Modified Gravity - a brief tour

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


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


TheoryObservations


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


TheoryObservations


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...


TheoryObservations


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...


TheoryObservations


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


TheoryObservations







? Extra dimensions:




� R



TheoryObservations







? Extra dimensions:




� R



TheoryObservations







? Extra dimensions:




� R



TheoryObservations


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


TheoryObservations












TheoryObservations












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/


TheoryObservations


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/


TheoryObservations


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)


TheoryObservations


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)


TheoryObservations


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


TheoryObservations


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


TheoryObservations


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).


TheoryObservations





a2 , δ ≡ δρρ



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





TheoryObservations





a2 , δ ≡ δρρ



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





TheoryObservations


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!


TheoryObservations


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


TheoryObservations














TheoryObservations














TheoryObservations














TheoryObservations














TheoryObservations


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

· · ·


TheoryObservations


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

· · ·


TheoryObservations


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


TheoryObservations


Backup Slides


TheoryObservations


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


TheoryObservations


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

Documents

Modified Gravity - a brief tour