Modified Gravity - a brief tour
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This document provides an overview of modified gravity theories as alternatives to dark energy and dark matter. It discusses that while general relativity is well-tested, there are theoretical motivations to consider modifications such as to address mysteries like dark energy. It reviews several examples of modified gravity theories involving additional fields, higher derivatives, or extra dimensions. Theories are constrained by solar system and laboratory tests but can be made compatible through screening mechanisms. Modified gravity can drive cosmic acceleration and affects the growth of structure, tested by cosmological observations of large-scale structure, peculiar velocities, and clusters.
![Theory Introduction
Observations Modified Gravities
Scalar-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](https://image.slidesharecdn.com/mogra-130205083421-phpapp01/85/Modified-Gravity-a-brief-tour-12-320.jpg)
![Theory Introduction
Observations Modified Gravities
Scalar-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](https://image.slidesharecdn.com/mogra-130205083421-phpapp01/85/Modified-Gravity-a-brief-tour-13-320.jpg)
![Theory Solar System
Observations Cosmology
Local 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](https://image.slidesharecdn.com/mogra-130205083421-phpapp01/85/Modified-Gravity-a-brief-tour-14-320.jpg)
![Theory Solar System
Observations Cosmology
Local 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](https://image.slidesharecdn.com/mogra-130205083421-phpapp01/85/Modified-Gravity-a-brief-tour-15-320.jpg)
Modified Gravity - a brief tour
- 1. Theory Observations Modified Gravity A brief tour Miguel Zumalac´rregui a Instituto de F´ ısica Te´rica IFT-UAM-CSIC o IFT-UAM Cosmology meeting IFT, February 2013, Madrid Miguel Zumalac´rregui a Modified Gravity
- 2. Theory Observations Outline 1 Theory Introduction Modified Gravities 2 Observations Solar System Cosmology 3) Conclusions Miguel Zumalac´rregui a Modified Gravity
- 3. Theory Introduction Observations Modified Gravities Introduction 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. Theory Introduction Observations Modified Gravities Introduction 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. Theory Introduction Observations Modified Gravities Einstein’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. Theory Introduction Observations Modified Gravities Einstein’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. Theory Introduction Observations Modified Gravities Beyond Einstein’s Theory: Examples Higher derivatives: f (R) gravity −→ Equivalent to h(φ)R + · · · Miguel Zumalac´rregui a Modified Gravity
- 8. Theory Introduction Observations Modified Gravities Beyond 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. Theory Introduction Observations Modified Gravities Beyond 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. Theory Introduction Observations Modified Gravities Beyond 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. Theory Introduction Observations Modified Gravities Scalar-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. Theory Introduction Observations Modified Gravities Scalar-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. Theory Introduction Observations Modified Gravities Scalar-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. Theory Solar System Observations Cosmology Local 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. Theory Solar System Observations Cosmology Local 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. Theory Solar System Observations Cosmology Screening 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. Theory Solar System Observations Cosmology Screening 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. Theory Solar System Observations Cosmology Cosmology 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. Theory Solar System Observations Cosmology Cosmology 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. Theory Solar System Observations Cosmology Linear 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. Theory Solar System Observations Cosmology Linear 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. Theory Solar System Observations Cosmology Linear 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. Theory Solar System Observations Cosmology Non-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. Theory Solar System Observations Cosmology Dynamical 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. Theory Solar System Observations Cosmology Dynamical 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. Theory Solar System Observations Cosmology Dynamical 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. Theory Solar System Observations Cosmology Dynamical 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. Theory Solar System Observations Cosmology Dynamical 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. Theory Solar System Observations Cosmology Theory 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. Theory Solar System Observations Cosmology Theory 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. Theory Solar System Observations Cosmology Conclusions 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. Theory Solar System Observations Cosmology Backup Slides Miguel Zumalac´rregui a Modified Gravity
- 33. Theory Solar System Observations Cosmology The 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. Theory Solar System Observations Cosmology Most 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