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Gravity and Scalar Fields Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham) µm 1AU 15Mpc Quantum Gravity General Relativity General unknown corrections? General Relativity plus General General Relativity plus General Dark Matter and Dark Energy Dark Matter General Relativity plus Dark Matter General 35 30 25 20 15 10 5 0 5 10 15 20 25 10− 10− 10− 10− 10− 10− 10− 10 10 10 10 10 10 [m] 7th Aegean Summer School, Paros, Sep 23rd 2013 Thomas P. Sotiriou GR + ?? GR GR + DM + DE 10 5 0 5 10 15 20 25 10− 10− 10 10 10 10 10 10 [m] We should not just wait for the right effective action to come from fundamental theory! Instead we should: “Interpret” experiments Combine them with theory (technical naturalness, effective field theory, ...) Construct interesting toy theories (classical alternatives to GR) Use them to “re-interpret” experiments and give feedback to QG model building There are also large scale puzzles that call for an answer! Can it be gravity? 7th Aegean Summer School, Paros, Sep 23rd 2013 Thomas P. Sotiriou General Relativity The action for general relativity is 1 S = d4x√ g (R 2Λ) + S (gµν ,ψ) 16πG − − m S m has to match the standard model in the local frame and minimal coupling with the metric is required by the equivalence principle. Gravitational action is uniquely determined thanks to: 1. Diffeomorphism invariance 2. Requirement to have second order equations 3. Requirement to have 4 dimensions 4. Requirement to have no other fields Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Beyond GR To go beyond GR one can Add extra fields Allow for more dimensions EFT equivalent: adding extra fields Allow for higher order equations Classically equivalent to adding extra fields Give up diffeomorphism invariance Classically equivalent to adding extra fields Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Taming the extra fields Extra fields can lead to problems Classical instabilities Quantum mechanical instabilities (negative energy) These are particularly hard problems when the fields are nonminimally coupled to gravity. Supposing that these have been tamed, the major problem becomes to hide the extra fields in regimes where GR works well make them re-appear when GR seems to fail Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Brans-Dicke theory The action of the theory is ω S = d4x√ g ϕR 0 µϕ ϕ V (ϕ)+L (g ,ψ) BD − − ϕ ∇ ∇µ − m µν and the corresponding field equations are ω 1 G = 0 ϕ ϕ g λϕ ϕ µν ϕ2 ∇µ ∇ν − 2 µν ∇ ∇λ 1 V (ϕ) + ( ϕ g ϕ) g ϕ ∇µ∇ν − µν − 2ϕ µν (2ω + 3) ϕ = ϕV 2V 0 − Solutions with constant ϕ are admissible and are GR solutions. Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Brans-Dicke theory However, they are not the only ones. E.g. for V = m2 (ϕ ϕ )2 − 0 around static, spherically symmetric stars a nontrivial configuration is necessary and 2ϕ0 2ω0 +3 exp[ mr] h 2ω0+3 γ ii|i=j = − − ≡ h00 2ϕ0 2ω0 +3+exp[ mr] − 2ω0+3 So, hiding the scalar requires, either a very large mass (short range) or a very large Brans-Dicke parameter Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Scalar-tensor theory The action for scalar-tensor theory is ω(ϕ) S = d4x√ g ϕR µϕ ϕ V (ϕ)+L (g , ψ) ST − − ϕ ∇ ∇µ − m µν Makes sense as an EFT for a scalar coupled nonminimally to gravity The generalization ω 0 ω ( ϕ ) does not really solve the problem with the range→ of the force. If the mass of the scalar is large then no large scale phenomenology There are interesting theories with large or zero mass though. Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Spontaneous scalarization Assume you have a theory without a potential which admits solutions such that ω(φ ) 0 →∞ Then the theory will admit GR solutions around matter! However they will not necessarily be the only ones... The non-GR configuration is preferred for sufficiently large central density T. Damour and G. Esposito-Farese, Phys. Rev. Lett. 70, 2220 (1993) Celebrated demonstration that strong field effects can be very important... Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Chameleons The scalar field satisfies the following equation 2 (2ω + 3) ϕ = ω( ϕ) + ϕV 2V + T − ∇ − ϕ is sourced by matter The mass is affected by the presence of matter Sun Potentials can be designed to have large mass locally Strong influence by the boundary conditions J. Khoury and A. Weltman, Phys. Rev. Lett. 93, 171104 (2004) Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Einstein frame Under a conformal transformation and a scalar field redefinition 2 gµν = A (φ)˜gµν φ = φ[ϕ] the action of scalar tensor theory becomes 1 S = d4x g˜ R˜ ˜ µφ ˜ φ U(φ)+L (g , ψ) ST − − 2∇ ∇µ − m µν and the equation of motion for φ is 3 ˜ φ U (φ)+A (φ) A(φ)T =0 − so there is an effective potential U U(φ) A4(φ)T/4 eff ≡ − Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Symmetrons In spherical symmetry 2 d 2 d 3 φ + φ = U + A Aρ dr2 r dr Assume that 1 1 1 U(φ)= µ2φ2 + λφ4 A(φ)=1+ φ2 −2 4 2M 2 Then the effective potential is 1 ρA3 1 U = µ2 φ2 + λφ4 eff 2 M 2 − 4 Spontaneous symmetry breaking when ρ =0 Vanishing VEV when ρA3 >M2µ2 K. Hinterbichler and J. Khoury, Phys. Rev. Lett. 104, 231301 (2010) Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Dynamical DE models Usual shortcomings: They usually don’t solve the old CC problem. Powerful no-go theorem by Weinberg! (modulo exceptions) S. Weinberg, Rev. Mod. Phys. 61, 1 (1989) They usually do not solve the new CC problem either! Turning a tiny cosmological constant to a tiny mass of a field, a tiny coupling for some extra terms, a strange potential, etc. is not really a solution It is only meaningful to “rename” the scale of the CC if this makes it (technically) natural! Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Generalized Galileons One can actually have terms in the action with more than 2 derivatives and still have second order equations: δ((∂φ)2 φ) = 2[(∂µ∂ν φ)(∂ ∂ φ) ( φ)2]δφ µ ν − Inspired by galileons: scalar that enjoy galilean symmetry Most general action given by Horndeski in 1974! It includes well-know terms, such as µν αβµν µν 2 ( µφ)( ν φ)G φ R R 4R R + R ∇ ∇ αβµν − µν The new terms are highly non-linear and can lead to Vainshtein screening A. Nicolis, R. Rattazzi and E. Trincherini, Phys. Rev. D 79, 064036 (2009) G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974) Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Ostrogradksi instability ∂L d ∂L d2 ∂L L = L(q, q,˙ q¨) + =0 → ∂q − dt ∂q˙ dt2 ∂q¨ Four pieces of initial data, four canonical coordinates ∂L d ∂L ∂L Q = q, P = ; Q =˙q, P = 1 1 ∂q˙ − dt ∂q¨ 2 2 ∂q¨ The corresponding hamiltonian is H = P1Q2 + P2q¨ + L(Q1,Q2, q¨)¨q =¨q(Q1,Q2,P2) Linear in P1 Unbound from below Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 f(R) gravity as an exception S = d4x√ gf(R) − can be written as S = d4x√ g [f(φ)+ϕ(R φ)] − − Variation with respect to φ yields ϕ = f (φ) One can then write 4 S = d x√ g [ϕR V (ϕ)] V (ϕ)=f(φ) f (φ)φ − − − Brans-Dicke theory with ω 0 =0 and a potential Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Symmetries and fields Consider the equation (in flat spacetime) flat φ =0 and the set of equations φ =0 Rµνκλ =0 The first equation is invariant only for inertial observers The second set is observer independent However, R =0 g = η µνκλ ⇒ µν µν All solutions for the new field break the symmetry! Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Einstein-aether theory The action of the theory is 1 S = d4x√ g( R M αβµν u u ) æ 16πG − − − ∇α µ∇β ν æ where αβµν αβ µν αµ βν αν βµ α β M = c1g g + c2g g + c3g g + c4u u gµν and the aether is implicitly assumed to satisfy the constraint µ u uµ =1 Most general theory with a unit timelike vector field which is second order in derivatives T. Jacobson and D. Mattingly, Phys. Rev. D 64, 024028 (2001) Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 Hypersurface orthogonality ∂ T Now assume u = α α µν g ∂µT ∂ν T and choose T as the time coordinate TT 1/2 uα = δαT (g )− = NδαT Replacing in the action and defining one gets ho 1 3 √ ij 2 (3) i Sæ = dT d xN h Kij K λK +ξ R + ηa ai 16πGH − with a i = ∂ i ln N and the parameter correspondence G 1 1+c c H = ξ = λ = 2 η = 14 G 1 c 1 c 1 c æ − 13 − 13 − 13 T. Jacobson, Phys. Rev. D 81, 101502 (2010) Thomas P. Sotiriou - 7th Aegean Summer School, Paros, Sep 23rd 2013 GR 5 0 5 10 10− 10 10 10 [m] 10 10 stellar and − intermediate mass neutron stars black holes 20 10− Gravity Probe B double pulsar Cavendish Curvature 30 experiment 10− lunar ranging Mercury’s precession 40 10− 2 [m− ] 7th Aegean Summer School, Paros, Sep 23rd 2013 Thomas P.
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