torsional, teleparallel and f(t) gravity and cosmologyviavca.in2p3.fr/presentations/torsional...1)...

Torsional, 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 – VIA web lecture, Feb 2014

2

Goal

n  We investigate cosmological scenarios in a universe governed by torsional modified gravity

n Note: A consistent or interesting cosmology

is not a proof for the consistency of the underlying gravitational theory


3

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

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

n  3) Perturbations and growth evolution

n  4) Bounce in f(T) cosmology

n  5) Non-minimal scalar-torsion theory

n  6) Black-hole solutions

n  7) Solar system and growth-index constraints

n  8) Conclusions-Prospects


4

Introduction

n  Einstein 1916: General Relativity: energy-momentum source of spacetime Curvature Levi-Civita connection: Zero Torsion n  Einstein 1928: Teleparallel Equivalent of GR: Weitzenbock connection: Zero Curvature

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

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


5

Introduction

n  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

n  Can we apply this to gravity?

( ) ( ) )1(23 USUSU ××


6

Introduction

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

n  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

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

obtaining SUGRA, conformal, Weyl, metric affine gauge theories of gravity

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

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


8

Introduction

n  1998: Universe acceleration Thousands of work in Modified Gravity

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

nonminimal derivative coupling, Galileons, Hordenski, massive etc)

n  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

n  1998: Universe acceleration Thousands of work in Modified Gravity

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

nonminimal derivative coupling, Galileons, Hordenski, massive etc)

n  Almost all in the curvature-based formulation of gravity

n  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)

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

n  Vierbeins : four linearly independent fields in the tangent space n  Use curvature-less Weitzenböck connection instead of torsion-less

Levi-Civita one: n  Torsion tensor:

)()()( xexexg BAAB νµµν η=

µAe

AA

W ee νµλλ

νµ ∂=Γ }{

( )AAA

WW eeeT µννµλλ

µνλνµ

λµν ∂−∂=Γ−Γ= }{}{ [Einstein 1928], [Pereira: Introduction to TG]


11

Teleparallel Equivalent of General Relativity (TEGR)

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

n  Vierbeins : four linearly independent fields in the tangent space n  Use curvature-less Weitzenböck connection instead of torsion-less

Levi-Civita one: n  Torsion tensor:

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

)()()( xexexg BAAB νµµν η=

µAe

AA

W ee νµλλ

νµ ∂=Γ }{

( )AAA

WW eeeT µννµλλ

µνλνµ

λµν ∂−∂=Γ−Γ= }{}{

νµν

ρρµνµρ

ρµνρµν

ρµν ΤΤ−+=≡ TTTTTL21

41

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

n  Completely equivalent with GR at the level of equations


12

f(T) Gravity and f(T) Cosmology

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

n  Equations of motion:

( )( ) }{1 4)]([41)(1 ΕΜ− Τ=+−∂+−+∂ ν

ρρν

µµνρ

ρνµρ

ρµλ

λµνρ

ρµ π AATTAATA GeTfTefTSeSTefSeee

[Bengochea, Ferraro PRD 79], [Linder PRD 82] [ ] mSTfTexdG

S ++= ∫ )(161 4

π


13

f(T) Gravity and f(T) Cosmology

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

n  Equations of motion:

n  f(T) Cosmology: Apply in FRW geometry:

(not unique choice)

n  Friedmann equations:

[ ] mSTfTexdG

S ++= ∫ )(161 4

π

( )( ) }{1 4)]([41)(1 ΕΜ− Τ=+−∂+−+∂ ν

ρρν

µµνρ

ρνµρ

ρµλ

λµνρ

ρµ π AATTAATA GeTfTefTSeSTefSeee

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

jiij

A dxdxtadtdsaaadiage δµ )(),,,1( 222 −=⇒=

22 26)(

38 HfTfGH Tm −−= ρπ

TTT

mm

fHfpGH 2121)(4

−++

−=ρπ!

n  Find easily 26HT −=


14

f(T) Cosmology: Background

n  Effective Dark Energy sector:

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

⎥⎦

⎤⎢⎣

⎡ +−= TDE fTfG 3683π

ρ

]2][21[2 2

TTTT

TTTDE TffTff

fTTffw−++

+−−=

[Linder PRD 82]


15

f(T) Cosmology: Perturbations

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

n  Obtain Perturbation Equations:

n  Focus on growth of matter overdensity go to Fourier modes:

)1(,)1(00 φαδψδ αµ

αµµµ −=+= ee ji

ij dxdxadtds δφψ )21()21( 222 −−+=⇒

SHRSHL .... =

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

[Dent, Dutta, Saridakis JCAP 1101]

m

m

ρδρ

δ ≡

( ) [ ] 0436)1)(/3(1213 42222 =+−+++−+ kmkTTTkTTT GfHfakHfHfH δρπφφ!

[Chen, Dent, Dutta, Saridakis PRD 83]


16


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

with and

CHdHH

GHHf Q += ∫ 216)(ρ

π )(2/2 φφρ VQ += ! 6/Τ−=Η



17


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


m

m

ρδρ

δ ≡


18

Bounce and Cyclic behavior

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

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

0<H 0=H 0>H0>H!

0>H 0=H 0<H0<H!


19

Bounce and Cyclic behavior in f(T) cosmology

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

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

0<H 0=H 0>H0>H!

0>H 0=H 0<H0<H!

22 26)(

38 HfTfGH Tm −−= ρπ

TTT

mm

fHfpGH 2121)(4

−++

−=ρπ!

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


20

Bounce in f(T) cosmology

n  Start with a bounching scale factor:

3/12

231)( ⎟

⎠

⎞⎜⎝

⎛ += tata B σ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ++−−±=⇒ 2

43

34

32

34)(

σσσ

σ TT

TTt

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

++

+=⇒ tsArcTan

ttM

tMtttf mB

pmB

p 236

326

)32(4)( 2

22

22 ρσσσρ

σ


21

Bounce in f(T) cosmology

n  Start with a bounching scale factor:

n  Examine the full perturbations:

with known in terms of and matter

n  Primordial power spectrum: n  Tensor-to-scalar ratio:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ++−−±=⇒ 2

43

34

32

34)(

σσσ

σ TT

TTt

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

++

+=⇒ tsArcTan

ttM

tMtttf mB

pmB

p 236

326

)32(4)( 2

22

22 ρσσσρ

σ

02

222 =+++ kskkk akc φφµφαφ !!! 22 ,, scµα TTT fffHH ,,,, !

22288 pMP

πσ

ζ =

01123 2

2

=+

−⎟⎟⎠

⎞⎜⎜⎝

⎛ ∇−+ h

ffHHh

ahHh

T

TTijijij

!!!!!

3108.2 −×≈r

⇒⇒

[Cai, Chen, Dent, Dutta, Saridakis CQG 28]

3/12

231)( ⎟

⎠

⎞⎜⎝

⎛ += tata B σ


22

Non-minimally coupled scalar-torsion theory

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

[Geng, Lee, Saridakis, Wu PLB704]

2^ϕξRR +)(RfR +

( ) ⎥⎦

⎤⎢⎣

⎡ +−+∂∂+= ∫ mLVTTexdS )(21

22

24 ϕϕξϕϕ

κµ

µ


23


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

n  Friedmann equations in FRW universe: with effective Dark Energy sector:

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

2^ϕξRR +)(RfR +

( ) ⎥⎦

⎤⎢⎣

⎡ +−+∂∂+= ∫ mLVTTexdS )(21

22

24 ϕϕξϕϕ

κµ

µ

( )DEmH ρρκ+=

3

22

( )DEDEmm ppH +++−= ρρκ2

2!

222

3)(2

ϕξϕϕ

ρ HVDE −+=!

( ) 222

234)(2

ϕξϕϕξϕϕ HHHVpDE !!!

+++−=

[Geng, Lee, Saridakis,Wu PLB 704]

[Geng, Lee, Saridakis, Wu PLB704]


24


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

n  Teleparallel Dark Energy:

[Geng, Lee, Saridakis, Wu PLB 704]


25

Observational constraints on Teleparallel Dark Energy

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

n  Include matter and standard radiation: n  We fit for various

ξ,0DEDE0M0 w,Ω,Ω )(ϕV

azaa rrMM /11,/,/ 40

30 =+== ρρρρ


26

Observational constraints on Teleparallel Dark Energy

Exponential potential

Quartic potential

[Geng, Lee, Saridkis JCAP 1201] E.N.Saridakis – VIA web lecture, Feb 2014

27

Phase-space analysis of Teleparallel Dark Energy

n  Transform cosmological system to its autonomous form:

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

),sgn(13

2222 ξ

ρ zyx=Hm

m +−−≡Ω⇒

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

,f(X)=X'⇒QUU =⇒ 'UX=X C +

0' =|XCX=X

Q

κϕξϕκϕκ

=== zHVy

Hx ,

3)(,

6!

)sgn(3

2222 ξ

ρ zyx=HDE

DE −+≡Ω


28

Phase-space analysis of Teleparallel Dark Energy

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

[Xu, Saridakis, Leon, JCAP 1207]

ξλ <2 1,1,1 −=−==Ω qwDEDE

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

1−≥DEw


29

Exact charged black hole solutions

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

n  Add E/M sector: with

n  Extract field equations:

[ ]Λ−+= ∫ 2)(21 TfTexdS D

κ

FFLF∗∧−=

21 µ

µdxAAdAF ≡= ,

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

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


30


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

n  Add E/M sector: with

n  Extract field equations:

n  Look for spherically symmetric solutions:

n  Radial Electric field: known

[ ]Λ−+= ∫ 2)(21 TfTexdS D

κ

FFLF∗∧−=

21 µ

µdxAAdAF ≡= ,

SHRSHL .... =

∑−

−−=⇒2

1

2222

222

)(1)(

D

idxrdrrG

dtrFds

!,,,,)(

1,)( 23

1210 rdxerdxedr

rGedtrFe ====

2−= Dr rQE 22 )(,)( rGrF⇒

[Gonzalez, Saridakis, Vasquez, JHEP 1207], [Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 1302] E.N.Saridakis – VIA web lecture, Feb 2014

31


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

n  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 µνρσµνρσ

⇒


32


[Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 1302] E.N.Saridakis – VIA web lecture, Feb 2014

33


n  More singularities in the curvature analysis than in torsion analysis! (some are naked)

n  The differences disappear in the f(T)=0 case, or in the uncharged case.

n  Should we go to quartic torsion invariants?

n  f(T) brings novel features.

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


34

Solar System constraints on f(T) gravity

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

)()( 32 TOTTf +=α

⎥⎦

⎤⎢⎣

⎡ Λ−−Λ−+

Λ−−=⇒

rcGM

rr

rcGMrF 22

22

2 4663

21)( α

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ −Λ−Λ−−Λ

+Λ

−−=⇒ 2222

22

22 882224

38

321)(

rrcGMr

rr

rcGMrG α


35

Solar System constraints on f(T) gravity

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

n  Use data from Solar System orbital motions:

T<<1 so consistent

n  f(T) divergence from TEGR is very small n  This was already known from cosmological observation constraints up to n  With Solar System constraints, much more stringent bound.

)()( 32 TOTTf +=α

⎥⎦

⎤⎢⎣

⎡ Λ−−Λ−+

Λ−−=⇒

rcGM

rr

rcGMrF 22

22

2 4663

21)( α

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ −Λ−Λ−−Λ

+Λ

−−=⇒ 2222

22

22 882224

38

321)(

rrcGMr

rr

rcGMrG α

10)( 102.6 −×≤Δ TfU

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

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


n  Perturbations: , clustering growth rate:

n  γ(z): Growth index.

36

Growth-index constraints on f(T) gravity

)(lnln

aad

dm

m γδΩ=

)('11Tf

Geff +=

mmeffmm GH δρπδδ 42 =+ !!!


37

Growth-index constraints on f(T) gravity

)(lnln

aad

dm

m γδΩ=

)('11Tf

Geff +=

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

mmeffmm GH δρπδδ 42 =+ !!!n  Perturbations: , clustering growth rate:

n  γ(z): Growth index.

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


38

Open issues of f(T) gravity

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

n  For nonlinear f(T), Lorentz invariance is not satisfied n  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]


39

Open issues of f(T) gravity

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

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

not equivalent (extra degrees of freedom)

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

n  Thermodynamics also raises issues

n  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)


40

Gravity modification in terms of torsion?

n  So can we modify gravity starting from its torsion formulation?

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

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

n  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?

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

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


41

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

of quantization. n  ii) Can we modify gravity based in its torsion formulation? n  iii) Simplest choice: f(T) gravity, i.e extension of TEGR n  iv) f(T) cosmology: Interesting phenomenology. Signatures in growth

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

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

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

modifications too.

2ξTT ϕ+

1010−


42

Outlook n  Many subjects are open. Amongst them:

n  i) Examine thermodynamics thoroughly.

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

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

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

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

n  vi) Convince people to work on the subject!


43

THANK YOU!


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