f(t) gravity and cosmology
DESCRIPTION
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 – Taiwan, Dec 2012. Goal. - PowerPoint PPT PresentationTRANSCRIPT
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 – Taiwan, Dec 2012
2
Goal
We investigate cosmological scenarios in a universe governed by f(T) gravity
Note: A consistent or interesting
cosmology is not a proof for the consistency of the underlying gravitational theory
E.N.Saridakis – Taiwan, Dec 2012
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) Black-hole solutions
7) Solar system constraints
8) Conclusions-Prospects
E.N.Saridakis – Taiwan, Dec 2012
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]
E.N.Saridakis – Taiwan, Dec 2012
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
E.N.Saridakis – Taiwan, Dec 2012
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)
E.N.Saridakis – Taiwan, Dec 2012
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. E.N.Saridakis – Taiwan, Dec
2012
8
Introduction
1998: Universe acceleration Thousands of work in Modified Gravity
(f(R), Gauss-Bonnet, Lovelock, nonminimal scalar coupling,
nonminimal derivative coupling, Galileons, Hordenski 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]
E.N.Saridakis – Taiwan, Dec 2012
9
Introduction
1998: Universe acceleration Thousands of work in Modified Gravity
(f(R), Gauss-Bonnet, Lovelock, nonminimal scalar coupling,
nonminimal derivative coupling, Galileons, Hordenski 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]
E.N.Saridakis – Taiwan, Dec 2012
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]
E.N.Saridakis – Taiwan, Dec 2012
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
E.N.Saridakis – Taiwan, Dec 2012
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
[Bengochea, Ferraro PRD 79], [Linder PRD 82] mSTfTexdG
S )(16
1 4
E.N.Saridakis – Taiwan, Dec 2012
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
[Bengochea, Ferraro PRD 79], [Linder PRD 82]
jiij
A dxdxtadtdsaaadiage )(),,,1( 222
22 26
)(
3
8Hf
TfGH Tm
TTT
mm
fHf
pGH
2121
)(4
Find easily26HT
E.N.Saridakis – Taiwan, Dec 2012
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]
E.N.Saridakis – Taiwan, Dec 2012
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]
E.N.Saridakis – Taiwan, Dec 2012
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]
E.N.Saridakis – Taiwan, Dec 2012
17
f(T) Cosmology: Perturbations
Application: Distinguish f(T) from quintessence 2) Examine evolution of matter overdensity
[Dent, Dutta, Saridakis JCAP 1101]
m
m
E.N.Saridakis – Taiwan, Dec 2012
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
E.N.Saridakis – Taiwan, Dec 2012
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]
E.N.Saridakis – Taiwan, Dec 2012
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
E.N.Saridakis – Taiwan, Dec 2012
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
E.N.Saridakis – Taiwan, Dec 2012
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 PLB704]
2^RR )(RfR
mLVTT
exdS )(2
1
22
24
E.N.Saridakis – Taiwan, Dec 2012
23
Non-minimally coupled scalar-torsion theory
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]
[Geng, Lee, Saridakis, Wu PLB704]
E.N.Saridakis – Taiwan, Dec 2012
24
Non-minimally coupled scalar-torsion theory
Main advantage: Dark Energy may lie in the phantom regime or/and experience the phantom-divide crossing
Teleparallel Dark Energy:
[Geng, Lee, Saridakis, Wu PLB 704]
E.N.Saridakis – Taiwan, Dec 2012
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
azaa rrMM /11,/,/ 40
30
,0DEDE0M0 w,Ω,Ω )(V
E.N.Saridakis – Taiwan, Dec 2012
26
Observational constraints on Teleparallel Dark Energy
Exponential potential
Quartic potential
[Geng, Lee, Saridkis JCAP 1201]E.N.Saridakis – Taiwan, Dec 2012
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 'UXX C
0|' CXXX
Q
z
H
Vy
Hx ,
3
)(,
6
)sgn(3
2222
zyx=
HDE
DE
E.N.Saridakis – Taiwan, Dec 2012
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
E.N.Saridakis – Taiwan, Dec 2012
29
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, 1210.1098, JHEP]
E.N.Saridakis – Taiwan, Dec 2012
30
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:
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, 1210.1098, JHEP]
E.N.Saridakis – Taiwan, Dec 2012
31
Exact charged black hole solutions
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, 1210.1098]
)(rRR
E.N.Saridakis – Taiwan, Dec 2012
32
Exact charged black hole solutions
[Capozzielo, Gonzalez, Saridakis, Vasquez 1210.1098]
E.N.Saridakis – Taiwan, Dec 2012
33
Exact charged black hole solutions
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.
Should we go to quartic torsion invariants?
f(T) brings novel features.
E/M in torsion formulation was known to be nontrivial (E/M in Einstein-Cartan and Poinaré theories)
E.N.Saridakis – Taiwan, Dec 2012
34
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
E.N.Saridakis – Taiwan, Dec 2012
35
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, 1203.5781] (to appear in MNRAS)
)1010( 21 O [Wu, Yu, PLB 693], [Bengochea PLB 695]
E.N.Saridakis – Taiwan, Dec 2012
36
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]
E.N.Saridakis – Taiwan, Dec 2012
37
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 and Solar System observations constraint f(T) very close to linear-in-T form
[Li, Sotiriou, Barrow PRD 83a],
[Li,Sotiriou,Barrow PRD 83c], [Li,Miao,Miao JHEP 1107]
[Capozzielo, Gonzalez, Saridakis, Vasquez 1210.1098]
[Bamba,Geng JCAP 1111], [Miao,Li,Miao JCAP 1111] [Bamba,Myrzakulov,Nojiri, Odintsov PRD 85]
[Geng,Lee,Saridakis,Wu PLB 704]
E.N.Saridakis – Taiwan, Dec 2012
38
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 Gauss Bonnet, Lovelock, Hordenski modifications)?
Can we modify gauge theories of gravity themselves? E.g. can we modify Poincaré gauge theory?
E.N.Saridakis – Taiwan, Dec 2012
39
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. vii) Exact black hole solutions. Curvature vs torsion analysis. viii) Solar system constraints: f(T) divergence from T less than ix) Many open issues. Need to search fro other torsion-based
modifications too.
2ξTT
1010
E.N.Saridakis – Taiwan, Dec 2012
40
Outlook Many subjects are open. Amongst them:
i) Examine thermodynamics thoroughly.
ii) Extend f(T) gravity in the braneworld.
iii) Understand the extra degrees of freedom and the extension to non-diagonal vierbeins.
iv) Try to modify TEGR using higher-order torsion invariants.
v) Try to modify Poincaré gauge theory (extremely hard!)
vi) Convince people to work on the subject! E.N.Saridakis – Taiwan, Dec 2012
41
THANK YOU!
E.N.Saridakis – Taiwan, Dec 2012