Theory Observations Modified Gravity A brief tour Miguel Zumalac´ arregui Instituto de F´ ısicaTe´oricaIFT-UAM-CSIC IFT-UAM Cosmology meeting IFT, February 2013, Madrid Miguel Zumalac´ arregui Modified Gravity
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
TheoryObservations
Outline
1 TheoryIntroductionModified Gravities
2 ObservationsSolar SystemCosmology
3) Conclusions
Miguel Zumalacarregui Modified Gravity
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
TheoryObservations
Solar SystemCosmology
Backup Slides
Miguel Zumalacarregui Modified Gravity
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
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