introduction to teleparallel and f(t) gravity and cosmology emmanuel n. saridakis physics...

Introduction to Teleparallel and f(T) Gravity

and Cosmology

Emmanuel N. Saridakis

Physics Department, National and Technical University of Athens, Greece

Physics Department, Baylor University, Texas, USA

E.N.Saridakis – IAP, Sept 2014

2

Goal We investigate cosmological

scenarios in a universe governed by torsional modified gravity

Note: A consistent or interesting cosmology is not a proof for

the consistency of the underlying gravitational theory


3

Talk Plan 1) Introduction: Gravity as a gauge theory, modified Gravity

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

3) Perturbations and growth evolution

4) Bounce in f(T) cosmology

5) Non-minimal scalar-torsion theory

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

7) Black-hole solutions

8) Solar system and growth-index constraints

9) Conclusions-ProspectsE.N.Saridakis – IAP, Sept 2014

4

Introduction

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

Curvature Levi-Civita connection: Zero Torsion

Einstein 1928: Teleparallel Equivalent of GR:

Weitzenbock connection: Zero Curvature

Einstein-Cartan theory: energy-momentum source of Curvature, spin source of Torsion

[Hehl, Von Der Heyde, Kerlick, Nester Rev.Mod.Phys.48]


5

Introduction

Gauge Principle: global symmetries replaced by local ones:

The group generators give rise to the compensating fields

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

Can we apply this to gravity?

)1(23 USUSU


6

Introduction

Formulating the gauge theory of gravity (mainly after 1960): Start from Special Relativity Apply (Weyl-Yang-Mills) gauge principle to its Poincaré- group symmetries Get 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]


7

Introduction

One could extend the gravity gauge group (SUSY, conformal, scale, metric affine transformations)

obtaining SUGRA, conformal, Weyl, metric affine 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.


8

Introduction

1998: Universe acceleration Thousands of work in Modified Gravity

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

nonminimal derivative coupling, Galileons, Hordenski, massive etc)

Almost all in the curvature-based formulation of gravity

[Copeland, Sami, Tsujikawa Int.J.Mod.Phys.D15], [Nojiri, Odintsov Int.J.Geom.Meth.Mod.Phys. 4]


9

Introduction

1998: Universe acceleration Thousands of work in Modified Gravity

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

nonminimal derivative coupling, Galileons, Hordenski, massive etc)

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], [Nojiri, Odintsov Int.J.Geom.Meth.Mod.Phys. 4]


10

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 Levi-Civita one:

Torsion tensor:

)()()( xexexg BAAB

Ae

AA

W ee

}{

AAA

WW eeeT

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

TG]


11

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 Levi-Civita one:

Torsion tensor:

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

)()()( xexexg BAAB

Ae

AA

W ee

}{

AAA

WW eeeT

}{}{

TTTTTL2

1

4

1

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

Completely equivalent with

GR at the level of equations


12

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:

}{1 4)]([4

1)(1

AATTAATA GeTfTefTSeSTefSeee

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

mSTfTexdG

S )(16

1 4

[Linder PRD 82]


13

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:

f(T) Cosmology: Apply in FRW geometry:

(not unique choice)

Friedmann equations:

mSTfTexdG

S )(16

1 4

}{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

[Ferraro, Fiorini PRD 78], [Bengochea, Ferraro PRD 79][Linder PRD 82]


14

f(T) Cosmology: Background

Effective Dark Energy sector:

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

TDE f

Tf

G 368

3

]2][21[

2 2

TTTT

TTTDE TffTff

fTTffw

[Linder PRD 82]


15

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

Obtain Perturbation Equations:

Focus on growth of matter overdensity go to Fourier modes:

)1(,)1(00

ee ji

ij dxdxadtds )21()21( 222

SHRSHL ....

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]


16

f(T) Cosmology: Perturbations

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/

[Dent, Dutta, Saridakis JCAP 1101]


17

f(T) Cosmology: Perturbations

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

[Dent, Dutta, Saridakis JCAP 1101]

m

m


18

Bounce and Cyclic behavior

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

Expanding ( ), turnaround ( ), contracting near and at the turnaround

0H 0H 0H0H

0H 0H 0H0H


19

Bounce and Cyclic behavior in f(T) cosmology

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

Expanding ( ), turnaround ( ), contracting near and at the turnaround

0H 0H 0H0H

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]


20

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

t

sArcTan

t

tM

tMt

ttf mB

pmB

p 2

36

32

6

)32(

4)(

2

22

22


21

Bounce in f(T) cosmology

Start with a bounching scale factor:

Examine the full perturbations:

with known in terms of and matter

Primordial power spectrum: Tensor-to-scalar ratio:

2

43

3

4

3

2

3

4)(

T

T

TTt

t

sArcTan

t

tM

tMt

ttf mB

pmB

p 2

36

32

6

)32(

4)(

2

22

22

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]

3/12

2

31)(

tata B


22

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


23


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

Friedmann equations in FRW universe:

with effective Dark Energy sector:

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

2^RR )(RfR

mLVTT

exdS )(2

1

22

24

DEmH

3

22

DEDEmm ppH 2

2

22

2

3)(2

HVDE

222

234)(2

HHHVpDE

[Geng, Lee, Saridakis,Wu PLB 704]



24


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

Teleparallel Dark Energy:



25

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

,0DEDE0M0 w,Ω,Ω )(V

azaa rrMM /11,/,/ 40

30


26

Observational constraints on Teleparallel Dark Energy

Exponential potential

Quartic potential

[Geng, Lee, Saridkis JCAP 1201]E.N.Saridakis – IAP, Sept 2014

27

Phase-space analysis of Teleparallel Dark Energy

Transform cosmological system to its autonomous form:

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

z

H

Vy

Hx ,

3

)(,

6

)sgn(3

2222

zyx=

HDE

DE


28

Phase-space analysis of Teleparallel Dark Energy

Apart from usual quintessence points, there exists an extra stable one for corresponding to

[Xu, Saridakis, Leon, JCAP 1207]

2 1,1,1 qwDEDE

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

1DEw


29

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]


30



Friedmann equations in FRW universe:


Different than non-minimal matter-curvature coupled theory

mLRf )(

mLTfTfTexdS )(1)(2

121

42

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



31


Interesting phenomenology

,)( 211 TTf


212 )( TTf ,)(1 Tf 2

112 )( TTTf


32



),( Rf

mLTfTexdS ),(2

1 42 [Harko, Lobo, Otalora, Saridakis, 1405.0519, to appear in

JCAP]


33



Friedmann equations in FRW universe ( ):


Different from gravity

),( Rf

mLTfTexdS ),(2

1 42

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, 1405.0519, to appear in JCAP]

mm p3

),( Rf


34


Interesting phenomenology

[Harko, Lobo, Otalora, Saridakis, 1405.0519, to appear in JCAP]

2),( TTf nTTf ),(


35

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 G .


36


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

gravity:

Different from and gravities

RRRRRG 42

abcdaaaa

fdc

eaf

aeb

aaa

fcd

eaf

aeb

aaa

fad

efc

aeb

aaa

fad

afc

eab

aeaG KKKKKKKKKKKKKKKKT

4321

4321432143214321

2 ,222

[Kofinas, Saridakis, 1404.2249 to appear in PRD]

),( GTf

,2 eTeTRe divergtoteTGe G .

mG STTfTexdS ),(2

1 42

),( GRf )(Tf

[Kofinas, Saridakis, 1408.0107 to appear in PRD][Kofinas, Leon, Saridakis, CQG 31]


37


Cosmological application:



[Kofinas, Leon, Saridakis, CQG 31]

GG TTGTDE fHfTfHf 32

22412

2

1

GGG TTGTGTTDE fHfTH

fTfHfHHfp 222

83

2434

2

1

26HT

2224 HHHTG

GG TTTTTf 22

1),( GG TTTTf 2

21),(


38

Exact charged black hole solutions

Extend f(T) gravity in D-dimensions (focus on D=3, D=4):

Add E/M sector: with

Extract field equations:

2)(2

1TfTexdS D

FFLF

2

1 dxAAdAF ,

SHRSHL .... [Gonzalez, Saridakis, Vasquez, JHEP 1207]

[Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 1302]


39


Extend f(T) gravity in D-dimensions (focus on D=3, D=4):

Add E/M sector: with

Extract field equations:

Look for spherically symmetric solutions:

Radial Electric field: known

2)(2

1TfTexdS D

FFLF

2

1 dxAAdAF ,

SHRSHL ....

2

1

2222

222

)(

1)(

D

idxrdrrG

dtrFds

,,,,)(

1,)( 2

31

210 rdxerdxedrrG

edtrFe

2Dr r

QE

22 )(,)( rGrF[Gonzalez, Saridakis, Vasquez, JHEP 1207], [Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 1302]


40


Horizon and singularity analysis:

1) Vierbeins, Weitzenböck connection, Torsion invariants: T(r) known obtain horizons and singularities

2) Metric, Levi-Civita connection, Curvature invariants: R(r) and Kretschmann known obtain horizons and singularities

[Gonzalez, Saridakis, Vasquez, JHEP1207], [Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 1302]

)(rRR


41


[Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 1302]


42


More singularities in the curvature analysis than in torsion analysis!

(some are naked) The differences disappear in the f(T)=0 case, or in the uncharged

case.

Going to quartic torsion invariants solves the problem.

f(T) brings novel features.

E/M in torsion formulation was known to be nontrivial (E/M in Einstein-Cartan and Poinaré theories)


43

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)(

2222

22

22 8

82

224

3

8

3

21)(

rrc

GMr

rr

rc

GMrG


44

Solar System constraints on f(T) gravity

Apply the black hole solutions in Solar System: Assume corrections to TEGR of the form

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.

)()( 32 TOTTf

rc

GM

rr

rc

GMrF

222

22 46

63

21)(

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.

45

Growth-index constraints on f(T) gravity

)(ln

lna

ad

dm

m

)('1

1

TfGeff

mmeffmm GH 42


46

Growth-index constraints on f(T) gravity

)(ln

lna

ad

dm

m

)('1

1

TfGeff

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

mmeffmm GH 42 Perturbations: , clustering growth rate:

γ(z): Growth index.

Viable f(T) models are practically indistinguishable from ΛCDM.


47

Open issues of f(T) gravity

f(T) cosmology is very interesting. But f(T) gravity and nonminially coupled teleparallel gravity has many open issues

For nonlinear f(T), Lorentz invariance is not satisfied Equivalently, the vierbein choices corresponding to the same

metric are not equivalent (extra degrees of freedom)

[Li, Sotiriou, Barrow PRD 83a],

[Li,Sotiriou,Barrow PRD 83c], [Li,Miao,Miao JHEP 1107]

[Geng,Lee,Saridakis,Wu PLB 704]


48

Open issues of f(T) gravity

f(T) cosmology is very interesting. But f(T) gravity and nonminially coupled teleparallel gravity has many open issues

For nonlinear f(T), Lorentz invariance is not satisfied Equivalently, the vierbein choices corresponding to the same

metric are not equivalent (extra degrees of freedom)

Black holes are found to have different behavior through curvature and torsion analysis

Thermodynamics also raises issues

Cosmological, Solar System and Growth Index observations constraint f(T) very close to linear-in-T form

[Li, Sotiriou, Barrow PRD 83a],

[Li,Miao,Miao JHEP 1107]

[Capozzielo, Gonzalez, Saridakis, Vasquez JHEP 1302]

[Bamba, Capozziello, Nojiri, Odintsov ASS 342], [Miao,Li,Miao JCAP 1111]

[Geng,Lee,Saridakis,Wu PLB 704]

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

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


49

Gravity modification in terms of torsion?

So can we modify gravity starting from its torsion formulation?

The simplest, a bit naïve approach, through f(T) gravity is interesting, but has open issues

Additionally, f(T) gravity is not in correspondence with f(R)

Even if we find a way to modify gravity in terms of torsion, will it be still in 1-1 correspondence with curvature-based modification?

What about higher-order corrections, but using torsion invariants (similar to Lovelock, Hordenski modifications)?

Can we modify gauge theories of gravity themselves? E.g. can we modify Poincaré gauge theory?


50

Conclusions i) Torsion appears in all approaches to gauge gravity, i.e to the first step of

quantization. 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

structure. v) We can obtain bouncing solutions vi) Non-minimal coupled scalar-torsion theory : Quintessence,

phantom or crossing behavior. Similarly in torsion-matter coupling and TEGB.

vii) Exact black hole solutions. Curvature vs torsion analysis. viii) Solar system constraints: f(T) divergence from T less than ix) Growth Index constraints: Viable f(T) models are practically

indistinguishable from ΛCDM. x) Many open issues. Need to search for other torsion-based modifications

too.

2ξTT

1010


51

Outlook Many subjects are open. Amongst them:

i) Examine thermodynamics thoroughly.

ii) Understand the extra degrees of freedom and the extension to non-diagonal vierbeins.

iii) Try to modify TEGR using higher-order torsion invariants.

iv) Try to modify Poincaré gauge theory (extremely hard!)

v) What to quantize? Metric, vierbeins, or connection?

vi) Convince people to work on the subject!


52

THANK YOU!


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