f(t) Gravity and Cosmology

Transcription

1 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 XXVII SRA, Dallas, Dec 013

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 E.N.Saridakis XXVII SRA, Dallas, Dec 013

3 Talk Plan 1) Introduction: Gravity as a gauge theory, modified Gravity ) 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 and growth-index constraints 8) Conclusions-Prospects 3 E.N.Saridakis XXVII SRA, Dallas, Dec 013

4 Introduction Einstein 1916: General Relativity: energy-momentum source of spacetime Curvature Levi-Civita connection: Zero Torsion Einstein 198: 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 XXVII SRA, Dallas, Dec 013

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 SU 3 SU U(1) Can we apply this to gravity? 5 E.N.Saridakis XXVII SRA, Dallas, Dec 013

6 Introduction Formulating the gauge theory of gravity (mainly after 1960): Start from Special Relativity Apply (Weyl-Yang-Mills) gauge principle to its Poincarégroup 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, 013] 6 E.N.Saridakis XXVII SRA, Dallas, Dec 013

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 XXVII SRA, Dallas, Dec 013

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) [Copeland, Sami, Tsujikawa Int.J.Mod.Phys.D15], [Nojiri, Odintsov Int.J.Geom.Meth.Mod.Phys. 4] Almost all in the curvature-based formulation of gravity 8 E.N.Saridakis XXVII SRA, Dallas, Dec 013

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) [Copeland, Sami, Tsujikawa Int.J.Mod.Phys.D15], [Nojiri, Odintsov Int.J.Geom.Meth.Mod.Phys. 4] 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 9 E.N.Saridakis XXVII SRA, Dallas, Dec 013

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 g e A A B ( x) e ( x) e ( x) AB Use curvature-less Weitzenböck connection instead of torsion-less W A Levi-Civita one: { } eae Torsion tensor: T e e e { W} { W} A A [Einstein 198], [Pereira: Introduction to TG] A 10 E.N.Saridakis XXVII SRA, Dallas, Dec 013

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 g e A A B ( x) e ( x) e ( x) AB Use curvature-less Weitzenböck connection instead of torsion-less W A Levi-Civita one: { } eae Torsion tensor: T { W} { W} e A e A e A Lagrangian (imposing coordinate, Lorentz, parity invariance, and up to nd order in torsion tensor) 1 1 Completely equivalent with L T T T T T 4 GR at the level of equations [Einstein 198], [Hayaski,Shirafuji PRD 19], [Pereira: Introduction to TG] 11 E.N.Saridakis XXVII SRA, Dallas, Dec 013

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: e T f ( T Sm S d x e G ) 16 [Bengochea, Ferraro PRD 79], [Linder PRD 8] 1 { } ee S 1 f e T S e S ( T ) f e [ T f ( T )] 4 Ge A T A A TT A A 4 1 E.N.Saridakis XXVII SRA, Dallas, Dec 013

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)) 1 4 Equations of motion: e T f ( T Sm S d x e G ) { } ee S 1 f e T S e S ( T ) f e [ T f ( T )] 4 Ge A T A A TT A A f(t) Cosmology: Apply in FRW geometry: [Bengochea, Ferraro PRD 79], [Linder PRD 8] 4 e A diag (1, a, a, a) ds dt a ( t) dx ij i dx j (not unique choice) Friedmann equations: H 8G m 3 4G( f ( T ) 6 p m m H 1 ft 1H ftt ) f T H Find easily T 6H 13 E.N.Saridakis XXVII SRA, Dallas, Dec 013

14 f(t) Cosmology: Background Effective Dark Energy sector: w DE DE 3 f T ft 8G 6 3 f TfT T [1 f Tf ][ f T TT ftt Tf T ] [Linder PRD 8] Interesting cosmological behavior: Acceleration, Inflation etc At the background level indistinguishable from other dynamical DE models 14 E.N.Saridakis XXVII SRA, Dallas, Dec 013

15 f(t) Cosmology: Perturbations Can I find imprints of f(t) gravity? Yes, but need to go to perturbation level e 0 0 (1 ) (1 ) Obtain Perturbation Equations: L. H. S R. H. S L. H. S R. H. S, e ds [Chen, Dent, Dutta, Saridakis PRD 83], [Dent, Dutta, Saridakis JCAP 1101] i (1 ) dt a (1 ) dx dx ij j m Focus on growth of matter overdensity go to Fourier modes: 3H 4 1 f 1H f (3H k / a )(1 f ) 36H f 4G 0 T TT k T m TT k m k [Chen, Dent, Dutta, Saridakis PRD 83] 15 E.N.Saridakis XXVII SRA, Dallas, Dec 013

16 f(t) Cosmology: Perturbations Application: Distinguish f(t) from quintessence 1) Reconstruct f(t) to coincide with a given quintessence scenario: f ( H) Q 16 GH dh CH with Q / V ( ) and H / 6 [Dent, Dutta, Saridakis JCAP 1101] 16 E.N.Saridakis XXVII SRA, Dallas, Dec 013

17 f(t) Cosmology: Perturbations Application: Distinguish f(t) from quintessence ) Examine evolution of matter overdensity [Dent, Dutta, Saridakis JCAP 1101] m m 17 E.N.Saridakis XXVII SRA, Dallas, Dec 013

18 Bounce and Cyclic behavior H 0 H 0 H 0 H 0 Contracting ( ), bounce ( ), expanding ( ) near and at the bounce H 0 H 0 H 0 H 0 Expanding ( ), turnaround ( ), contracting near and at the turnaround 18 E.N.Saridakis XXVII SRA, Dallas, Dec 013

19 Bounce and Cyclic behavior in f(t) cosmology H 0 H 0 H 0 H 0 Contracting ( ), bounce ( ), expanding ( ) near and at the bounce H 0 H 0 H 0 H 0 Expanding ( ), turnaround ( ), contracting H 8G m 3 near and at the turnaround 4G( f ( T ) f 6 p m m H 1 ft 1H ftt Bounce and cyclicity can be easily obtained [Cai, Chen, Dent, Dutta, Saridakis CQG 8] ) T H 19 E.N.Saridakis XXVII SRA, Dallas, Dec 013

20 0 Bounce in f(t) cosmology Start with a bounching scale factor: 1/ ) ( t a t a B ) ( T T T T t t s ArcTan t tm t M t t t f mb p mb p ) 3 ( 4 ) ( E.N.Saridakis XXVII SRA, Dallas, Dec 013

21 Bounce in f(t) cosmology Start with a bounching scale factor: T t( T ) 3T 3 3T f 4t 6tM p mb t) 6 ( 3t ) M t 3t Examine the full perturbations: Primordial power spectrum: Tensor-to-scalar ratio: ArcTan ( mb p k a k k k cs 0 k, s h ij a 3Hh ij h ij 1HHf 1 f T TT 3s t with, c known in terms of H, H, f, f, f and matter T h 0 [Cai, Chen, Dent, Dutta, Saridakis CQG 8] P r / 3 3 ( t ) a 1 t a B 88 M p 3 TT 1 E.N.Saridakis XXVII SRA, Dallas, Dec 013

22 Non-minimally coupled scalar-torsion theory In curvature-based gravity, apart from R f (R) one can use Let s do the same in torsion-based gravity: R R^ T V ( L 4 T 1 S d x e ) m [Geng, Lee, Saridakis, Wu PLB704] E.N.Saridakis XXVII SRA, Dallas, Dec 013

23 Non-minimally coupled scalar-torsion theory In curvature-based gravity, apart from R f (R) one can use Let s do the same in torsion-based gravity: R R^ T V ( L 4 T 1 S d x e ) m [Geng, Lee, Saridakis, Wu PLB704] Friedmann equations in FRW universe: H 3 m DE H m p m DE p with effective Dark Energy sector: DE DE V ( ) 3H p DE V ( ) 4H 3H H Different than non-minimal quintessence! (no conformal transformation in the present case) [Geng, Lee, Saridakis,Wu PLB 704] 3 E.N.Saridakis XXVII SRA, Dallas, Dec 013

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] 4 E.N.Saridakis XXVII SRA, Dallas, Dec 013

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: Ω V ( ) We fit M0,ΩDE0, wde0, for various 3 4 M M 0 / a, r r0 / a,1 z 1/ a 5 E.N.Saridakis XXVII SRA, Dallas, Dec 013

26 Observational constraints on Teleparallel Dark Energy Exponential potential Quartic potential [Geng, Lee, Saridkis JCAP 101] 6 E.N.Saridakis XXVII SRA, Dallas, Dec 013

27 Phase-space analysis of Teleparallel Dark Energy Transform cosmological system to its autonomous form: V ( ) x, y, z 6H 3H m w m = 1 x y z sgn( ), 3H DE = w DE x, y, z, X' = f(x), Linear Perturbations: Q X ' = 0 X X=X C = X DE = x y z sgn( ) 3H C [Xu, Saridakis, Leon, JCAP 107] U' QU Eigenvalues of determine type and stability of C.P DE U 7 E.N.Saridakis XXVII SRA, Dallas, Dec 013

28 Phase-space analysis of Teleparallel Dark Energy Apart from usual quintessence points, there exists an extra stable one for corresponding to 1, 1, q 1 DE w DE w DE 1 At the critical points however during the evolution it can lie in quintessence or phantom regimes, or experience the phantomdivide crossing! [Xu, Saridakis, Leon, JCAP 107] 8 E.N.Saridakis XXVII SRA, Dallas, Dec 013

29 Exact charged black hole solutions Extend f(t) gravity in D-dimensions (focus on D=3, D=4): 1 D S d x et f ( T ) Add E/M sector: F F with L F 1 F da, A A dx Extract field equations: L. H. S R. H. S [Gonzalez, Saridakis, Vasquez, JHEP 107] [Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 130] 9 E.N.Saridakis XXVII SRA, Dallas, Dec 013

30 Exact charged black hole solutions Extend f(t) gravity in D-dimensions (focus on D=3, D=4): 1 D S d x et f ( T ) Add E/M sector: F F with L F 1 F da, A A dx Extract field equations: L. H. S R. H. S Look for spherically symmetric solutions: e 0 ds F( r) dt, e dr, e rdx1,, e rdx, G( r) F( r) dt 1 G( r) dr r D 1 dx i Q r Radial Electric field: Er D F( r), G( r) known [Gonzalez, Saridakis, Vasquez, JHEP 107], [Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 130] 30 E.N.Saridakis XXVII SRA, Dallas, Dec 013

31 Exact charged black hole solutions Horizon and singularity analysis: 1) Vierbeins, Weitzenböck connection, Torsion invariants: T(r) known obtain horizons and singularities ) Metric, Levi-Civita connection, Curvature invariants: R(r) and Kretschmann obtain horizons and singularities R R (r) known [Gonzalez, Saridakis, Vasquez, JHEP107], [Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 130] 31 E.N.Saridakis XXVII SRA, Dallas, Dec 013

32 Exact charged black hole solutions [Capozzielo, Gonzalez, Saridakis, Vasquez, JHEP 130] E.N.Saridakis XXVII SRA, Dallas, Dec 013 3

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) 33 E.N.Saridakis XXVII SRA, Dallas, Dec 013

34 34 Solar System constraints on f(t) gravity Apply the black hole solutions in Solar System: Assume corrections to TEGR of the form ) ( ) ( 3 O T T T f r c GM r r r c GM r F ) ( ) ( r r c GM r r r r c GM r G E.N.Saridakis XXVII SRA, Dallas, Dec 013

35 Solar System constraints on f(t) gravity Apply the black hole solutions in Solar System: Assume corrections to TEGR of the form GM 6 ( r) 1 r 6 c r 3 r F 4 Use data from Solar System orbital motions: T<<1 so consistent GM c r f(t) divergence from TEGR is very small f ( T) T GM 8 4 GM 8 G( r) 1 r r 8 c r 3 3 r c r r 10 U f ( T ) O( T This was already known from cosmological observation constraints up to 1 O (10 10 ) [Wu, Yu, PLB 693], [Bengochea PLB 695] With Solar System constraints, much more stringent bound. [Iorio, Saridakis, Mon.Not.Roy.Astron.Soc 47) 3 ) 35 E.N.Saridakis XXVII SRA, Dallas, Dec 013

36 Growth-index constraints on f(t) gravity H 4G Perturbations: m m eff m m, clustering growth rate: γ(z): Growth index. G eff 1 1 f '( T ) d d ln m m ln a ( a) 36 E.N.Saridakis XXVII SRA, Dallas, Dec 013

37 Growth-index constraints on f(t) gravity H 4G Perturbations: m m eff m m, clustering growth rate: γ(z): Growth index. G eff 1 1 f '( T ) d d ln m m ln a ( a) Viable f(t) models are practically indistinguishable from ΛCDM. [Nesseris, Basilakos, Saridakis, Perivolaropoulos, PRD 88] 37 E.N.Saridakis XXVII SRA, Dallas, Dec 013

38 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 [Li, Sotiriou, Barrow PRD 83a], [Geng,Lee,Saridakis,Wu PLB 704] Equivalently, the vierbein choices corresponding to the same metric are not equivalent (extra degrees of freedom) [Li,Sotiriou,Barrow PRD 83c], [Li,Miao,Miao JHEP 1107] 38 E.N.Saridakis XXVII SRA, Dallas, Dec 013

39 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 [Capozzielo, Gonzalez, Saridakis, Vasquez JHEP 130] Thermodynamics also raises issues [Li, Sotiriou, Barrow PRD 83a], [Geng,Lee,Saridakis,Wu PLB 704] [Li,Sotiriou,Barrow PRD 83c], [Li,Miao,Miao JHEP 1107] [Bamba,Geng JCAP 1111], [Miao,Li,Miao JCAP 1111] Cosmological, Solar System and Growth Index observations constraint f(t) very close to linear-in-t form 39 E.N.Saridakis XXVII SRA, Dallas, Dec 013

40 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? 40 E.N.Saridakis XXVII SRA, Dallas, Dec 013

41 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 T ξt : 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) Growth Index constraints: Viable f(t) models are practically indistinguishable from ΛCDM. x) Many open issues. Need to search for other torsion-based modifications too. 41 E.N.Saridakis XXVII SRA, Dallas, Dec 013

42 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) What to quantize? Metric, vierbeins, or connection? vii) Convince people to work on the subject! 4 E.N.Saridakis XXVII SRA, Dallas, Dec 013

43 THANK YOU! 43