1. Dark Energy in f(R) Gravity Nikodem J. Popławski Indiana University 16 th Midwest Relativity Meeting 18 XI MMVI

2. Cosmic acceleration NASA / WMAP We are living in an accelerating universe! References: A. G. Riess et al., Astron. J. 116, 1009 (1998) S. Perlmutter et al., Astrophys. J. 517, 565 (1999)  Cosmological constant ΛCDM model Agrees with observations

3. Dark energy Hypothetical form of energy with strong negative pressure EXPLANATIONS Cosmological constant Quintessence – dynamical field Alternative gravity theories (talks of G. Mathews and G. J. Olmo) NATURE OF DARK ENERGY homogeneous not very dense not known to interact nongravitationally

4. Dark energy Dark Force = –▼Dark Energy Hypothetical form of energy with strong negative pressure EXPLANATIONS Cosmological constant Quintessence – dynamical field Alternative gravity theories NATURE OF DARK ENERGY homogeneous not very dense not known to interact nongravitationally

5. Variable cosmological constant Cosmological constant problem – why is it so small? No known natural way to derive it from particle physics Possible solution: dark energy decays  Cosmological constant is not constant (Bronstein, 1933) energy Λ matter  Dark energy interact with matter Current interaction rate very small Phenomenological models of decaying Λ relate it to: t -2, a -2, H 2, q, R etc. (Berman, 1991; Ozer and Taha, 1986; Chen and Wu, 1990; Lima and Carvalho, 1994)  lack covariance and/or variational derivation

6. f(R) gravity Lagrangian – function of curvature scalar R R -1 or other negative powers of R → current acceleration Positive powers of R → inflation Minimal coupling in Jordan (original) frame (JF)

7. f(R) gravity Lagrangian – function of curvature scalar R R -1 or other negative powers of R → current acceleration Positive powers of R → inflation Fully covariant theory based on the principle of least action f(R) usually polynomial in R Variable gravitational coupling and cosmological term Solar system and cosmological constraints  polynomial coefficients very small Minimal coupling in Jordan (original) frame (JF) G. J. Olmo, W. Komp, gr-qc/0403092

8. Variational principles I f(R) gravity field equations: vary total action for both the field & matter Two approaches: metric and metric-affine

9. Variational principles I f(R) gravity field equations: vary total action for both the field & matter Two approaches: metric and metric-affine METRIC (Einstein–Hilbert) variational principle: action varied with respect to the metric affine connection given by Christoffel symbols (Levi-Civita connection)

10. Variational principles I f(R) gravity field equations: vary total action for both the field & matter Two approaches: metric and metric-affine METRIC (Einstein–Hilbert) variational principle: action varied with respect to the metric affine connection given by Christoffel symbols (Levi-Civita connection) METRIC–AFFINE (Palatini) variational principle: action varied with respect to the metric and connection metric and connection are independent if f(R)=R  metric and metric-affine give the same field equations: variation with respect to connection  connection = Christoffel symbols E. Schrödinger, Space-time structure, Cambridge (1950)

11. METRIC variational principle: connection: Christoffel symbols of metric tensor  metric compatibility fourth-order differential field equations mathematically equivalent to Brans–Dicke (BD) gravity with ω=0 1/R gravity unstable – but instabilities disappear with additional positive powers of R potential inconsistencies with cosmological evolution need to transform to the Einstein conformal frame to avoid violations of the dominant energy condition (DEC)  EF is physical Variational Principles: Metric

12. METRIC–AFFINE variational principle: no a priori relation between metric and connection second-order differential equations of field mathematically equivalent to BD gravity with ω=−3/2 field equations in vacuum reduce to GR with cosmological constant no instabilities no inconsistencies with cosmological evolution both the Jordan and Einstein frame obey DEC Variational Principles: Metric–Affine Work presented here uses metric–affine formulation

13. Jordan frame Variation of connection  Assume action for matter is independent of connection (good for cosmology)  connection = Christoffel symbols of :

14. Jordan frame Variation of connection  Variation of metric  Dynamical energy-momentum (EM) tensor generated by metric: Assume action for matter is independent of connection (good for cosmology)  connection = Christoffel symbols of Writing and allows interpretation of Θ as additional source and brings EOF into GR form :

15. Helmholtz Lagrangian The action in the Jordan frame is dynamically equivalent to the Helmholtz action The scalar degree of freedom corresponding to nonlinear terms in the Lagrangian is transformed into an auxiliary nondynamical scalar field p (or φ ) provided GR limit and Solar System constraints under debate Scalar – tensor gravity (STG) T. P. Sotiriou, Class. Quantum Grav. 23, 5117 (2006) V. Faraoni, Phys. Rev. D 74, 023529 (2006)

16. Einstein frame Conformal transformation of metric: Effective potential Non-minimal coupling in Einstein frame (EF)

17. Einstein frame G. Magnano, L. M. Sokołowski, Phys. Rev. D 50, 5039 (1994) Conformal transformation of metric: Effective potential Non-minimal coupling in Einstein frame (EF) If minimal coupling in Einstein frame  GR with cosmological constant Both JF and EF are equivalent in vacuum Coupling matter–gravity different in conformally related frames Principle of equivalence violated in EF → constraints on f(R) gravity Experiments should verify which frame (JF or EF) is physical

18. Equations of field and motion Variation of : Structural equation

19. Equations of field and motion Variation of : Structural equation If T=0 (vacuum or radiation)  algebraic equation for φ → φ =const  GR with cosmological constant Gravitational coupling and cosmological term vary The energy-momentum tensor is not covariantly conserved If the EM tensor generated by the EF metric tensor is physical  constancy of V( φ ) → GR with cosmological constant NJP, Class. Quantum Grav. 23, 2011 (2006)

20. Dark energy–momentum tensor Non-conservation of EM tensors for matter and DE separately Total EM for matter + DE conserved  interaction

21. Dark energy–momentum tensor Non-conservation of EM tensors for matter and DE separately Total EM for matter + DE conserved  interaction Continuity equation with interaction term Q: Interaction rate Γ =Q/ ε Λ Nondimensional rate γ = Γ /H Assume homogeneous and isotropic universe NJP, Phys. Rev. D 74, 084032 (2006)

22. Cosmological parameters NJP, Class. Quantum Grav. 23, 4819 (2006); Phys. Lett. B 640, 135 (2006) Hubble parameterDeceleration parameter Higher derivatives of scale factor (jerk and snap) more complicated More nondimensional parameters: deceleration-to-acceleration transition redshift z t, dq/dz| 0 etc. Redshift  H(z) Omega(L=f)

23. Cosmological term Palatini f(R) gravity in Einstein frame predicts (p=0)

24. Cosmological term Palatini f(R) gravity in Einstein frame predicts (p=0) NJP, Phys. Rev. D 74, 084032 (2006) Resembles simple phenomenological models of variable cosmological constant Unlike them, it arises from least-action-principle based theory Duh! ΛCDM model says so But: ΛCDM – constant Λ relates H and q f(R) gravity – variable Λ depends on H and q

25. R-1/R gravity The simplest f(R) that produces current cosmic acceleration Deceleration-to-acceleration transition:

26. R-1/R gravity Unification of inflation and current cosmic acceleration T=0  2 de Sitter phases: D. N. Vollick, Phys. Rev. D 68, 063510 (2003) S. M. Carroll, V. Duvvuri, M. Trodden, M. S. Turner, Phys. Rev. D 70, 043528 (2004) S. Nojiri, S. D. Odintsov, Phys. Rev. D 68, 123512 (2003); NJP, CQG 23, 2011 (2006) Simplest f(R) that produces current cosmic acceleration Deceleration-to-acceleration transition:

27. R-1/R gravity Unification of inflation and current cosmic acceleration T=0  2 de Sitter phases: D. N. Vollick, Phys. Rev. D 68, 063510 (2003) S. M. Carroll, V. Duvvuri, M. Trodden, M. S. Turner, Phys. Rev. D 70, 043528 (2004) S. Nojiri, S. D. Odintsov, Phys. Rev. D 68, 123512 (2003); NJP, CQG 23, 2011 (2006) The simplest f(R) that produces current cosmic acceleration Deceleration-to-acceleration transition: β/α ~10 120 ?

28. Compatibility with observations I A. G. Riess et al., Astrophys. J. 607, 665 (2004) f(R)observations  Use SNLS X clusters Gold ΛCDM j=1 Z t =-0.56+0.07-0.04

29. Compatibility with observations II A. G. Riess et al., Astrophys. J. 607, 665 (2004) f(R)observations  Use SNLS X clusters Gold ΛCDM j=1 Z t =-0.56+0.07-0.04

30. Compatibility with observations III Current interaction rate Interaction between matter and dark energy is weak At deceleration-to-acceleration transition P. Wang, X. H. Meng, CQG 22, 283 (2005) ε ~ a 3-n f(R): n=0.04 Observations  n<0.1

31. Conclusions f(R) gravity provides possible explanation for present cosmic acceleration Dark energy interacts with matter in EF – decaying Λ R-1/R model is nice – simple, nondimensional cosmological parameters do not depend on α We need stronger constraints from astronomical observations FUTURE WORK Compare with JF Generalize to p≠0 (inflation and radiation epochs) Solar system constraints and Newtonian limit? THANK YOU!

32. Back-up Slides

33. Conservation of matter Bianchi identity Homogeneous and isotropic universe with no pressure (comoving frame) Time evolution of φ NJP, Class. Quantum Grav. 23, 2011 (2006)

34. Dark energy density in f(R) Matter energy density Dark energy density NJP, Phys. Rev. D 74, 084032 (2006)

35. More cosmological parameters NJP, Class. Quantum Grav. 23, 4819 (2006); Phys. Lett. B 640, 135 (2006) Deceleration parameter slope Jerk parameter