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Theory          Observations        Modified Gravity               A brief tour       Miguel Zumalac´rregui                ...
Theory                             ObservationsOutline   1   Theory         Introduction         Modified Gravities   2   O...
Theory    Introduction                            Observations   Modified GravitiesIntroduction    Why Modified Gravity?    ...
Theory    Introduction                             Observations   Modified GravitiesIntroduction    Why Modified Gravity?   ...
Theory    Introduction                             Observations   Modified GravitiesEinstein’s Theory   Lovelock’s Theorem ...
Theory    Introduction                              Observations   Modified GravitiesEinstein’s Theory   Lovelock’s Theorem...
Theory    Introduction                            Observations   Modified GravitiesBeyond Einstein’s Theory: Examples    Hi...
Theory    Introduction                             Observations   Modified GravitiesBeyond Einstein’s Theory: Examples    H...
Theory    Introduction                             Observations   Modified GravitiesBeyond Einstein’s Theory: Examples    H...
Theory    Introduction                              Observations   Modified GravitiesBeyond Einstein’s Theory: Examples    ...
Theory    Introduction                             Observations   Modified GravitiesScalar-Tensor Theories    Scalar fields ...
Theory    Introduction                             Observations   Modified GravitiesScalar-Tensor Theories    Scalar fields ...
Theory    Introduction                             Observations   Modified GravitiesScalar-Tensor Theories    Scalar fields ...
Theory    Solar System                             Observations   CosmologyLocal Gravity Tests                            ...
Theory    Solar System                                  Observations   CosmologyLocal Gravity Tests                       ...
Theory    Solar System                               Observations   CosmologyScreening Mechanisms   §                     ...
Theory    Solar System                               Observations   CosmologyScreening Mechanisms   §                     ...
Theory    Solar System                            Observations   CosmologyCosmology    Scalars can source cosmic accelerat...
Theory      Solar System                            Observations     CosmologyCosmology    Scalars can source cosmic accel...
Theory    Solar System                              Observations   CosmologyLinear Perturbations   Quasi-static approximat...
Theory    Solar System                                   Observations   CosmologyLinear Perturbations   Quasi-static appro...
Theory    Solar System                                   Observations   CosmologyLinear Perturbations   Quasi-static appro...
Theory    Solar System                              Observations   CosmologyNon-Linear Perturbations   - Higher order PT v...
Theory    Solar System                            Observations   CosmologyDynamical Observables: Matter and Light    Large...
Theory    Solar System                            Observations   CosmologyDynamical Observables: Matter and Light    Large...
Theory    Solar System                            Observations   CosmologyDynamical Observables: Matter and Light    Large...
Theory    Solar System                            Observations   CosmologyDynamical Observables: Matter and Light    Large...
Theory    Solar System                            Observations   CosmologyDynamical Observables: Matter and Light    Large...
Theory    Solar System                            Observations   CosmologyTheory vs Observations      No pure test of grav...
Theory    Solar System                            Observations   CosmologyTheory vs Observations      No pure test of grav...
Theory    Solar System                                Observations   CosmologyConclusions        Many possible modification...
Theory    Solar System         Observations   Cosmology          Backup SlidesMiguel Zumalac´rregui              a        ...
Theory    Solar System                                 Observations   CosmologyThe Frontiers of Gravity   What is the most...
Theory    Solar System                             Observations   CosmologyMost General Scalar-Tensor theory   Horndenski’...
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Modified Gravity - a brief tour

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very brief overview of alternative theories of gravity and the means to test them

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Modified Gravity - a brief tour

  1. 1. Theory Observations Modified Gravity A brief tour Miguel Zumalac´rregui aInstituto de F´ ısica Te´rica IFT-UAM-CSIC o IFT-UAM Cosmology meeting IFT, February 2013, Madrid Miguel Zumalac´rregui a Modified Gravity
  2. 2. Theory ObservationsOutline 1 Theory Introduction Modified Gravities 2 Observations Solar System Cosmology 3) Conclusions Miguel Zumalac´rregui a Modified Gravity
  3. 3. Theory Introduction Observations Modified GravitiesIntroduction Why Modified Gravity? Mystery: Λ and CDM problems Observational Outliers (LSS bulk motions, halo profiles, satellite galaxies...) Testing General Relativity ⇒ Model independence of cosmological probes Miguel Zumalac´rregui a Modified Gravity
  4. 4. Theory Introduction Observations Modified GravitiesIntroduction Why Modified Gravity? Mystery: Λ and CDM problems Observational Outliers (LSS bulk motions, halo profiles, satellite galaxies...) Testing General Relativity ⇒ Model independence of cosmological probes Main Points Many different scenarios for modified gravity Need to analyze in a (sufficiently) self consistent way Miguel Zumalac´rregui a Modified Gravity
  5. 5. Theory Introduction Observations Modified GravitiesEinstein’s Theory Lovelock’s Theorem (1971) gµν + Local + 4-D + Lorentz Theory with 2nd order Eqs∗ √ 1 −g (R − 2Λ) 16πG ∗ Theories with higher time derivatives unstable: E → −∞ (Ostrogradski’s Theorem) Miguel Zumalac´rregui a Modified Gravity
  6. 6. Theory Introduction Observations Modified GravitiesEinstein’s Theory Lovelock’s Theorem (1971) gµν + Local + 4-D + Lorentz Theory with 2nd order Eqs∗ √ 1 −g (R − 2Λ) 16πG ∗ Theories with higher time derivatives unstable: E → −∞ (Ostrogradski’s Theorem) Acceptable modifications (Clifton et al. 1106.2476): Higher derivatives Additional fields Extra dimensions Weird stuff: Lorentz violation, non-local, non-metric... Miguel Zumalac´rregui a Modified Gravity
  7. 7. Theory Introduction Observations Modified GravitiesBeyond Einstein’s Theory: Examples Higher derivatives: f (R) gravity −→ Equivalent to h(φ)R + · · · Miguel Zumalac´rregui a Modified Gravity
  8. 8. Theory Introduction Observations Modified GravitiesBeyond Einstein’s Theory: Examples Higher derivatives: f (R) gravity −→ Equivalent to h(φ)R + · · · § ¤ Additional fields: Scalar: φ ¦ ¥ - Vector: Aµ , e.g. TeVeS (alternative to DM) - Tensor: hµν Massive gravity −→ scalar φ in decoupling limit Miguel Zumalac´rregui a Modified Gravity
  9. 9. Theory Introduction Observations Modified GravitiesBeyond Einstein’s Theory: Examples Higher derivatives: f (R) gravity −→ Equivalent to h(φ)R + · · · § ¤ Additional fields: Scalar: φ ¦ ¥ - Vector: Aµ , e.g. TeVeS (alternative to DM) - Tensor: hµν Massive gravity −→ scalar φ in decoupling limit Extra dimensions: - DGP → φ = brane location in extra dim. - Kaluza-Klein → φ ∝ volume of compact dim. Miguel Zumalac´rregui a Modified Gravity
  10. 10. Theory Introduction Observations Modified GravitiesBeyond Einstein’s Theory: Examples Higher derivatives: f (R) gravity −→ Equivalent to h(φ)R + · · · § ¤ Additional fields: Scalar: φ ¦ ¥ - Vector: Aµ , e.g. TeVeS (alternative to DM) - Tensor: hµν Massive gravity −→ scalar φ in decoupling limit Extra dimensions: - DGP → φ = brane location in extra dim. - Kaluza-Klein → φ ∝ volume of compact dim. 2 − /M∗ Weird stuff: Non-local ⊃ R e R - Lorentz violation: Horava-Lifschitz gravity ¨ ξ → −ξ + 4ξ Miguel Zumalac´rregui a Modified Gravity
  11. 11. Theory Introduction Observations Modified GravitiesScalar-Tensor Theories Scalar fields arise in many contexts: geometry of extra dimensions f (R), decoupling limit of massive gravity, etc... Isotropy friendly → no prefered directions Miguel Zumalac´rregui a Modified Gravity
  12. 12. Theory Introduction Observations Modified GravitiesScalar-Tensor Theories Scalar fields arise in many contexts: geometry of extra dimensions f (R), decoupling limit of massive gravity, etc... Isotropy friendly → no prefered directions Most general: Horndenski’s Theory → 4 functions of φ, (∂φ)2 : L2 = K[φ, (∂φ)2 ] → no φ ↔ Rµν interaction (dark energy) L3 , L4 , L5 explicit couplings φ ↔ Rµν (modified gravity) Miguel Zumalac´rregui a Modified Gravity
  13. 13. Theory Introduction Observations Modified GravitiesScalar-Tensor Theories Scalar fields arise in many contexts: geometry of extra dimensions f (R), decoupling limit of massive gravity, etc... Isotropy friendly → no prefered directions Most general: Horndenski’s Theory → 4 functions of φ, (∂φ)2 : L2 = K[φ, (∂φ)2 ] → no φ ↔ Rµν interaction (dark energy) L3 , L4 , L5 explicit couplings φ ↔ Rµν (modified gravity) Also interacing DM: scalar couples only to DM Miguel Zumalac´rregui a Modified Gravity
  14. 14. Theory Solar System Observations CosmologyLocal Gravity Tests √ −gR Transform to Einstein-frame: L = 16πG + Lm (˜µν [φ]) +Lφ g matter metric Matter follows geodesic of gµν rather than gµν ˜ ⇒ φ mediates an additional force F ∝ φ Miguel Zumalac´rregui a Modified Gravity
  15. 15. Theory Solar System Observations CosmologyLocal Gravity Tests √ −gR Transform to Einstein-frame: L = 16πG + Lm (˜µν [φ]) +Lφ g matter metric Matter follows geodesic of gµν rather than gµν ˜ ⇒ φ mediates an additional force F ∝ φ Constrained by laboratory and Solar System tests: Perihelion precession, Lunar laser ranging... → massive bodies Gravitational light bending, time delay... → light geodesics e.g. http://relativity.livingreviews.org/Articles/lrr-2001-4/ Miguel Zumalac´rregui a Modified Gravity
  16. 16. Theory Solar System Observations CosmologyScreening Mechanisms § ¤ Non-linear interactions → Hide φ around massive bodies ¦ ¥ Screening from V (φ) −mφ r e Chameleon: ρ dependent field range: φ ∝ r Symmetron: ρ dependent coupling to matter Only surface contribution from screened objects: Qφ QG . (Lam Hui’s lectures: www.slideshare.net/CosmoAIMS/hui-modified-gravity) Miguel Zumalac´rregui a Modified Gravity
  17. 17. Theory Solar System Observations CosmologyScreening Mechanisms § ¤ Non-linear interactions → Hide φ around massive bodies ¦ ¥ Screening from V (φ) −mφ r e Chameleon: ρ dependent field range: φ ∝ r Symmetron: ρ dependent coupling to matter Only surface contribution from screened objects: Qφ QG . Screening from φ 1 rs 3 Vainshtein: interaction suppressed for r rV = m2 ∗ significant scalar force for r > rV : Qφ ≈ QG Disformal: field evolution independent of ρ (if ρ m4 ) ∗ (Lam Hui’s lectures: www.slideshare.net/CosmoAIMS/hui-modified-gravity) Miguel Zumalac´rregui a Modified Gravity
  18. 18. Theory Solar System Observations CosmologyCosmology Scalars can source cosmic acceleration: Effective Cosmological Constant: Λ → V (φ) + 1 (∂φ)2 2 Self-acceleration: H ≈ constant is solution. Miguel Zumalac´rregui a Modified Gravity
  19. 19. Theory Solar System Observations CosmologyCosmology Scalars can source cosmic acceleration: Effective Cosmological Constant: Λ → V (φ) + 1 (∂φ)2 2 Self-acceleration: H ≈ constant is solution. Einstein frame: Energy transfer µν µν µ Tm =− µ Tφ = −Qφ,ν Geometric measurements (DL , DA ) can’t distinguish dark energy (Q = 0) from modified gravity (Q = 0) Perturbations: Additional force if Q = 0 Miguel Zumalac´rregui a Modified Gravity
  20. 20. Theory Solar System Observations CosmologyLinear Perturbations Quasi-static approximation on sub-horizon scales k2 δρ Neglect time derivatives, keep terms ∝ a2 ,δ ≡ ρ ¨ ˙ δ + 2H δ ≈ 4π Geff (k, t) ρm δ (effective gravitational constant) Φ = − η(k, t) Ψ (anisotropic parameter) Miguel Zumalac´rregui a Modified Gravity
  21. 21. Theory Solar System Observations CosmologyLinear Perturbations Quasi-static approximation on sub-horizon scales k2 δρ Neglect time derivatives, keep terms ∝ a2 ,δ ≡ ρ ¨ ˙ δ + 2H δ ≈ 4π Geff (k, t) ρm δ (effective gravitational constant) Φ = − η(k, t) Ψ (anisotropic parameter) Geff 1 1 + 4(f /f )(k/a)2 1 + 2(f /f )(k/a)2 f (R) gravity: = , η= G f 1 + 3(f /f )(k/a)2 1 + 4(f /f )(k/a)2 4 3 enhancement on small scales (De Felice et al. 1108.4242). Miguel Zumalac´rregui a Modified Gravity
  22. 22. Theory Solar System Observations CosmologyLinear Perturbations Quasi-static approximation on sub-horizon scales k2 δρ Neglect time derivatives, keep terms ∝ a2 ,δ ≡ ρ ¨ ˙ δ + 2H δ ≈ 4π Geff (k, t) ρm δ (effective gravitational constant) Φ = − η(k, t) Ψ (anisotropic parameter) Geff 1 1 + 4(f /f )(k/a)2 1 + 2(f /f )(k/a)2 f (R) gravity: = , η= G f 1 + 3(f /f )(k/a)2 1 + 4(f /f )(k/a)2 4 3 enhancement on small scales (De Felice et al. 1108.4242). Parameterized Post-Friedmann framework (PPF) General treatment of linear perturbations → O(20) free functions (e.g. Baker et al. 1209.2117). Miguel Zumalac´rregui a Modified Gravity
  23. 23. Theory Solar System Observations CosmologyNon-Linear Perturbations - Higher order PT very hard, especially beyond GR from M. Baldi 1109.5695 - N-body simulations computationally expensive: Non-linear equation for φ(x, t): Solve on a grid. ˙ ¨ Usually assume quasi-static field evolution φ, φ ∼ 0 yet necessary to access small scales! Miguel Zumalac´rregui a Modified Gravity
  24. 24. Theory Solar System Observations CosmologyDynamical Observables: Matter and Light Large Scale Structure: P (k) → linear & non-linear, limited by bias d log(δ) Peculiar velocities/RSD → f = d log(a) (linear) Miguel Zumalac´rregui a Modified Gravity
  25. 25. Theory Solar System Observations CosmologyDynamical Observables: Matter and Light Large Scale Structure: P (k) → linear & non-linear, limited by bias d log(δ) Peculiar velocities/RSD → f = d log(a) (linear) Bispectrum → non-linear Cluster abundances & profiles → non-linear scales! Miguel Zumalac´rregui a Modified Gravity
  26. 26. Theory Solar System Observations CosmologyDynamical Observables: Matter and Light Large Scale Structure: P (k) → linear & non-linear, limited by bias d log(δ) Peculiar velocities/RSD → f = d log(a) (linear) Bispectrum → non-linear Cluster abundances & profiles → non-linear scales! Voids → test low ρ environments Miguel Zumalac´rregui a Modified Gravity
  27. 27. Theory Solar System Observations CosmologyDynamical Observables: Matter and Light Large Scale Structure: P (k) → linear & non-linear, limited by bias d log(δ) Peculiar velocities/RSD → f = d log(a) (linear) Bispectrum → non-linear Cluster abundances & profiles → non-linear scales! Voids → test low ρ environments Cosmic Microwave Background ˙ ˙ Integrated Sachs Wolfe → measures Φ − Ψ, small statistics Miguel Zumalac´rregui a Modified Gravity
  28. 28. Theory Solar System Observations CosmologyDynamical Observables: Matter and Light Large Scale Structure: P (k) → linear & non-linear, limited by bias d log(δ) Peculiar velocities/RSD → f = d log(a) (linear) Bispectrum → non-linear Cluster abundances & profiles → non-linear scales! Voids → test low ρ environments Cosmic Microwave Background ˙ ˙ Integrated Sachs Wolfe → measures Φ − Ψ, small statistics Weak gravitational lensing: Shear → measures Φ + Ψ, complementary to P (k), non-linear scales, systematics Miguel Zumalac´rregui a Modified Gravity
  29. 29. Theory Solar System Observations CosmologyTheory vs Observations No pure test of gravity: probes sensitive to several effects (expansion, neutrinos, primordial non-Gaussianity...) ⇒ Complementarity is essential Miguel Zumalac´rregui a Modified Gravity
  30. 30. Theory Solar System Observations CosmologyTheory vs Observations No pure test of gravity: probes sensitive to several effects (expansion, neutrinos, primordial non-Gaussianity...) ⇒ Complementarity is essential Ideally: self consistent analysis → assume MG on all steps or at least keep track of assumptions: Poisson eq. Φ = 4πk 2 Gρk Matter geodesics xi = − i Φ ¨ Galaxy bias Calibration with simulations ··· Miguel Zumalac´rregui a Modified Gravity
  31. 31. Theory Solar System Observations CosmologyConclusions Many possible modifications of gravity (not only f (R)!) Scalar-tensor encompass many of them in some limit Screening mechanisms to pass local gravity tests Cosmology: need dynamical data to distinguish DE from MG (LSS, CMB, lensing...) Theory vs Data: exploit complementarity and bear assumptions in mind Doubts? check the Bible of modified gravity: - Clifton et al. 2011 ”Modified Gravity and Cosmology” 1106.2476 Miguel Zumalac´rregui a Modified Gravity
  32. 32. Theory Solar System Observations Cosmology Backup SlidesMiguel Zumalac´rregui a Modified Gravity
  33. 33. Theory Solar System Observations CosmologyThe Frontiers of Gravity What is the most general possible theory of gravity? Ostrogradski’s Theorem (1850) ∂nq Theories with L ⊃ , n ≥ 2 are unstable∗ ∂tn ∂L d ∂L d2 ∂L q(t), L(q, q, q ) → ˙ ¨ − + =0 ∂q dt ∂ q ˙ dt2 ∂ q ¨ ... ¨ q, q, q , q → Q1 , Q2 , P1 , P2 ˙ ¨ P1,2 ≡ ∂L/∂ Q1,2 H = P1 Q2 + terms independent of P1 ∗ ... .... If no q , q in the Equations ⇒ Loophole Miguel Zumalac´rregui a Modified Gravity
  34. 34. Theory Solar System Observations CosmologyMost General Scalar-Tensor theory Horndenski’s Theory (1974) £   gµν + φ + Local + 4-D + Lorentz Theory with 2nd order Eqs. ¢ ¡ 1 ⇒ ∃ 4 free functions of φ, X ≡ − 2 φ,µ φ,µ L2 = G2 (X, φ) −→ No φ ↔ gµν interaction L3 = −G3 φ −→ eqs ⊃ G3,X Rµν φ,µ φ,ν L4 = G4 R + G4,X ( φ)2 − φ;µν φ;µν L5 = G5 Gµν φ;µν − 1 G5,X ( φ)3 − 3( φ)φ;µν φ;µν + 2φ;µ;ν φ;ν ;λ φ;λ;µ 6 Miguel Zumalac´rregui a Modified Gravity

very brief overview of alternative theories of gravity and the means to test them

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