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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
E.N.Saridakis – GWiMG, NTUA, March 2018
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
E.N.Saridakis – GWiMG, NTUA, March 2018
Why Modified Gravity?
Knowledge of Physics: Standard Model
E.N.Saridakis – SEMFE, NTUA, March 2016
E.N.Saridakis – GWiMG, NTUA, March 2018
Why Modified Gravity?
Knowledge of Physics: Standard Model + General Relativity
E.N.Saridakis – SEMFE, NTUA, March 2016
E.N.Saridakis – GWiMG, NTUA, March 2018
Why Modified Gravity?Universe History:
E.N.Saridakis – GWiMG, NTUA, March 2018
Why Modified Gravity?So can our knowledge of Physics describes all these?
E.N.Saridakis – GWiMG, NTUA, March 2018
Why Modified Gravity?So can our knowledge of Physics describes all these?
E.N.Saridakis – GWiMG, NTUA, March 2018
Why Modified Gravity?
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
E.N.Saridakis – GWiMG, NTUA, March 2018
Modified Gravity
10
Non-minimal gravity-matter coupling
(Gen. Proca)
E.N.Saridakis – GWiMG, NTUA, March 2018
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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
E.N.Saridakis – GWiMG, NTUA, March 2018
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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.
E.N.Saridakis – GWiMG, NTUA, March 2018
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
nonminimal derivative coupling, Galileons, Hordenski, massive etc)
Almost all in the curvature-based formulation of gravity
[Copeland, Sami, Tsujikawa Int.J.Mod.Phys.D15], [Capozziello, De Laurentis, Phys. Rept. 509]
E.N.Saridakis – GWiMG, NTUA, March 2018
Introduction
1998: Universe accelerationThousands of work in Modified Gravity
(f(R), Gauss-Bonnet, Lovelock, nonminimal scalar coupling,
nonminimal derivative coupling, Galileons, Hordenski, massive etc)
16
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], [Capozziello, De Laurentis, Phys. Rept. 509]
E.N.Saridakis – GWiMG, NTUA, March 2018
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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
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
E.N.Saridakis – GWiMG, NTUA, March 2018
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
Equations of motion:
}{1 4)]([4
1)(1
AATTAATA GeTfTefTSeSTefSeee
E.N.Saridakis – GWiMG, NTUA, March 2018
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:
mSTfTexdG
S )(16
1 4
[Ferraro, Fiorini PRD 78], [Bengochea, Ferraro PRD 79]
[Linder PRD 82]
20
Equations of motion:
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
E.N.Saridakis – GWiMG, NTUA, March 2018
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
E.N.Saridakis – GWiMG, NTUA, March 2018
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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/
23
[Dent, Dutta, Saridakis JCAP 1101]
E.N.Saridakis – GWiMG, NTUA, March 2018
f(T) Cosmology: Perturbations
Application: Distinguish f(T) from quintessence 2) Examine evolution of matter overdensity
[Dent, Dutta, Saridakis JCAP 1101]
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
E.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
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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
E.N.Saridakis – GWiMG, NTUA, March 2018
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
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:
2^RR )(RfR
mLVTT
exdS )(2
1
22
24
[Geng, Lee, Saridakis, Wu PLB 704]
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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:
31
[Geng, Lee, Saridakis, Wu PLB 704]
E.N.Saridakis – GWiMG, NTUA, March 2018
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
E.N.Saridakis – GWiMG, NTUA, March 2018
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
E.N.Saridakis – GWiMG, NTUA, March 2018
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
E.N.Saridakis – GWiMG, NTUA, March 2018
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
E.N.Saridakis – GWiMG, NTUA, March 2018
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
37
Friedmann equations in FRW universe:
with effective Dark Energy sector:
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
[Harko, Lobo, Otalora, Saridakis, PRD 89]
E.N.Saridakis – GWiMG, NTUA, March 2018
Non-minimally matter-torsion coupled theory
Interesting phenomenology
38
,)( 211 TTf
[Harko, Lobo, Otalora, Saridakis, PRD 89]
212 )( TTf ,)(1 Tf 2
112 )( TTTf
E.N.Saridakis – GWiMG, NTUA, March 2018
Non-minimally matter-torsion coupled theory
In curvature-based gravity, one can use coupling Let’s do the same in torsion-based gravity:
),( Rf
mLTfTexdS ),(2
1 42 [Harko, Lobo, Otalora, Saridakis, JCAP 1412]
39
2
E.N.Saridakis – GWiMG, NTUA, March 2018
Non-minimally matter-torsion coupled theory
In curvature-based gravity, one can use coupling Let’s do the same in torsion-based gravity:
),( Rf
mLTfTexdS ),(2
1 42
40
Friedmann equations in FRW universe ( ):
with effective Dark Energy sector:
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
E.N.Saridakis – GWiMG, NTUA, March 2018
Non-minimally matter-torsion coupled theory
Interesting phenomenology
41[Harko, Lobo, Otalora, Saridakis, JCAP 1412]
2),( TTf nTTf ),(
E.N.Saridakis – GWiMG, NTUA, March 2018
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 .
E.N.Saridakis – GWiMG, NTUA, March 2018
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 .
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
Teleparallel Equivalent of Gauss-Bonnet and f(T,T_G) gravity
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),(
E.N.Saridakis – GWiMG, NTUA, March 2018
Torsional Gravity with higher derivatives
),(,)(,2
1 242 m
Am eSTTTFexdS
45[Otalora, Saridakis, PRD 94]
E.N.Saridakis – GWiMG, NTUA, March 2018
Torsional Modified Gravity
E.N.Saridakis – GWiMG, NTUA, March 2018
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
E.N.Saridakis – GWiMG, NTUA, March 2018
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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
49E.N.Saridakis – GWiMG, NTUA, March 2018
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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)]
54E.N.Saridakis – GWiMG, NTUA, March 2018
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
<- linear evolution of perturbations
<- 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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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
57
the extrinsic torsion is defined as
with the orthogonal to t=cont. surfaces unitary vector =
Using the projection operator we can express
[Cai, Li, Saridakis, Xue, 1801.05827, Li, Cai, Cai, Saridakis, in preparation]
E.N.Saridakis – GWiMG, NTUA, March 2018
The (EFT) approach to torsional gravity
We perturb the previous tensors, and we finally obtain:
58
where the time-dependent functions are determined by the background solution.
[Cai, Li, Saridakis, Xue, 1801.05827, Li, Cai, Cai, Saridakis, in preparation]
E.N.Saridakis – GWiMG, NTUA, March 2018
The (EFT) approach to torsional gravity
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 ,
[Cai, Li, Saridakis, Xue, 1801.05827, Li, Cai, Cai, Saridakis, in preparation]
E.N.Saridakis – GWiMG, NTUA, March 2018
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
The (EFT) approach to f(T) gravity: Background
For the case of f(T) gravity, at the background level, we have:
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where by comparison:
Performing variation we obtain the background equations of motion (Friedmann Eqs):
[Li, Cai, Cai, Saridakis, in preparation]
E.N.Saridakis – GWiMG, NTUA, March 2018
The (EFT) approach to f(T) gravity: Background
These can be written as:
with
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with
and thus:
The same equations with standard approach![Li, Cai, Cai, Saridakis, in preparation]
E.N.Saridakis – GWiMG, NTUA, March 2018
For tensor perturbations: i.e.
The (EFT) approach to f(T) gravity: Tensor Perturbations
63E.N.Saridakis – GWiMG, NTUA, March 2018
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]
E.N.Saridakis – GWiMG, NTUA, March 2018
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.
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[Cai, Li, Saridakis, Xue, 1801.05827]
E.N.Saridakis – GWiMG, NTUA, March 2018
Choosing the ansatz:
We obtain the dispersion relation:
The (EFT) approach to f(T) gravity: Gravitational Waves
where we can express:
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[Cai, Li, Saridakis, Xue, 1801.05827]
E.N.Saridakis – GWiMG, NTUA, March 2018
Choosing the ansatz:
We obtain the dispersion relation:
The (EFT) approach to f(T) gravity: Gravitational Waves
where we can express:
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.
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[Cai, Li, Saridakis, Xue, 1801.05827]
E.N.Saridakis – GWiMG, NTUA, March 2018
Choosing the ansatz:
We obtain the dispersion relation:
The (EFT) approach to f(T) gravity: Gravitational Waves
where we can express:
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.
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[Cai, Li, Saridakis, Xue, 1801.05827]
E.N.Saridakis – GWiMG, NTUA, March 2018
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
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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.
E.N.Saridakis – GWiMG, NTUA, March 2018
Outlook Many subjects are open. Amongst them:
i) Examine higher-order perturbations to look for further polarizations.[Farugia, Gakis, Jackson, Saridakis, in preparation]
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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]
E.N.Saridakis – GWiMG, NTUA, March 2018
“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...
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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.
E.N.Saridakis – GWiMG, NTUA, March 2018
“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
Vierbeins : four linearly independent fields in the tangent space
Connection:
Curvature tensor:
)()()( xexexg BAAB
Ae
ABCCB
AC
CB
AC
AB
AB
ABR ,,
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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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