo prglilhgjudylw\ +ljk dffxudf\ dgydqflqj *: dvwurqrp\ riihuv$
Gravitational Waves in Torsional Modified Gravity

Emmanuel N. SaridakisEmmanuel N. Saridakis

Physics Department, National and Technical University of Athens, Greece

Physics Department, Baylor University, Texas, USA

E.N.Saridakis – GWiMG, NTUA, March 2018

Goal

We investigate the propagation of gravitational waves (GW) in a universe governed by torsional

2

universe governed by torsional modified gravity

High accuracy advancing GW astronomy offers a new window in testing Modified Gravity


Talk Plan 1) Introduction: Why Modified Gravity

2) Teleparallel Equivalent of General Relativity and f(T) modification

3) Non-minimal scalar-torsion theories

3

4) Teleparallel Equivalent of Gauss-Bonnet and f(T,T_G) modification

5) Solar system, growth-index, baryogenesis and BBN constraints

6) The EFT approach to torsional gravity

7) Background solutions

8) Gravitational Waves and observational signatures

9) Conclusions-Prospects


Why Modified Gravity?

Knowledge of Physics: Standard Model

E.N.Saridakis – SEMFE, NTUA, March 2016



Knowledge of Physics: Standard Model + General Relativity

E.N.Saridakis – SEMFE, NTUA, March 2016


Why Modified Gravity?Universe History:


Why Modified Gravity?So can our knowledge of Physics describes all these?



Einstein 1916: General Relativity: energy-momentum source of spacetime Curvature

,21 44 gLxdRgxdS m

9

with

,2

16gLxdRgxd

GS m

TGgRgR 82

1

g

L

gT m

2


Modified Gravity

10

Non-minimal gravity-matter coupling

(Gen. Proca)


Introduction

Einstein 1916: General Relativity: energy-momentum source of spacetime CurvatureLevi-Civita connection: Zero Torsion

11

Einstein 1928: Teleparallel Equivalent of GR:

Weitzenbock connection: Zero Curvature

[Cai, Capozziello, De Laurentis, Saridakis, Rept.Prog.Phys. 79]


Introduction

Gauge Principle: global symmetries replaced bylocal ones:The group generators give rise to the compensating fields

12

fieldsIt works perfect for the standard model of strong, weak and E/M interactions

Can we apply this to gravity?

)1(23 USUSU


Introduction

Formulating the gauge theory of gravity (mainly after 1960):

Start from Special Relativity Apply (Weyl-Yang-Mills) gauge principle to its Poincaré-

13

Apply (Weyl-Yang-Mills) gauge principle to its Poincaré-group symmetriesGet Poinaré gauge theory:Both curvature and torsion appear as field strengths

Torsion is the field strength of the translational group(Teleparallel and Einstein-Cartan theories are subcases of Poincaré theory)

[Blagojevic, Hehl, Imperial College Press, 2013]


Introduction

One could extend the gravity gauge group (SUSY, conformal, scale, metric affine transformations)obtaining SUGRA, conformal, Weyl, metric affine gauge theories of gravity

14

gauge theories of gravity

In all of them torsion is always related to the gauge structure.

Thus, a possible way towards gravity quantization would need to bring torsion into gravity description.


Introduction

1998: Universe accelerationThousands of work in Modified Gravity

(f(R), Gauss-Bonnet, Lovelock, nonminimal scalar coupling,

nonminimal derivative coupling, Galileons, Hordenski, massive etc)

15


Almost all in the curvature-based formulation of gravity

[Copeland, Sami, Tsujikawa Int.J.Mod.Phys.D15], [Capozziello, De Laurentis, Phys. Rept. 509]


Introduction

1998: Universe accelerationThousands of work in Modified Gravity

(f(R), Gauss-Bonnet, Lovelock, nonminimal scalar coupling,


16


Almost all in the curvature-based formulation of gravity

So question: Can we modify gravity starting from its torsion-based formulation?torsion gauge quantization

modification full theory quantization ?

?

[Copeland, Sami, Tsujikawa Int.J.Mod.Phys.D15], [Capozziello, De Laurentis, Phys. Rept. 509]


Teleparallel Equivalent of General Relativity (TEGR)

Let’s start from the simplest tosion-based gravity formulation, namely TEGR:

Vierbeins : four linearly independent fields in the tangent space

Use curvature-less Weitzenböck connection instead of torsion-less

)()()( xexexg BAAB

Ae

17

Use curvature-less Weitzenböck connection instead of torsion-lessLevi-Civita one:

Torsion tensor:

AA

W ee

}{

AAA

WW eeeT

}{}{ [Einstein 1928], [Pereira: Introduction to TG]


Teleparallel Equivalent of General Relativity (TEGR)

Let’s start from the simplest tosion-based gravity formulation, namely TEGR:


Use curvature-less Weitzenböck connection instead of torsion-less

)()()( xexexg BAAB

Ae

18

Use curvature-less Weitzenböck connection instead of torsion-lessLevi-Civita one:

Torsion tensor:

Lagrangian (imposing coordinate, Lorentz, parity invariance, and up to 2nd order in torsion tensor)

AA

W ee

}{

AAA

WW eeeT

}{}{

TTTTTL2

1

4

1

[Einstein 1928], [Hayaski,Shirafuji PRD 19], [Pereira: Introduction to TG]

Completely equivalent withGR at the level of equations


f(T) Gravity and f(T) Cosmology

f(T) Gravity: Simplest torsion-based modified gravity Generalize T to f(T) (inspired by f(R))

Equations of motion:

[Ferraro, Fiorini PRD 78], [Bengochea, Ferraro PRD 79] mSTfTexdG

S )(16

1 4

[Linder PRD 82]

19


}{1 4)]([4

1)(1

AATTAATA GeTfTefTSeSTefSeee


f(T) Gravity and f(T) Cosmology

f(T) Gravity: Simplest torsion-based modified gravity Generalize T to f(T) (inspired by f(R))


mSTfTexdG

S )(16

1 4

[Ferraro, Fiorini PRD 78], [Bengochea, Ferraro PRD 79]

[Linder PRD 82]

20


f(T) Cosmology: Apply in FRW geometry:

(not unique choice)

Friedmann equations:

}{1 4)]([4

1)(1

AATTAATA GeTfTefTSeSTefSeee

jiij

A dxdxtadtdsaaadiage )(),,,1( 222

22 26

)(

3

8Hf

TfGH Tm

TTT

mm

fHf

pGH

2121

)(4

Find easily26HT


f(T) Cosmology: Background

Effective Dark Energy sector:

TDE f

Tf

G 368

3

]2][21[

2 2

TTTT

TTTDE TffTff

fTTffw

[Linder PRD 82]

21

Interesting cosmological behavior: Acceleration, Inflation etc At the background level indistinguishable from other dynamical DE models


f(T) Cosmology: Perturbations

Can I find imprints of f(T) gravity? Yes, but need to go to perturbation level

Obtain Perturbation Equations:

)1(,)1(00

ee ji

ij dxdxadtds )21()21( 222

SHRSHL ....

22

Focus on growth of matter overdensity go to Fourier modes:

SHRSHL .... [Chen, Dent, Dutta, Saridakis PRD 83],

[Dent, Dutta, Saridakis JCAP 1101]

m

m

0436)1)(/3(1213 42222 kmkTTTkTTT GfHfakHfHfH

[Chen, Dent, Dutta, Saridakis PRD 83]



Application: Distinguish f(T) from quintessence 1) Reconstruct f(T) to coincide with a given quintessence scenario:

with and CHdHH

GHHf Q 216)(

)(2/2 VQ

6/

23




Application: Distinguish f(T) from quintessence 2) Examine evolution of matter overdensity


m

m

24E.N.Saridakis – GWiMG, NTUA, March 2018

Bounce and Cyclic behavior in f(T) cosmology

Contracting ( ), bounce ( ), expanding ( )near and at the bounce

Expanding ( ), turnaround ( ), contracting

0H 0H 0H0H

0H 0H 0H

25

Expanding ( ), turnaround ( ), contractingnear and at the turnaround

0H 0H 0H0H


Bounce and Cyclic behavior in f(T) cosmology

Contracting ( ), bounce ( ), expanding ( )near and at the bounce

Expanding ( ), turnaround ( ), contracting

0H 0H 0H0H

0H 0H 0H

26

Expanding ( ), turnaround ( ), contractingnear and at the turnaround

0H 0H 0H0H

22 26

)(

3

8Hf

TfGH Tm

TTT

mm

fHf

pGH

2121

)(4

Bounce and cyclicity can be easily obtained[Cai, Chen, Dent, Dutta, Saridakis CQG 28]


Bounce in f(T) cosmology

Start with a bounching scale factor:3/1

2

2

31)(

tata B

2

43

3

4

3

2

3

4)(

T

T

TTt

ts

ArcTantMt

tf pmB 36

64)(

22

27

t

sArcTan

t

tM

tMt

ttf mB

pmB

p 2

36

32

6

)32(

4)(

222


Bounce in f(T) cosmology

Start with a bounching scale factor:

2

43

3

4

3

2

3

4)(

T

T

TTt

ts

ArcTantMt

tf pmB 36

64)(

22

3/12

2

31)(

tata B

28

Examine the full perturbations:

with known in terms of and matter

Primordial power spectrum: Tensor-to-scalar ratio:

t

sArcTan

t

tM

tMt

ttf mB

pmB

p 2

36

32

6

)32(

4)(

222

02

222 kskkk a

kc

22 ,, sc TTT fffHH ,,,,

22288 pMP

01

123

2

2

h

f

fHHh

ahHh

T

TTijijij

3108.2 r

[Cai, Chen, Dent, Dutta, Saridakis CQG 28]E.N.Saridakis – GWiMG, NTUA, March 2018

Non-minimally coupled scalar-torsion theory

In curvature-based gravity, apart from one can use Let’s do the same in torsion-based gravity:

[Geng, Lee, Saridakis, Wu PLB 704]

2^RR )(RfR

mLVTT

exdS )(2

1

22

24

29

22



In curvature-based gravity, apart from one can use Let’s do the same in torsion-based gravity:

2^RR )(RfR

mLVTT

exdS )(2

1

22

24


30

Friedmann equations in FRW universe:

with effective Dark Energy sector:

Different than non-minimal quintessence!(no conformal transformation in the present case)

22

DEmH

3

22

DEDEmm ppH 2

2

222

3)(2

HVDE

222

234)(2

HHHVpDE

[Geng, Lee, Saridakis,Wu PLB 704]



Main advantage: Dark Energy may lie in the phantom regime or/and experience the phantom-divide crossing

Teleparallel Dark Energy:

31



Observational constraints on Teleparallel Dark Energy

Use observational data (SNIa, BAO, CMB) to constrain the parameters of the theory

Include matter and standard radiation: We fit for various ,w,Ω,Ω )(V

azaa rrMM /11,/,/ 40

30

32

We fit for various ,0DEDE0M0 w,Ω,Ω )(V


Observational constraints on Teleparallel Dark Energy

Exponential potential

33

Quartic potential

[Geng, Lee, Saridkis JCAP 1201]E.N.Saridakis – GWiMG, NTUA, March 2018

Phase-space analysis of Teleparallel Dark Energy

Transform cosmological system to its autonomous form:

),sgn(1 222 zyx=m

zH

Vy

Hx ,

3

)(,

6

)sgn(222 zyx=DE

34

Linear Perturbations: Eigenvalues of determine type and stability of C.P

),sgn(13

2222 zyx=

Hm

m

[Xu, Saridakis, Leon, JCAP 1207] ,,, zyxw=w DEDE

,f(X)=X'QUU 'UX=X C

0' =|XCX=X

Q

)sgn(3

2222

zyx=HDE

DE


Phase-space analysis of Teleparallel Dark Energy

Apart from usual quintessence points, there exists an extrastable one for corresponding to 2 1,1,1 qwDEDE

35

[Xu, Saridakis, Leon, JCAP 1207]

At the critical points however during the evolution it can lie in quintessence or phantomregimes, or experience the phantom-divide crossing!

1DEw


Non-minimally matter-torsion coupled theory

In curvature-based gravity, one can use coupling Let’s do the same in torsion-based gravity:

mLRf )(

mLTfTfTexdS )(1)(2

121

42

[Harko, Lobo, Otalora, Saridakis, PRD 89]

36

2




mLRf )(

mLTfTfTexdS )(1)(2

121

42

37

Friedmann equations in FRW universe:


Different than non-minimal matter-curvature coupled theory

2

DEmH

3

22

DEDEmm ppH 2

2

22

212

121212

2

1fHffHf mDE

22

221

21

22

22

12

1

22

2 122

12

122121

121fHf

fHf

fHffHf

fHfpp m

mmmDE




Interesting phenomenology

38

,)( 211 TTf


212 )( TTf ,)(1 Tf 2

112 )( TTTf




),( Rf

mLTfTexdS ),(2

1 42 [Harko, Lobo, Otalora, Saridakis, JCAP 1412]

39

2




),( Rf

mLTfTexdS ),(2

1 42

40

Friedmann equations in FRW universe ( ):


Different from gravity

2

DEmH

3

22

DEDEmm ppH 2

2

mmTDE pffHf

2122

1 22

mmTTmmmTTT

mmDE pffHffddpdHdHfHf

fpp

2122

11

/31/121

/1 222

2

[Harko, Lobo, Otalora, Saridakis, JCAP 1412]

mm p3

),( Rf



Interesting phenomenology

41[Harko, Lobo, Otalora, Saridakis, JCAP 1412]

2),( TTf nTTf ),(


Teleparallel Equivalent of Gauss-Bonnet and f(T,T_G) gravity

In curvature-based gravity, one can use higher-order invariants like the Gauss-Bonnet one

Let’s do the same in torsion-based gravity: Similar to we construct with

RRRRRG 42

2 eTeTRe divergtoteTGe .

42

Similar to we construct with ,2 eTeTRe divergtoteTGe G .



In curvature-based gravity, one can use higher-order invariants like the Gauss-Bonnet one

Let’s do the same in torsion-based gravity: Similar to we construct with

RRRRRG 42

2 eTeTRe divergtoteTGe .

43

Similar to we construct with

gravity:

Different from and gravities

abcdaaaa

fdc

eaf

aeb

aaa

fcd

eaf

aeb

aaa

fad

efc

aeb

aaa

fad

afc

eab

aeaG KKKKKKKKKKKKKKKKT

4321

4321432143214321

2 ,222

[Kofinas, Saridakis, PRD 90a]

),( GTf

,2 eTeTRe divergtoteTGe G .

mG STTfTexdS ),(2

1 42

),( GRf )(Tf

[Kofinas, Saridakis, PRD 90b]

[Kofinas, Leon, Saridakis, CQG 31]



Cosmological application:

GG TTGTDE fHfTfHf 322

24122

1

GGG TTGTGTTDE fHfTH

fTfHfHHfp 222

83

2434

2

1

26HT

2224 HHHTG

44

[Kofinas, Saridakis, PRD 90a]

[Kofinas, Saridakis, PRD 90b]

[Kofinas, Leon, Saridakis, CQG 31]

GG TTTTTf 22

1),( GG TTTTf 2

21),(


Torsional Gravity with higher derivatives

),(,)(,2

1 242 m

Am eSTTTFexdS

45[Otalora, Saridakis, PRD 94]


Torsional Modified Gravity


Solar System constraints on f(T) gravity

Apply the black hole solutions in Solar System: Assume corrections to TEGR of the form )()( 32 TOTTf

rc

GM

rr

rc

GMrF

222

22 46

63

21)(

822482 GMGM

47

2222

22

22 8

82

224

3

8

3

21)(

rrc

GMr

rr

rc

GMrG


Solar System constraints on f(T) gravity

Apply the black hole solutions in Solar System: Assume corrections to TEGR of the form )()( 32 TOTTf

rc

GM

rr

rc

GMrF

222

22 46

63

21)(

822482 GMGM

48

Use data from Solar System orbital motions:

T<<1 so consistent

f(T) divergence from TEGR is very small This was already known from cosmological observation constraints up to

With Solar System constraints, much more stringent bound.

2222

22

22 8

82

224

3

8

3

21)(

rrc

GMr

rr

rc

GMrG

10)( 102.6 TfU

[Iorio, Saridakis, Mon.Not.Roy.Astron.Soc 427)

)1010( 21 O [Wu, Yu, PLB 693], [Bengochea PLB 695]


Perturbations: , clustering growth rate:

γ(z): Growth index.

Growth-index constraints on f(T) gravity

)(ln

lna

ad

dm

m

)('1

1

TfGeff

mmeffmm GH 42


Growth-index constraints on f(T) gravity

)(ln

lna

ad

dm

m

)('1

1

TfGeff

mmeffmm GH 42 Perturbations: , clustering growth rate:

γ(z): Growth index.

50[Nesseris, Basilakos, Saridakis, Perivolaropoulos, PRD 88]

Viable f(T) models are practically indistinguishable from ΛCDM.[Nunes, Pan, Saridakis, JCAP 1608] E.N.Saridakis – GWiMG, NTUA, March 2018

Baryon-anti-baryon asymmetry through CP violating term:

Baryogenesis and BBN constraints on f(T) gravity

JTfexd

M)(

1 42

BBN constraints:

51

510 f

GRT

f

f

q

H

[Oikonomou, Saridakis, PRD 94]

[Capozziello, Lambiase, Saridakis, EPJC77]


Covariant formulation of f(T) gravity

In standard f(T) gravity spin connection is set to zero.

However vierbein transformations must be accompanied by connection ones:BA

BA ee

CB

AC

DB

CD

AC

AB [Krssak, Pereira EPJC 75]

52

BCBDCB [Krssak, Pereira EPJC 75]


Covariant formulation of f(T) gravity

In standard f(T) gravity spin connection is set to zero.

However vierbein transformations must be accompanied by connection ones:BA

BA ee

CB

AC

DB

CD

AC

AB [Krssak, Pereira EPJC 75]

53

Example: FRW geometryor

On the other hand, if one assumes/imposes then only “peculiar” forms of vierbeins will be allowed.

Lorentz invariance has been restored in f(T) gravity[Krssak, Saridakis CQG 33]

0AB

),,,1( aaadiage A )sin,,,1( raraadiage A

BCBDCB

cos,sin,1 23

13

12

0AB

[Krssak, Pereira EPJC 75]


The Effective Field Theory (EFT) approach

The EFT approach allows to ignore the details of the underlying theory and write an action for the perturbations around a time-dependent background solution.

One can systematically analyze the perturbations separately from the background evolution. [Arkani-Hamed, Cheng JHEP0405 (2004)]


The Effective Field Theory (EFT) approach

The EFT approach allows to ignore the details of the underlying theory and write an action for the perturbations around a time-dependent background solution.

One can systematically analyze the perturbations separately from the background evolution. [Arkani-Hamed, Cheng JHEP0405 (2004)]

55

<- background

<- linear evolution of perturbations



<- 2nd-order evolution of perturbations

The functions Ψ(t), Λ(t), b(t), are determined by the background solution

[Gubitosi, Piazza, Vernizzi, JCAP1302]


The (EFT) approach to torsional gravity

Application of the EFT approach to torsional gravity leads to include terms:

i) Invariant under 4D diffeomorphisms: e.g. R,T multiplied by functions of time.

ii) Invariant under spatial diffeomorphisms: e.g.

ii) Invariant under spatial diffeomorphisms: e.g. , , , the extrinsic torsion is defined as

56

the extrinsic torsion is defined as

with the orthogonal to t=cont. surfaces unitary vector =

[Cai, Li, Saridakis, Xue, 1801.05827, Li, Cai, Cai, Saridakis, in preparation]



Application of the EFT approach to torsional gravity leads to include terms:

i) Invariant under 4D diffeomorphisms: e.g. R,T multiplied by functions of time.

ii) Invariant under spatial diffeomorphisms: e.g.

ii) Invariant under spatial diffeomorphisms: e.g. , , , the extrinsic torsion is defined as

57

the extrinsic torsion is defined as

with the orthogonal to t=cont. surfaces unitary vector =

Using the projection operator we can express




We perturb the previous tensors, and we finally obtain:

58

where the time-dependent functions are determined by the background solution.




Finally, the EFT action of torsional gravity becomes:

59

The perturbation part contains:

i) Terms present in curvature EFT action

ii) Pure torsion terms such as ,

iii) Terms that mix curvature and torsion, such as ,



The (EFT) approach to f(T) gravity: Background

For the case of f(T) gravity, at the background level, we have:

60

where by comparison:

[Li, Cai, [Cai, Saridakis, in preparation]



For the case of f(T) gravity, at the background level, we have:

61

where by comparison:

Performing variation we obtain the background equations of motion (Friedmann Eqs):

[Li, Cai, Cai, Saridakis, in preparation]



These can be written as:

with

62

with

and thus:

The same equations with standard approach![Li, Cai, Cai, Saridakis, in preparation]


For tensor perturbations: i.e.

The (EFT) approach to f(T) gravity: Tensor Perturbations


For tensor perturbations: i.e.

We obtain:

The (EFT) approach to f(T) gravity: Tensor Perturbations

We obtain:

And finally:

64[Cai, Li, Saridakis, Xue, 1801.05827]


Varying the action and going to Fourier space we get the equation for GWs:

The (EFT) approach to f(T) gravity: Gravitational Waves

with

An immediate result: The speed of GWs is equal to the speed of light!

GW170817 constraints that

are trivially satisfied.

65

[Cai, Li, Saridakis, Xue, 1801.05827]


Choosing the ansatz:

We obtain the dispersion relation:


where we can express:

66







In GR and TEGR is zero. Thus, if a non-zero is measured in future observations, it could be the smoking gun of modified gravity. Very important since f(T) gravity has the same polarization modes with GR.

67







In GR and TEGR is zero. Thus, if a non-zero is measured in future observations, it could be the smoking gun of modified gravity. Very important since f(T) gravity has the same polarization modes with GR.

The effect of f(T) gravity on GWs comes through its effect on the background solutions itself, since at linear perturbation order f(T) gravity is effectively TEGR.

68



Conclusions i) Many cosmological and theoretical arguments favor modified gravity.

ii) Can we modify gravity based in its torsion formulation?

iii) Simplest choice: f(T) gravity, i.e extension of TEGR

iv) f(T) cosmology: Interesting phenomenology. Signatures in growth

69

iv) f(T) cosmology: Interesting phenomenology. Signatures in growth structure.

v) Non-minimal coupled scalar-torsion theory: Quintessence, phantom or crossing behavior. Similarly in torsion-matter coupling and TEGB.

vi) EFT approach allows for a systematic study of perturbations

vii) Observational signatures in the dispersion relation of GWs

viii) No further polarization modes.


Outlook Many subjects are open. Amongst them:

i) Examine higher-order perturbations to look for further polarizations.[Farugia, Gakis, Jackson, Saridakis, in preparation]

70

ii) Extend the analysis to other torsional modified gravity.

iii) Try to break the various degeneracies and find a signature of this particular class of modified gravity

vi) Convince people to work on the subject!

[Farugia, Gakis, Jackson, Saridakis, in preparation]


“There are the ones that invent occultfluids to understand the Laws of Nature.They come to conclusions, but they nowrun out into dreams and chimerasneglecting the true constitutions of thethings...

71

things...However there are those that from thesimplest observation of Nature, theyreproduce New Forces”…

From the Preface of PRINCIPIA (II edition) 1687 by Isaac Newton, written by Mr. Roger Cotes.


“There are the ones that invent occultfluids to understand the Laws of Nature.They come to conclusions, but they nowrun out into dreams and chimerasneglecting the true constitutions of thethings...

7272

things...However there are those that from thesimplest observation of Nature, theyreproduce New Forces”…

From the Preface of PRINCIPIA (II edition) 1687 by Isaac Newton, written by Mr. Roger Cotes.

THANK YOU!E.N.Saridakis – GWiMG, NTUA, March 2018

Curvature and Torsion


Connection:

Curvature tensor:

)()()( xexexg BAAB

Ae

ABCCB

AC

CB

AC

AB

AB

ABR ,,

74

Curvature tensor:

Torsion tensor:

Levi-Civita connection and Contorsion tensor:

Curvature and Torsion Scalars:

BCBCBBBR ,,

BAB

BAB

AAA eeeeT ,,

ABCABCABC K BACABCBCACABABC KTTTK

2

1

;2 TTRR

TTTTT2

1

4

1

RgRgR

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Transcript of *RDO · 2018. 4. 3. · *rdo :h lqyhvwljdwh wkh sursdjdwlrq ri judylwdwlrqdo zdyhv *: lq d xqlyhuvh...

RDO · 2018. 4. 3. · rdo :h lqyhvwljdwh wkh sursdjdwlrq ri judylwdwlrqdo zdyhv *: lq d xqlyhuvh...

Transcript of RDO · 2018. 4. 3. · rdo :h lqyhvwljdwh wkh sursdjdwlrq ri judylwdwlrqdo zdyhv *: lq d xqlyhuvh...