Modified Gravity - a brief tour
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TheoryObservations
Modified GravityA brief tour
Miguel Zumalacarregui
Instituto de Fısica Teorica IFT-UAM-CSIC
IFT-UAM Cosmology meeting
IFT, February 2013, Madrid
Miguel Zumalacarregui Modified Gravity
![Page 2: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/2.jpg)
TheoryObservations
Outline
1 TheoryIntroductionModified Gravities
2 ObservationsSolar SystemCosmology
3) Conclusions
Miguel Zumalacarregui Modified Gravity
![Page 3: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/3.jpg)
TheoryObservations
IntroductionModified 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 Zumalacarregui Modified Gravity
![Page 4: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/4.jpg)
TheoryObservations
IntroductionModified 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 Zumalacarregui Modified Gravity
![Page 5: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/5.jpg)
TheoryObservations
IntroductionModified Gravities
Einstein’s Theory
Lovelock’s Theorem (1971)
gµν + Local + 4-D + Lorentz Theory with 2nd order Eqs∗
√−g 1
16πG(R− 2Λ)
∗ 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 Zumalacarregui Modified Gravity
![Page 6: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/6.jpg)
TheoryObservations
IntroductionModified Gravities
Einstein’s Theory
Lovelock’s Theorem (1971)
gµν + Local + 4-D + Lorentz Theory with 2nd order Eqs∗
√−g 1
16πG(R− 2Λ)
∗ 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 Zumalacarregui Modified Gravity
![Page 7: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/7.jpg)
TheoryObservations
IntroductionModified 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.
? Weird stuff: Non-local ⊃ R e−�/M2∗
� R
- Lorentz violation: Horava-Lifschitz gravity �ξ → −ξ + ~∇4ξ
Miguel Zumalacarregui Modified Gravity
![Page 8: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/8.jpg)
TheoryObservations
IntroductionModified 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.
? Weird stuff: Non-local ⊃ R e−�/M2∗
� R
- Lorentz violation: Horava-Lifschitz gravity �ξ → −ξ + ~∇4ξ
Miguel Zumalacarregui Modified Gravity
![Page 9: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/9.jpg)
TheoryObservations
IntroductionModified 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.
? Weird stuff: Non-local ⊃ R e−�/M2∗
� R
- Lorentz violation: Horava-Lifschitz gravity �ξ → −ξ + ~∇4ξ
Miguel Zumalacarregui Modified Gravity
![Page 10: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/10.jpg)
TheoryObservations
IntroductionModified 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.
? Weird stuff: Non-local ⊃ R e−�/M2∗
� R
- Lorentz violation: Horava-Lifschitz gravity �ξ → −ξ + ~∇4ξ
Miguel Zumalacarregui Modified Gravity
![Page 11: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/11.jpg)
TheoryObservations
IntroductionModified 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 Zumalacarregui Modified Gravity
![Page 12: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/12.jpg)
TheoryObservations
IntroductionModified 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 Zumalacarregui Modified Gravity
![Page 13: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/13.jpg)
TheoryObservations
IntroductionModified 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 Zumalacarregui Modified Gravity
![Page 14: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/14.jpg)
TheoryObservations
Solar SystemCosmology
Local Gravity Tests
Transform to Einstein-frame: L =√−gR
16πG+ Lm (gµν [φ])︸ ︷︷ ︸
matter metric
+Lφ
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 Zumalacarregui Modified Gravity
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TheoryObservations
Solar SystemCosmology
Local Gravity Tests
Transform to Einstein-frame: L =√−gR
16πG+ Lm (gµν [φ])︸ ︷︷ ︸
matter metric
+Lφ
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 Zumalacarregui Modified Gravity
![Page 16: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/16.jpg)
TheoryObservations
Solar SystemCosmology
Screening Mechanisms�� ��Non-linear interactions → Hide φ around massive bodies
Screening from V (φ)
Chameleon: ρ dependent field range: φ ∝ e−mφr
r
Symmetron: ρ dependent coupling to matter
Only surface contribution from screened objects: Qφ � QG.
Screening from ∇∇φ
Vainshtein: interaction suppressed for r � rV =(rsm2∗
) 13
significant scalar force for r > rV : Qφ ≈ QGDisformal: field evolution independent of ρ (if ρ� m4
∗)
(Lam Hui’s lectures: www.slideshare.net/CosmoAIMS/hui-modified-gravity)
Miguel Zumalacarregui Modified Gravity
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TheoryObservations
Solar SystemCosmology
Screening Mechanisms�� ��Non-linear interactions → Hide φ around massive bodies
Screening from V (φ)
Chameleon: ρ dependent field range: φ ∝ e−mφr
r
Symmetron: ρ dependent coupling to matter
Only surface contribution from screened objects: Qφ � QG.
Screening from ∇∇φ
Vainshtein: interaction suppressed for r � rV =(rsm2∗
) 13
significant scalar force for r > rV : Qφ ≈ QGDisformal: field evolution independent of ρ (if ρ� m4
∗)
(Lam Hui’s lectures: www.slideshare.net/CosmoAIMS/hui-modified-gravity)
Miguel Zumalacarregui Modified Gravity
![Page 18: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/18.jpg)
TheoryObservations
Solar SystemCosmology
Cosmology
? Scalars can source cosmic acceleration:
Effective Cosmological Constant: Λ→ V (φ) + 12(∂φ)2
Self-acceleration: H ≈ constant is solution.
Einstein frame: Energy transfer
∇µTµνm = −∇µTµνφ = −Qφ,ν
? Geometric measurements (DL, DA) can’t distinguish
dark energy (Q = 0) from modified gravity (Q 6= 0)
? Perturbations: Additional force if Q 6= 0
Miguel Zumalacarregui Modified Gravity
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TheoryObservations
Solar SystemCosmology
Cosmology
? Scalars can source cosmic acceleration:
Effective Cosmological Constant: Λ→ V (φ) + 12(∂φ)2
Self-acceleration: H ≈ constant is solution.
Einstein frame: Energy transfer
∇µTµνm = −∇µTµνφ = −Qφ,ν
? Geometric measurements (DL, DA) can’t distinguish
dark energy (Q = 0) from modified gravity (Q 6= 0)
? Perturbations: Additional force if Q 6= 0
Miguel Zumalacarregui Modified Gravity
![Page 20: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/20.jpg)
TheoryObservations
Solar SystemCosmology
Linear Perturbations
Quasi-static approximation on sub-horizon scales
Neglect time derivatives, keep terms ∝ k2
a2 , δ ≡ δρρ
δ + 2Hδ ≈ 4π Geff(k, t) ρmδ (effective gravitational constant)
Φ = − η(k, t) Ψ (anisotropic parameter)
f(R) gravity:Geff
G=
1
f ′1 + 4(f ′′/f ′)(k/a)2
1 + 3(f ′′/f ′)(k/a)2, η =
1 + 2(f ′′/f ′)(k/a)2
1 + 4(f ′′/f ′)(k/a)2
43
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 Zumalacarregui Modified Gravity
![Page 21: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/21.jpg)
TheoryObservations
Solar SystemCosmology
Linear Perturbations
Quasi-static approximation on sub-horizon scales
Neglect time derivatives, keep terms ∝ k2
a2 , δ ≡ δρρ
δ + 2Hδ ≈ 4π Geff(k, t) ρmδ (effective gravitational constant)
Φ = − η(k, t) Ψ (anisotropic parameter)
f(R) gravity:Geff
G=
1
f ′1 + 4(f ′′/f ′)(k/a)2
1 + 3(f ′′/f ′)(k/a)2, η =
1 + 2(f ′′/f ′)(k/a)2
1 + 4(f ′′/f ′)(k/a)2
43
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 Zumalacarregui Modified Gravity
![Page 22: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/22.jpg)
TheoryObservations
Solar SystemCosmology
Linear Perturbations
Quasi-static approximation on sub-horizon scales
Neglect time derivatives, keep terms ∝ k2
a2 , δ ≡ δρρ
δ + 2Hδ ≈ 4π Geff(k, t) ρmδ (effective gravitational constant)
Φ = − η(k, t) Ψ (anisotropic parameter)
f(R) gravity:Geff
G=
1
f ′1 + 4(f ′′/f ′)(k/a)2
1 + 3(f ′′/f ′)(k/a)2, η =
1 + 2(f ′′/f ′)(k/a)2
1 + 4(f ′′/f ′)(k/a)2
43
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 Zumalacarregui Modified Gravity
![Page 23: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/23.jpg)
TheoryObservations
Solar SystemCosmology
Non-Linear Perturbations
- Higher order PT very hard, especially beyond GR
fromM. 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 Zumalacarregui Modified Gravity
![Page 24: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/24.jpg)
TheoryObservations
Solar SystemCosmology
Dynamical Observables: Matter and Light
? Large Scale Structure:
P (k) → linear & non-linear, limited by bias
Peculiar velocities/RSD → f = d log(δ)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 Zumalacarregui Modified Gravity
![Page 25: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/25.jpg)
TheoryObservations
Solar SystemCosmology
Dynamical Observables: Matter and Light
? Large Scale Structure:
P (k) → linear & non-linear, limited by bias
Peculiar velocities/RSD → f = d log(δ)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 Zumalacarregui Modified Gravity
![Page 26: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/26.jpg)
TheoryObservations
Solar SystemCosmology
Dynamical Observables: Matter and Light
? Large Scale Structure:
P (k) → linear & non-linear, limited by bias
Peculiar velocities/RSD → f = d log(δ)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 Zumalacarregui Modified Gravity
![Page 27: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/27.jpg)
TheoryObservations
Solar SystemCosmology
Dynamical Observables: Matter and Light
? Large Scale Structure:
P (k) → linear & non-linear, limited by bias
Peculiar velocities/RSD → f = d log(δ)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 Zumalacarregui Modified Gravity
![Page 28: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/28.jpg)
TheoryObservations
Solar SystemCosmology
Dynamical Observables: Matter and Light
? Large Scale Structure:
P (k) → linear & non-linear, limited by bias
Peculiar velocities/RSD → f = d log(δ)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 Zumalacarregui Modified Gravity
![Page 29: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/29.jpg)
TheoryObservations
Solar SystemCosmology
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 stepsor at least keep track of assumptions:
Poisson eq. Φ 6= 4πk2GρkMatter geodesics xi 6= −∇iΦGalaxy biasCalibration with simulations
· · ·
Miguel Zumalacarregui Modified Gravity
![Page 30: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/30.jpg)
TheoryObservations
Solar SystemCosmology
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 stepsor at least keep track of assumptions:
Poisson eq. Φ 6= 4πk2GρkMatter geodesics xi 6= −∇iΦGalaxy biasCalibration with simulations
· · ·
Miguel Zumalacarregui Modified Gravity
![Page 31: Modified Gravity - a brief tour](https://reader034.fdocuments.us/reader034/viewer/2022042602/5597deeb1a28ab52388b471f/html5/thumbnails/31.jpg)
TheoryObservations
Solar SystemCosmology
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 bearassumptions in mind
Doubts? check the Bible of modified gravity:
- Clifton et al. 2011 ”Modified Gravity and Cosmology” 1106.2476
Miguel Zumalacarregui Modified Gravity
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TheoryObservations
Solar SystemCosmology
Backup Slides
Miguel Zumalacarregui Modified Gravity
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TheoryObservations
Solar SystemCosmology
The Frontiers of Gravity
What is the most general possible theory of gravity?
Ostrogradski’s Theorem (1850)
Theories with L ⊃ ∂nq
∂tn, n ≥ 2 are unstable∗
q(t), L(q, q, q)→ ∂L
∂q− d
dt
∂L
∂q+
d2
dt2∂L
∂q= 0
q, q, q,...q → Q1, Q2, P1, P2
(P1,2 ≡ ∂L/∂Q1,2
)H = P1Q2 + terms independent of P1
∗ If no...q ,
....q in the Equations ⇒ Loophole
Miguel Zumalacarregui Modified Gravity
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TheoryObservations
Solar SystemCosmology
Most General Scalar-Tensor theory
Horndenski’s Theory (1974)
gµν +�� ��φ + Local + 4-D + Lorentz Theory with 2nd order Eqs.
⇒ ∃ 4 free functions of φ, X ≡ −12φ,µφ
,µ
L2 = G2(X,φ) −→ No φ↔ gµν interaction
L3 = −G3�φ −→ eqs ⊃ G3,XRµνφ,µφ,ν
L4 = G4R +G4,X
[(�φ)2 − φ;µνφ
;µν]
L5 = G5Gµνφ;µν
− 16G5,X
[(�φ)3 − 3(�φ)φ;µνφ
;µν + 2φ ;ν;µ φ ;λ
;ν φ ;µ;λ
]Miguel Zumalacarregui Modified Gravity