F(T) gravity from higher dimensional
theories and its cosmology
Main reference: arXiv:1304.6191 [gr-qc]. To appear in Phys. Lett. B.

KMI-IEEC Joint International Workshop-- Inflation, Dark Energy, and Modified Gravity
in the PLANCK Era -3rd August, 2013
Kobayashi-Maskawa Institute for the Origin ofParticles and the Universe (KMI), ES635
Science Symposia
Presenter :Collaborators :Kazuharu Bamba (KMI, Nagoya University)
Shin'ichi Nojiri (KMI and Dep. of Phys., Nagoya University),
Sergei D. Odintsov (ICREA and CSIC-IEEC, Spain)

2Contents
I. IntroductionCurrent cosmic accelerated expansion
II. F(T) gravity
III. From the Kaluza-Klein (KK) theory
IV. From the Randall-Sundrum (RS) theory
V. Summary

3I. Introduction
Current cosmic acceleratedexpansion

4・ Recent observations of Type Ia Supernova(SNe Ia) has supported that the current
expansion of the universe is accelerating.
[Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J.
517, 565 (1999)][Riess et al. [Supernova Search Team Collaboration], Astron. J. 116, 1009 (1998)]

2011 Nobel Prize in Physics

5

PLANCK 2013 results of SNLS
Magnitude residuals of the Λ CDM modelthat best fits the SNLS combined sample

Λ CDMmodelz : Redshift
From [Ade et al. [Planck Collaboration], arXiv:1303.5076 [astro-ph.CO]].

6・ Suppose that the universe is strictly homogeneousand isotropic.

There are two approaches to explain the current
accelerated expansion of the universe.
Reviews: E.g.,[Copeland, Sami and Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006)][Nojiri and Odintsov, Phys. Rept. 505, 59 (2011); Int. J. Geom. Meth. Mod.
Phys. 4, 115 (2007)][Capozziello and Faraoni, Beyond Einstein Gravity (Springer, 2010)][Clifton, Ferreira, Padilla and Skordis, Phys. Rept. 513, 1 (2012)][KB, Capozziello, Nojiri and Odintsov, Astrophys. Space Sci. 342, 155 (2012)]

5Gravitational field equation2Gö÷ = ô T ö÷Gravity

MatterGö÷ : Einstein tensor
Tö÷ : Energy-momentum tensor: Planck mass

(1) General relativistic approach(2) Extension of gravitational theory

Dark Energy

6(1) Candidates for dark energy
Cosmological constant, Scalar field, Fluid(2) Extension of gravitational theory・ F(R) gravity

F (R ) : Arbitrary function of the Ricci scalar R・ DGP braneworld scenario
・ Galileon gravity

・ Massive gravity

・ Extended teleparallel gravity (F(T) gravity)
F(T) : Arbitrary function of the torsion scalar T

Condition for accelerated expansion

7Flat Friedmann-Lema tre-Robertson-Walker (FLRW) space-timea(> 0) : Scale factor
Equation of a (t ) for a single perfect fluidú : Energy densityô2ä=1+3àw) ú(a6P : Pressureä > 0 : Accelerated expansion
P1w ñ ú <à3Cf. w = à 1: Cosmological constantw : Equation of state (EoS) parameter

PLANCK data for the current w
Marginalized posterior distribution

10

From [Ade etal. [PlanckCollaboration],
arXiv:1303.507
6 [astroph.CO]].

w = constant
WP: WMAP
BAO: Baryon
AcousticOscillation

10* Hubble constant ( H0 ) measurement

PLANCK data for the time-dependent w

12

2D Marginalized posterior distribution
From [Ade et al.
[Planck Collaboration],
arXiv:1303.5076[astro-ph.CO]].

(68% CL)(95% CL)

13Motivation and Subject
It is meaningful to investigate theoreticalfeatures of modified gravity theories.
We concentrate on cosmological aspectsof F(T) gravity.
・ We explore the four-dimensional effective
F(T) gravity originating from higherdimensional space-time theories, inparticular the Kaluza-Klein (KK) andRandall-Sundrum (RS) theories.

II. F(T) gravity

15

Teleparallelism
ñAB : Minkowski metric・eA(xö) : Orthonormal tetrad componentsú(W)・ T úö÷ ñ Γú(W)Γà=ö÷÷ö: Torsion tensor

ú(W)Γö÷ñ eúA∂ öeA÷ : Weitzenböck connection

coordinate indices on the manifold and also run over 0, 1, 2, 3,* ö and ÷ areö
and eA(x ) forms the tangent vector of the manifold.

A runs over 0, 1, 2, 3 for the tangent space at each
* An index
öpoint x of the manifold.

16： Torsion scalar: Contorsion tensor[Hehl, Von Der Heyde, Kerlick and Nester, Rev. Mod. Phys. 48, 393 (1976)][Hayashi and Shirafuji, Phys. Rev. D 19, 3529 (1979)[Addendum-ibid. D 24, 3312 (1981)]]

17

Extended teleparallel gravity
Action： F(T) gravity: Matter Lagrangian:Energy-momentum
tensor of matter

Gravitational field equation

F0F0F 00F

* A prime denotes a derivative with respect to T .
[Bengochea and Ferraro, Phys. Rev. D 79, 124019 (2009)]

18･ Gravitational field equation in F(T) gravity is
2nd order, while it is 4th order in F(R) gravity.

･ For the flat FLRW space-time with the metric:

açHña: Hubble parameter

19

Gravitational field equations

úDE : Dark energy density
P DE : Pressure of dark energyú M, P M: Energy density andpressure of dark energy(à T à F + 2 TF 0)[4(1 à F 0 à 2TF 00)Hç à T à F + 2TF 0]
Continuity equation

Example of F(T) gravity model

20

F(T) = T +u(> 0) : Positive constant,
ô2・ The model contains only one parameter u
if one has the value of Ω(0)m .
[KB, Geng, Lee and Luo, JCAP 1101, 021 (2011)]

Cosmological evolutions of wDE

21

From [KB, Geng, Lee andLuo, JCAP 1101, 021 (2011)].

u = 0.5(dash-dottedline)

u = 0.8(dashed line)u = 1 (solid line)

wDE = à 1
Crossing of thephantom divideCosmological evolutions of ΩDE , Ωm and Ωr
ΩDE22

u=1Ωr
Dark energy
dominatedstageΩm
From [KB, Geng, Lee andLuo, JCAP 1101, 021 (2011)].

III. From Kaluza-Klein (KK)theory24

Action in five-dimensional space-time*run over.

* “5” denotes the component of thefifth coordinate.

・ The superscript or subscript of (5) or 5 mean the quantities
in the five-dimensional space-time.
[Capozziello, Gonzalez, Saridakis and Vasquez, JHEP 1302, 039 (2013)]

25

Original KK compactification scenario
・ One of the dimensions of space is compactified toa small circle and the four-dimensional space-time
is extended infinitely.
・ The radius of the fifth dimension is taken to beof order of the Planck length in order for the KK
effects not to be seen.
The size of the circle is so small that phenomena
in sufficiently low energies cannot be detected.
[Appelquist, Chodos and Freund, Modern Kaluza-Klein Theories(Addison-Wesley, Reading, 1987)][Fujii and Maeda, The Scalar-Tensor Theory of Gravitation(Cambridge University Press, Cambridge, United Kingdom, 2003)]

26

Metric in five-dimensional space-time,: Dimensionlesshomogeneous scalar fieldϕã : Fiducial value of ϕ
R : Rradius of the compactified spaceò : Dimensionless coordinates such as an angle: Determinant of the metric corresponding to
the pure geometrical part represented by ò: Compactified space volume

27

Effective action in the four-dimensional space-time

・ Our KK reduced action is compatible with the results inthe following reference:[Fiorini, Gonzalez and Vasquez, arXiv:1304.1912 [gr-qc]].

28Case of teleparallelism with a positive
cosmological constant・・ We define û as,: Cosmological constant,

Canonical kinetic term

29

Cosmology in the flat FLRW space-time
Gravitational field equations

Equation of motion of û

Cf. [Geng, Lee, Saridakis and Wu, Phys. Lett. B 704, 384 (2011)]

30

Solutionb1, b2(> 0), t1
・ In the limit: Constants, we can find approximate expressions,An exponential inflation can be realized.

IV. From the Randall-Sundrum(RS) theory

32

The RS type-I and II models
・ In the RS type-I model, there are a positive tension
brane at y = 0 and a negative one at y = s , where yis the fifth direction.

dsà2,: Warp factorΛ 5(< 0) : Negative cosmologicalconstant in the bulks→∞

・ In the RS type-II model, there is only one brane withthe positive tension floating in the AdS bulk space.
[Randall and Sundrum, Phys. Rev. Lett. 83, 3370 (1999); 4690 (1999)]
Cf. [Garriga and Tanaka, Phys. Rev. Lett. 84, 2778 (2000)]

33Settings in the RS type-II model・ We start with the equation in the five-dimensional space-timewith the brane whose tension is a positive constant.

・ We consider that the vacuum solution in the five-dimensional
space-time is the AdS one, and that the brane configuration isconsistent with the equation in the five-dimensional space-time.
This implies that the brane configuration with a positive
constant tension connecting two vacuum solutions in thefive-dimensional space-time, namely, the condition of theconfiguration is nothing but the equation for the brane.

34

Procedures in the RS type-II model
Pioneering work:[Shiromizu, Maeda and Sasaki, Phys. Rev. D 62, 024012 (2000)]

Application to teleparallelism:[Nozari, Behboodi and Akhshabi, Phys. Lett. B 723, 201 (2013)](i) The corresponding Gauss-Codazzi equations inteleparallelism, namely, the induced equations on thebrane, is examined by using the projection vierbeinof the five-dimensional space-time quantities into thefour-dimensional space-time brane.

35(ii) The Israel's junction conditions to connect the left-side andright-side bulk spaces sandwiching the brane are investigated.

・ The first junction condition is that the vierbeins inducedon the brane from the left side and right side of the brane
should be the same with each other.

・ Moreover, the second junction condition is that thedifference of the tensor
between the left side andright side of the brane comes from the energy-momentum
tensor of matter, which is confined in the brane.
(iii) Provided that there exists Z2 symmetry, i.e., y ↔ à y ,in the five-dimensional space-time, the quantities on theleft and right sides of the brane are explored.

36・ The difference between the scalar curvature and the torsion
scalar is a total derivative of the torsion tensor.
This may affect the boundary.

・ It has been shown that in comparison with the gravitational
field equations in general relativity, the induced gravitational
field equations on the brane have new terms, which comesfrom the additional degrees of freedom in teleparallelism.

・ These extra terms correspond to the projection on the braneof the vector portion of the torsion tensor in the bulk.

37

Cosmology in the flat FLRW space-timeFriedmann equation on the brane

includes the contributions
from teleparallelism, which do
not exist in general relativity.
: Eeffective cosmological constant in the brane: the tension of the brane38

Case (1)* At the dark energy completely
dominated stage, we canconsider.

,An approximate de Sitter solution
on the brane can be realized.
Cf. Other solution
For,, and,[Astashenok, Elizalde, de Haro, Odintsov and Yurov, arXiv:1301.6344 [gr-qc]]

39

Case (2): Mass scaleë: Constant,Similar approximate de Sitter solution
on the brane can be obtained.

V. Summary

41･ Four-dimensional effective F(T) gravity coming fromthe five-dimensional KK and RS space-time theories
have been studied.
･ With the KK reduction, the four-dimensional effective
theory of F(T) gravity coupling to a scalar field has
been built.
･ For the RS type-II model, the contribution of F(T)gravity appears on the four-dimensional FLRW brane.
･ Inflation and the dark energy dominated stage canbe realized in the KK and RS theories, respectively,
due to the effect of only the torsion in teleparallelism
without that of the curvature.

Backup Slides

General relativistic approachNo. 6(i) Cosmological constant(ii) Scalar field :･ x-matter, Quintessence

Canonical field[Chiba, Sugiyama and Nakamura, Mon. Not. Roy. Astron. Soc. 289, L5 (1997)][Caldwell, Dave and Steinhardt, Phys. Rev. Lett. 80, 1582 (1998)]
Cf. Pioneering work: [Fujii, Phys. Rev. D 26, 2580 (1982)]･ Phantom

Wrong sign kinetic term
[Caldwell, Phys. Lett. B 545, 23 (2002)]･ K-essence

Non canonical kinetic term
[Chiba, Okabe and Yamaguchi, Phys. Rev. D 62, 023511 (2000)][Armendariz-Picon, Mukhanov and Steinhardt, Phys. Rev. Lett. 85, 4438 (2000)]･ Tachyon

String theories * The mass squared is negative.
[Padmanabhan, Phys. Rev. D 66, 021301 (2002)]

5Distanceestimator

SNLS dataFlat Λcosmology

Pure mattercosmology

z: Redshift

From [Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)].

Baryon acoustic oscillation (BAO)

No. 14

Specialpattern in thelarge-scalecorrelation
function ofSloan Digital
Sky Survey(SDSS)
luminous red
galaxies

Pure cold dark matter (CDM)
model: “No peak”

From [Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005)].
Cf. [Yamamoto, astro-ph/0110596; Astrophys. J. 595, 577 (2003)][Matsubara and Szalay, Phys. Rev. Lett. 90, 021302 (2003)]

9-year WMAP data of currentw

No. 15[Hinshaw et al., arXiv:1212.5226 [astro-ph.CO]]For constant w :(68% CL)(From.)* Hubble constant ( H0) measurement

Time-dependentw

No. 16(68% CL)(95% CL)
From [Hinshaw et al.,arXiv:1212.5226 [astro-ph.CO]].

w0 : Current value of w
(From.)For the flat universe:,

No. 7(iii) Fluid :･ (Generalized) Chaplygin gas
Equation of state (EoS):P = à A/úuA > 0, u : Constants

ú : Energy density

P : Pressure[Kamenshchik, Moschella and Pasquier, Phys. Lett. B 511, 265 (2001)][Bento, Bertolami and Sen, Phys. Rev. D 66, 043507 (2002)](u = 1)Extension of gravitational theory･ F(R) gravity

No. 8

F (R ) : Arbitrary function ofthe Ricci scalar R

Cf. Application to inflation: [Starobinsky, Phys. Lett. B 91, 99 (1980)][Capozziello, Cardone, Carloni and Troisi, Int. J. Mod. Phys. D 12, 1969 (2003)][Carroll, Duvvuri, Trodden and Turner, Phys. Rev. D 70, 043528 (2004)][Nojiri and Odintsov, Phys. Rev. D 68, 123512 (2003)]･ Scalar-tensor theoriesfi(þ) (i = 1, 2)

f1(þ)R: Arbitrary function of a scalar field þ[Boisseau, Esposito-Farese, Polarski and Starobinsky, Phys. Rev. Lett. 85, 2236 (2000)][Gannouji, Polarski, Ranquet and Starobinsky, JCAP 0609, 016 (2006)]

No. 9･ Ghost condensates scenario[Arkani-Hamed, Cheng, Luty and Mukohyama, JHEP 0405, 074 (2004)]･ Higher-order curvature term
Gauss-Bonnet invariant with a coupling toa scalar field: f 2(þ)GG ñ R2 à: Ricci curvature tensor: Gauss-Bonnet invariant: Riemann tensor[Nojiri, Odintsov and Sasaki, Phys. Rev. D 71, 123509 (2005)]･ f(G) gravity

R+ f(G)2ô 2

ô2 ñ 8ùG

G : Gravitational constant[Nojiri and Odintsov, Phys. Lett. B 631, 1 (2005)]

No. 10･ DGP braneworld scenario[Dvali, Gabadadze and Porrati, Phys. Lett B 485, 208 (2000)][Deffayet, Dvali and Gabadadze, Phys. Rev. D 65, 044023 (2002)]･ Non-local gravity

12ô 2f (R) : Quantum effects

à1[Deser and Woodard, Phys. Rev. Lett. 99, 111301 (2007)]: Covariant d'Alembertian･ F(T) gravity[Nojiri and Odintsov, Phys. Lett. B 659, 821 (2008)]

Extended teleparallel Lagrangian
described by the torsion scalar T .
[Bengochea and Ferraro, Phys. Rev. D 79, 124019 (2009)][Linder, Phys. Rev. D 81, 127301 (2010) [Erratum-ibid. D 82, 109902 (2010)]]* “Teleparallelism” : One could use the Weitzenböck connection, which has nocurvature but torsion, rather than the curvature defined bythe Levi-Civita connection.

[Hayashi and Shirafuji, Phys. Rev. D 19, 3524 (1979) [Addendum-ibid. D 24, 3312 (1982)]]

No. 11･ Galileon gravity

þ (∂ ö þ∂ ö þ )Longitudinal graviton (a branebending mode þ )[Nicolis, Rattazzi and Trincherini, Phys. Rev. D 79, 064036 (2009)]･ Massive gravity

Graviton with a non-zero mass[de Rham and Gabadadze, Phys. Rev. D 82, 044020 (2010)][de Rham and Gabadadze and Tolley, Phys. Rev. Lett. 106, 231101 (2011)]
Review: [Hinterbichler, Rev. Mod. Phys. 84, 671 (2012)]