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

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