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Introduction to Teleparallel and f(T) Gravity and Cosmology

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. Goal.

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Introduction to Teleparallel and f(T) Gravity and Cosmology

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  1. 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 2 E.N.Saridakis – IAP, Sept 2014

  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 inf(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-Prospects 3 E.N.Saridakis – IAP, Sept 2014

  4. Introduction • Einstein 1916: General Relativity: energy-momentum source of spacetime Curvature • Levi-Civitaconnection: 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] 4 E.N.Saridakis – IAP, Sept 2014

  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? 5 E.N.Saridakis – IAP, Sept 2014

  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] 6 E.N.Saridakis – IAP, Sept 2014

  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. 7 E.N.Saridakis – IAP, Sept 2014

  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-basedformulation of gravity [Copeland, Sami, Tsujikawa Int.J.Mod.Phys.D15], [Nojiri, Odintsov Int.J.Geom.Meth.Mod.Phys. 4] 8 E.N.Saridakis – IAP, Sept 2014

  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-basedformulation 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] 9 E.N.Saridakis – IAP, Sept 2014

  10. Teleparallel Equivalent of General Relativity (TEGR) • Let’s start from the simplesttosion-based gravity formulation, namely TEGR: • Vierbeins : four linearly independent fields in the tangent space • Use curvature-lessWeitzenböckconnection instead of torsion-lessLevi-Civita one: • Torsion tensor: [Einstein 1928], [Pereira: Introduction to TG] 10 E.N.Saridakis – IAP, Sept 2014

  11. Teleparallel Equivalent of General Relativity (TEGR) • Let’s start from the simplesttosion-based gravity formulation, namely TEGR: • Vierbeins : four linearly independent fields in the tangent space • Use curvature-lessWeitzenböckconnection instead of torsion-lessLevi-Civita one: • Torsion tensor: • Lagrangian(imposing coordinate, Lorentz, parity invariance, and up to 2nd order in torsion tensor) • Completely equivalent with GR at the level of equations 11 [Einstein 1928], [Hayaski,Shirafuji PRD 19], [Pereira: Introduction to TG] E.N.Saridakis – IAP, Sept 2014

  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: [Ferraro, Fiorini PRD 78], [Bengochea, Ferraro PRD 79] [Linder PRD 82] 12 E.N.Saridakis – IAP, Sept 2014

  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: [Ferraro, Fiorini PRD 78], [Bengochea, Ferraro PRD 79] [Linder PRD 82] • Find easily 13 E.N.Saridakis – IAP, Sept 2014

  14. f(T) Cosmology: Background • Effective Dark Energy sector: • Interesting cosmological behavior: Acceleration, Inflationetc • At the background level indistinguishable from other dynamical DE models [Linder PRD 82] 14 E.N.Saridakis – IAP, Sept 2014

  15. f(T) Cosmology: Perturbations • Can I find imprints of f(T) gravity? Yes, but need to go to perturbation level • ObtainPerturbation Equations: • Focus on growth of matter overdensitygo to Fourier modes: [Chen, Dent, Dutta, Saridakis PRD 83], [Dent, Dutta, Saridakis JCAP 1101] [Chen, Dent, Dutta, Saridakis PRD 83] 15 E.N.Saridakis – IAP, Sept 2014

  16. f(T) Cosmology: Perturbations • Application: Distinguishf(T) from quintessence • 1) Reconstruct f(T) to coincide with a given quintessence scenario: • with and [Dent, Dutta, Saridakis JCAP 1101] 16 E.N.Saridakis – IAP, Sept 2014

  17. f(T) Cosmology: Perturbations • Application: Distinguishf(T) from quintessence • 2) Examine evolution ofmatter overdensity [Dent, Dutta, Saridakis JCAP 1101] 17 E.N.Saridakis – IAP, Sept 2014

  18. Bounce and Cyclic behavior • Contracting ( ), bounce ( ), expanding ( ) • near and at the bounce • Expanding ( ), turnaround ( ), contracting near and at the turnaround 18 E.N.Saridakis – IAP, Sept 2014

  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 • Bounce and cyclicity can be easily obtained [Cai, Chen, Dent, Dutta, Saridakis CQG 28] 19 E.N.Saridakis – IAP, Sept 2014

  20. Bounce in f(T) cosmology • Start with a bounching scale factor: 20 E.N.Saridakis – IAP, Sept 2014

  21. Bounce in f(T) cosmology • Start with a bounching scale factor: • Examine the fullperturbations: with known in terms of and matter • Primordialpower spectrum: • Tensor-to-scalarratio: 21 [Cai, Chen, Dent, Dutta, Saridakis CQG 28] E.N.Saridakis – IAP, Sept 2014

  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] 22 E.N.Saridakis – IAP, Sept 2014

  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) [Geng, Lee, Saridakis, Wu PLB 704] [Geng, Lee, Saridakis,Wu PLB 704] 23 E.N.Saridakis – IAP, Sept 2014

  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] 24 E.N.Saridakis – IAP, Sept 2014

  25. Observational constraintson 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 25 E.N.Saridakis – IAP, Sept 2014

  26. Observational constraintson Teleparallel Dark Energy Exponential potential Quartic potential 26 [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 [Xu, Saridakis, Leon, JCAP 1207] 27 E.N.Saridakis – IAP, Sept 2014

  28. Phase-space analysis of Teleparallel Dark Energy • Apart from usual quintessence points, there exists an extrastable one for corresponding to • At the critical points however during the evolution it can lie in quintessence or phantom regimes, or experience the phantom-divide crossing! [Xu, Saridakis, Leon, JCAP 1207] 28 E.N.Saridakis – IAP, Sept 2014

  29. Non-minimally matter-torsion coupled theory • In curvature-based gravity, one can use coupling • Let’s do the same in torsion-based gravity: [Harko, Lobo, Otalora, Saridakis, PRD 89] 29 E.N.Saridakis – IAP, Sept 2014

  30. Non-minimally matter-torsion coupled theory • In curvature-based gravity, one can use coupling • Let’s do the same in torsion-based gravity: • Friedmann equations in FRW universe: with effective Dark Energy sector: • Different than non-minimal matter-curvature coupled theory [Harko, Lobo, Otalora, Saridakis, PRD 89] 30 E.N.Saridakis – IAP, Sept 2014

  31. Non-minimally matter-torsion coupled theory • Interesting phenomenology 31 [Harko, Lobo, Otalora, Saridakis, PRD 89] E.N.Saridakis – IAP, Sept 2014

  32. Non-minimally matter-torsion coupled theory • In curvature-based gravity, one can use coupling • Let’s do the same in torsion-based gravity: [Harko, Lobo, Otalora, Saridakis, 1405.0519, to appear in JCAP] 32 E.N.Saridakis – IAP, Sept 2014

  33. Non-minimally matter-torsion coupled theory • In curvature-based gravity, one can use coupling • Let’s do the same in torsion-based gravity: • Friedmann equations in FRW universe ( ): with effective Dark Energy sector: • Different from gravity [Harko, Lobo, Otalora, Saridakis, 1405.0519, to appear in JCAP] 33 E.N.Saridakis – IAP, Sept 2014

  34. Non-minimally matter-torsion coupled theory • Interesting phenomenology 34 [Harko, Lobo, Otalora, Saridakis, 1405.0519, to appear in JCAP] E.N.Saridakis – IAP, Sept 2014

  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 35 E.N.Saridakis – IAP, Sept 2014

  36. 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 • gravity: • Different from and gravities [Kofinas, Saridakis, 1404.2249 to appear in PRD] [Kofinas, Saridakis, 1408.0107 to appear in PRD] [Kofinas, Leon, Saridakis, CQG 31] 36 E.N.Saridakis – IAP, Sept 2014

  37. Teleparallel Equivalent of Gauss-Bonnet and f(T,T_G) gravity • Cosmological application: [Kofinas, Saridakis, 1408.0107 to appear in PRD] [Kofinas, Saridakis, 1404.2249 to appear in PRD] [Kofinas, Leon, Saridakis, CQG 31] 37 E.N.Saridakis – IAP, Sept 2014

  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: [Gonzalez, Saridakis, Vasquez, JHEP 1207] [Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 1302] 38 E.N.Saridakis – IAP, Sept 2014

  39. 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 39 • [Gonzalez, Saridakis, Vasquez, JHEP 1207], [Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 1302] E.N.Saridakis – IAP, Sept 2014

  40. Exact charged black hole solutions • Horizon andsingularityanalysis: • 1) Vierbeins, Weitzenböck connection, Torsion invariants: T(r) known obtain horizons and singularities • 2) Metric, Levi-Civita connection, Curvature invariants: R(r) and Kretschmannknown obtain horizons and singularities 40 • [Gonzalez, Saridakis, Vasquez, JHEP1207], [Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 1302] E.N.Saridakis – IAP, Sept 2014

  41. Exact charged black hole solutions 41 [Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 1302] E.N.Saridakis – IAP, Sept 2014

  42. Exact charged black hole solutions • Moresingularities in the curvature analysis than in torsion analysis! (some are naked) • The differencesdisappear in the f(T)=0 case, or in the uncharged case. • Going to quartic torsion invariants solves the problem. • f(T) brings novelfeatures. • E/M in torsion formulation was known to be nontrivial (E/M in Einstein-Cartan and Poinaré theories) 42 E.N.Saridakis – IAP, Sept 2014

  43. Solar System constraints on f(T) gravity • Apply the black hole solutions in Solar System: • Assume corrections to TEGR of the form 43 E.N.Saridakis – IAP, Sept 2014

  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 stringentbound. [Iorio, Saridakis, Mon.Not.Roy.Astron.Soc 427) [Wu, Yu, PLB 693], [Bengochea PLB 695] 44 E.N.Saridakis – IAP, Sept 2014

  45. Growth-index constraints on f(T) gravity • Perturbations: , clustering growth rate: • γ(z): Growth index. 45 E.N.Saridakis – IAP, Sept 2014

  46. Growth-index constraints on f(T) gravity • Perturbations: , clustering growth rate: • γ(z): Growth index. • Viablef(T) models are practically indistinguishable from ΛCDM. 46 [Nesseris, Basilakos, Saridakis, Perivolaropoulos, PRD 88] E.N.Saridakis – IAP, Sept 2014

  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 samemetric are not equivalent (extra degrees of freedom) [Li, Sotiriou, Barrow PRD 83a], [Geng,Lee,Saridakis,Wu PLB 704] [Li,Sotiriou,Barrow PRD 83c], [Li,Miao,Miao JHEP 1107] 47 E.N.Saridakis – IAP, Sept 2014

  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 samemetric 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 Systemand Growth Index observationsconstraintf(T) very close to linear-in-T form [Li, Sotiriou, Barrow PRD 83a], [Geng,Lee,Saridakis,Wu PLB 704] [Li,Miao,Miao JHEP 1107] [Capozzielo, Gonzalez, Saridakis, Vasquez JHEP 1302] [Bamba, Capozziello, Nojiri, Odintsov ASS 342], [Miao,Li,Miao JCAP 1111] [Iorio, Saridakis, Mon.Not.Roy.Astron.Soc 427) [Nesseris, Basilakos, Saridakis, Perivolaropoulos, PRD 88] 48 E.N.Saridakis – IAP, Sept 2014

  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 modifyPoincaré gauge theory? 49 E.N.Saridakis – IAP, Sept 2014

  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: Viablef(T) models are practically indistinguishable from ΛCDM. • x) Many open issues. Need to search for other torsion-based modifications too. 50 E.N.Saridakis – IAP, Sept 2014

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