Modifications of gravity Constantinos Skordis (Perimeter Institute) - PowerPoint PPT Presentation

Modifications of gravity Constantinos Skordis (Perimeter Institute) GGI, 4 Mar 2009

Tensor-Vector-Scalar theory (Sanders 1997, Bekenstein 2004) Ingredients • Tensor field (metric) ˜ g ab • Unit-timelike Vector field A a • Scalar field φ Physical g ab = e − 2 φ (˜ g ab + A a A b ) − e 2 φ A a A b metric • toy-theory (phenomenological) • gives MOND in the non-relativistic limit • good platform for studying alternatives to CDM Λ See topical review in CQG ( C.S. on arXiv next week)

Growth of structure in CDM and TeVeS CDM Λ growth in CDM sources baryons small Φ − Ψ TeVeS A i = � ∇ i α growth in vector sources baryons large Φ − Ψ (C.S., D. Mota, P . Ferreira, C.Boehm, 2005) (S.Dodelson & M.Liguori, 2006)

Lessons from TeVeS • gravity may depend on additional fields • these fields may mimic dark matter • however, some other effect different from dark matter may appear example : Both CDM and TeVeS give same P ( k ) Λ But Φ − Ψ is very different Can be distinguished from combinations of other observables

How special is General Relativity? • A principle theory (SEP , diffeo-invariance) • Lovelock-Grigore theorem : In any dimension, the only local diffeomorphism invariant action leading to 2nd order field equations and which depends only on a metric is a linear combination of the E-H action with a cosmological constant up to a total derivative. Any other theory must : (at least one applies) • be non-local (Sousa-Woodard, Dvali-Gabadaze-Porrati, etc) • Have absolute elements (stratified theories) • Depend on other fields (JFBD, TeVeS, etc) • have higher than 2nd order field equations (Weyl gravity)

What is/not gravity? • “f(R) and scalar-tensor theories are not modified gravity because they are equivalent to GR + scalar” • “Scalar-tensor and TeVeS theories are not modified gravity because they depend on extra fields” • TeVeS is not modified gravity because it can be written in a single-metric frame without coupling to matter. d 4 x √− gR − d 4 x √− g K abcd ( g, B c ) ∇ a B b ∇ c B d + S m [ g ] 1 � � S = 16 πG • “If then we have modified gravity” Φ − Ψ � = 0 these are certainly all incorrect statements

Cosmological tests of gravity How do we treat complicated theories of gravity? g cd , Riem, Ric, R, φ A , . . . T cd , g cd , Ric, R, φ A , . . . � � � � f ab = 8 πGh ab extra fields matter metric curvature add G ab to both sides Trick : add and subtract on RHS 8 πGT ab G ab = 8 πGT ( known ) Collect terms + U ab ab where U ab = 8 πG [ h ab − T ab ] + G ab − f ab Field equations for φ A Bianchi identity : ∇ a U a b = 0 C.S. (arXiv:0806.1238)

What is gravity “The natural phenomenon of attraction between physical objects with mass or energy.” m Gravity: generated by mass, affects mass m Fluid: contributes only to total energy density Other force: generated by/affects other q q charges To distinguish gravity from fluids or other forces we must specify the field content

Can we distinguish modified gravity from matter at the FRW level? ds 2 = − dt 2 + a 2 d� 2 FRW K G 0 3 H 2 + 3 K : � a 2 = 8 πG ρ i + X 0 i G i − 2¨ : a a − H 2 = 8 πG � P i + Y i i ˙ Bianchi identity gives X + 3 H ( X + Y ) = 0

Can we distinguish modified gravity from matter at the FRW level? ds 2 = − dt 2 + a 2 d� 2 FRW K G 0 3 H 2 + 3 K : � a 2 = 8 πG ρ i + X 0 i G i − 2¨ : a a − H 2 = 8 πG � P i + Y i i ˙ Bianchi identity gives X + 3 H ( X + Y ) = 0 the answer is therefore : NO

Linear perturbation level C.S. (arXiv:0806.1238) , , , Metric has 4 scalar dof : ζ Ψ Φ ν Given a vector field s.t. ξ µ = a ( − ξ, � ξ a ∇ i ψ ) reduced to 2 by gauge transformations Distinguishing gravity from fluids : field content • Linearized equations must be gauge form-invariant. � � δG µ ν = O i ∆ i + [ FRWeq. ] ξ O i ∆ i → i i • Holds iff background equations satisfied. • Fixes all gauge non-invariant terms • Bianchi identity holds (local energy conservation)

Parameterizing field equations ∇ 2 + 3 K )(Φ − � a 2 a ( ˙ 2( � 3 � ∇ 2 ν ) − 6 ˙ Φ − 1 ∇ 2 ζ ) − 6 ˙ a 2 Ψ = 8 π Ga 2 ρδ a gauge transform ∇ 2 +3 K )(Φ � − � a 2 Ψ � = 8 πGa 2 ρδ � +[ FRWeq. ] ξ Φ � − 1 a 2 2( � a ( ˙ 3 � ∇ 2 ν � ) − 6 ˙ ∇ 2 ζ � ) − 6 ˙ a • Parameterizations in a fixed gauge are inconsistent • Parameterizations using gauge invariant combinations are consistent but may be too arbitrary (from Stewart-Walker lemma) • All parameterizations must take into account the field content.

Distinguishing gravity from fluids at the linear level • Must specify field content • Specify the parameterization • Determine the force between 2 well separated masses in vacuum m m This requires writing an action leading to the parameterized equations

The extended CDM model Λ C.S. (arXiv:0806.1238) Background : CDM Λ No additional fields No higher than 2 time derivatives No Gauge Non-Invariant terms allowed δU a and derivatives contains Φ GI Ψ GI b ζ − ˙ a ν − ˙ ν + ζ ) contains 2nd derivatives Ψ GI = Ψ − ¨ a ( ˙ Φ GI = Φ + 1 ∇ 2 ν + ˙ a � contains 1st derivatives a ( ˙ ν + ζ ) 3

Constructing the tensor U − G ab = 8 πGT ( known ) + U ab ab Constraints : 1st derivatives δU 0 0 = A Φ GI δ U 0 i = B Φ GI Propagation : 2nd derivatives i = C 1 Φ GI + C 2 ˙ δ U i Φ GI + C 3 Ψ GI j − 1 δU i 3 δU k k δ i j = D 1 Φ GI + D 2 ˙ Φ GI + D 3 Ψ GI

Bianchi identity gives B = 1 3 C 2 + 2 A = − ˙ � � a � ∇ 2 + 3 K C 3 = D 3 = 0 D 2 a C 2 3 a B + 2 ˙ a B − 1 3 C 1 − 2 A + ˙ ∇ 2 B + ˙ � � a a ∇ 2 + 3 K � a A − � ˙ ˙ a C 1 = 0 D 1 = 0 3 Corollary 1: If D 1 = D 2 = 0 Then no shear : no modification U a b = 0 Corollary II: If A = B = 0 Then U a b = 0 no constraints : no modification Corollary III: If D 2 = B = 0 i.e. Φ GI − Φ GI = D 1 Φ GI Then U a b = 0 no modification

A simple model : Φ GI − Ψ GI = D 2 ˙ Φ GI B = C 1 = D 1 = 0 C 2 = − βH 2 D 2 = β H 2 A = β H 2 1 0 0 0 � a 2˙ a a ∇ 2 ˙ non-locality new parameter β C.S. (arXiv:0806.1238)

Further consistency requirements (in progress) • Action for parameterized perturbed cosmological equations • Quantize on de Sitter • Eliminate ghosts : Should impose further constraints on the allowed terms • Initial conditions : e.g. Inflation • Modified gravity at the non-linear level (non- linear completion)

The end • Detecting not enough Φ − Ψ � = 0 • Parametrizing only in terms of the potentials can ignore important physics. Gravity may depend on additional fields. • Constraints depend on the field content • Field content important for distinguishing gravity from fluids.