Dark Energy The Modified Gravity Perspective Part II: Beyond GR with Screening David F . Mota Institute of Theoretical Astrophysics University of Oslo 2015
Basic Observational Requirements of Modifying Gravity
Modified Gravity as alternative to Dark Energy The need of screening Mechanisms (assuming homogeneity and isotropy) 1 F ( g ) 8 GT ( ) R g R 8 π GT + + π φ − = µ ν µ ν µ ν µ ν µ ν 2 gravity matter Modifying Gravity Sector Modifying Matter Sector (dark matter / dark energy) Often equivalent via conformal or disformal transformations
Scalar-Tensor Theories Extra scalar degree of freedom (fifth force)
What do local tests mean? E.g. scalar-tensor theory - new scalar graviton Theory is GR on local scales Cassini:
Tensor - scalar theories 2 } + S matter [ matter , g µ n A 2 ( j ) g µ n ] S = 16 p G Ú - g { R - 2 ( µ j ) 1 * * * spin 2 spin 0 physical metric ln A( j ) = a 0 ( j – j 0 ) + 1 b 0 ( j – j 0 ) 2 + … matter 2 j j matter | a 0 | j ... j j j ln A( j ) j perihelion 0.035 shift 0.030 curvature b 0 0.025 slope a 0 LLR 0.020 j j 0 VLBI 0.015 0.010 2 ) G eff = G ( 1 + a 0 LLR graviton scalar Cassini 0.005 a 0 a 0 g PPN – 1 a 0 2 b 0 b 0 - 6 - 4 - 2 0 2 4 6 matter j b PPN – 1 a 0 2 b 0 j a 0 a 0 General Relativity 2 1 Vertical axis ( b 0 = 0) : Jordan–Fierz–Brans–Dicke theory a 0 = 2 w BD + 3 Horizontal axis ( a 0 = 0) : perturbatively equivalent to G.R.
Extremely tight constraints on Modified Gravity from experiments at small scales ! 10 -3 cm 1AU 1kpc 1Mpc 1000Mpc GR extra dimensions? MOND? Modified Gravity? Cassini Probe Lunar Laser Range
Modified Gravity as Dark Energy 4 [ ] ( ) d x g f R + L ∫ matter ( ) 4 2 3 d x g R + R R ... + + ∫ ✓ f(R) models are simple ✓ easy to produce acceleration (first inflationary model) ✓ high-energy corrections to gravity likely to introduce higher-order terms ✓ particular case of scalar-tensor and extra-dimensional theory Negative Pressure!
Large Scale Structure Formation: deviations from GR must be small Supernovae + Large Scale Structures + CMBR Supernovae + Large Scale Structures + CMBR + Baryon Oscillations Supernovae Supernovae + Large Scale Structures High-Z Supernovae Search Team Decelerated Expansion Almost LCDM! SDSS Accelerated Expansion PLANCK
How to Modify Gravity and evade constraints? Solar System Bounds (Modified Gravity) Cosmological Bounds Laboratory Bounds (Coupling to Dark Matter) (Coupling to ordinary matter) Amendola PRD (1999)
Screening Mechanisms!
Scalar-Tensor Theories Extra scalar degree of freedom (fifth force)
Quantum Picture: Yukawa Interaction an interaction between a scalar field ϕ and a Dirac field ψ of the type ϕ g g ψ ψ mass of the scalar boson
Yukawa Potential Feynman amplitude of the diagram ϕ g g ψ ψ mass of the scalar boson
Classical Picture: Fifth Force Yukawa potential is equivalent to a scalar field profile (think of Photons and Electromagnetic field) ϕ F φ = �r V Yukawa φ ( r ) = − g 2 e − kmr F φ = �r φ r What sources a scalar field profile? How to compute the scale field profile?
Scalar-Tensor Theories Gravity experiments should see the total force!? GM F total = F G + F φ r / ( ) r (1 e ) − λ Ψ = − + α r range coupling
Screening mechanisms key elements GM r / ( ) r (1 e ) − λ Scalar bosons lead to Yukawa correction to Newton potential: Ψ = − + α r coupling coupling range range α ∼ O (1) ⇒ λ ∼ 0 . 1 mm If extra scalar for gravity, then: Either coupling becomes very small in Solar System or… the range becomes very short in Solar System scale dependent coupling/range! Long et al. Nature (2004)
Range of Fifth Force on Scalar-Tensor Gravity GM r / ( ) r (1 e ) − λ Ψ = − + α range r
Chameleon Screening: range of fifth force depends on local density } Nonlinear mass/range Mpc mm
Chameleon mechanism Range of dark force depends on local environment (Khoury & Weltman 2004) Mpc Kpc mm
Symmetron Screening: coupling of fifth force depends on local density } Nonlinear coupling GM r / ( ) r (1 e ) − λ Ψ = − + α r coupling No coupling! coupled! coupled! V eff H Φ L V eff H Φ L vev vev Φ Φ
Symmetron mechanism Strength of dark force depends on local environment (Hinterbicheler & Khoury 2010)
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