Dark Energy in f(R) Gravity

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

2. Cosmic acceleration Cosmological constant ΛCDM model Agrees with observations 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)

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

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

5. Λ matter energy 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)  Dark energy interact with matter Current interaction rate very small Phenomenological models of decaying Λ relate it to: t-2, a-2, H2, 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. Lagrangian – function of curvature scalar R • R-1 or other negative powers of R → current acceleration • Positive powers of R → inflation f(R) gravity Minimal coupling in Jordan (original) frame (JF) • 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 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. Variational Principles: Metric • 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

12. Variational Principles: Metric–Affine • METRIC–AFFINE variational principle: • noa 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 Work presented here uses metric–affine formulation

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

14. : Assume action for matter is independent of connection (good for cosmology) Jordan frame Variation of connection  connection = Christoffel symbols of Variation of metric  Dynamical energy-momentum (EM) tensor generated by metric: and Writing 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 provided Scalar – tensor gravity (STG) GR limit and Solar System constraints under debate The scalar degree of freedom corresponding to nonlinear terms in the Lagrangian is transformed into an auxiliary nondynamical scalar field p (orφ) 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 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 G. Magnano, L. M. Sokołowski, Phys. Rev. D 50, 5039 (1994)

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

19. Equations of field and motion Variation of : 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 Assume homogeneous and isotropic universe Continuity equation with interaction term Q: Interaction rate Γ=Q/εΛ Nondimensional rate γ=Γ/H NJP, Phys. Rev. D 74, 084032 (2006)

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

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) Duh! ΛCDM model says so But: ΛCDM – constant Λrelates H and q f(R) gravity – variable Λdepends on H and q • Resembles simple phenomenological models of variable cosmological • constant • Unlike them, it arises from least-action-principle based theory NJP, Phys. Rev. D 74, 084032 (2006)

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

26. R-1/R gravity Simplest f(R) that produces current cosmic acceleration Deceleration-to-acceleration transition: 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)

27. R-1/R gravity The simplest f(R) that produces current cosmic acceleration Deceleration-to-acceleration transition: Unification of inflation and current cosmic acceleration β/α ~10120 ? 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)

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

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

30. Compatibility with observations III Current interaction rate At deceleration-to-acceleration transition Interaction between matter and dark energy is weak ε ~ a3-n f(R): n=0.04 Observations  n<0.1 P. Wang, X. H. Meng, CQG 22, 283 (2005)

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