An all-scale exploration of alternative theories of gravity. Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste

Transcription

1 An all-scale exploration of alternative theories of gravity Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste

2 General Outline Beyond GR: motivation and pitfalls Alternative theories of gravity: theory and phenomenology Testing gravity with compact objects

3 Alternative theories of gravity: theory and phenomenology Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste

4 General Relativity The action for general relativity is S = 1 d 4 x g (R 2Λ) + S m (g µν,ψ) 16πG 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

5 Scalar-tensor theory The action for scalar-tensor theory is S ST = d 4 x g ϕr ω(ϕ) ϕ µ ϕ µ ϕ V (ϕ)+l m (g µν, ψ) 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.

6 Chameleons The scalar field satisfies the following equation (2ω + 3)ϕ = ω ( ϕ) 2 + ϕv 2V + T ϕ is sourced by matter Sun The mass is affected by the presence of matter Potentials can be designed to have large mass locally Strong influence by the boundary conditions J. Khoury and A. Weltman, Phys. Rev. Lett. 93, (2004)

7 Einstein frame Under a conformal transformation and a scalar field redefinition g µν = A 2 (φ) g µν φ = φ[ϕ] the action of scalar tensor theory becomes S ST = d 4 x 1 g R 2 µ φ µ φ U(φ)+L m (g µν, ψ) and the equation of motion for φ φ U (φ)+a 3 (φ) A (φ)t =0 so there is an effective potential is U eff U(φ) A 4 (φ)t/4

8 Symmetrons In spherical symmetry Assume that d 2 dr 2 φ + 2 r d dr φ = U + A 3 A ρ U(φ) = 1 2 µ2 φ λφ4 A(φ) =1+ 1 2M 2 φ2 Then the effective potential is U eff = 1 ρa 3 2 M 2 µ2 φ λφ4 Spontaneous symmetry breaking when Vanishing VEV when ρa 3 >M 2 µ 2 ρ =0 K. Hinterbichler and J. Khoury, Phys. Rev. Lett. 104, (2010)

9 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)( µ ν φ) (φ) 2 ]δφ Inspired by galileons: scalar that enjoy galilean symmetry Most general action given by Horndeski in 1974! It includes well-know terms, such as ( µ φ)( ν φ)g µν φ R αβµν R αβµν 4R µν R µν + R 2 The new terms are highly non-linear and can lead to Vainshtein screening A. Nicolis, R. Rattazzi and E. Trincherini, Phys. Rev. D 79, (2009) G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974)

10 Ostrogradksi instability L = L(q, q, q) L q d dt L q + d2 dt 2 L q =0 Four pieces of initial data, four canonical coordinates Q 1 = q, P 1 = L q d dt L q ; Q 2 = q, P 2 = L q The corresponding hamiltonian is H = P 1 Q 2 + P 2 q + L(Q 1,Q 2, q) q = q(q 1,Q 2,P 2 ) Linear in P 1 Unbound from below

11 f(r) gravity as an exception S = d 4 x gf(r) can be written as S = d 4 x g [f(φ)+ϕ(r φ)] Variation with respect to φ yields ϕ = f (φ) One can then write S = d 4 x g [ϕr V (ϕ)] V (ϕ) =f(φ) f (φ)φ Brans-Dicke theory with ω 0 =0 and a potential

12 Symmetries and fields Consider the equation (in flat spacetime) and the set of equations flat φ =0 φ =0 R µνκλ =0 The first equation is invariant only for inertial observers The second set is observer independent However, R µνκλ =0 g µν = η µν All solutions for the new field break the symmetry!

13 Einstein-aether theory The action of the theory is 1 S æ = d 4 x g( R M αβµν α u µ β u ν ) 16πG æ where M αβµν = c 1 g αβ g µν + c 2 g αµ g βν + c 3 g αν g βµ + c 4 u α u β g µν and the aether is implicitly assumed to satisfy the constraint u µ u µ =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, (2001).

14 Hypersurface orthogonality Now assume u α = α T g µν µ T ν T and choose T as the time coordinate u α = δ αt (g TT ) 1/2 = Nδ αt Replacing in the action and defining one gets Sæ ho 1 = dt d 3 xn h K ij K ij λk 2 +ξ (3) R + ηa i a i 16πG H with a i = i ln N G H G æ = ξ = and the parameter correspondence 1 1 c 13 λ = 1+c 2 1 c 13 η = c 14 1 c 13 T. Jacobson, Phys. Rev. D 81, (2010).

15 Horava-Lifshitz gravity The action of the theory is 1 S HL = dt d 3 xn h( L πG H M 2 where L 2 = K ij K ij λk 2 + ξ (3) R + ηa i a i L M 4 L 6 ) L 4 : contains all 4th order terms constructed with the induced metric and h ij a i L 6 : contains all 6th order terms constructed in the same way P. Hořava, Phys. Rev. D 79, (2009) D. Blas, 0. Pujolas and S. Sibiryakov, Phys. Rev. Let. 104, (2010) T.P.S., J. Phys. Conf. Ser. 283, (2011), arxiv: [hep-th]

16 Horava-Lifshitz gravity Higher order terms contain higher order spatial derivatives: higher order dispersion relations! They modify the propagator and render the theory power-counting renormalizable All terms consistent with the symmetries will be generated by radiative corrections This version of the theory is viable so far Low energy limit is h.o. Einstein-aether theory!

17 Covariance and fine tuning Equivalence restricted to low energy action, but extendible Construct the most general action for ae-theory at a given order Lower order equivalence gives you the prescription of how to introduce the preferred foliation Go to the next order, identify the terms that match, set the rest of the couplings to zero/desired value Effectively, one can covariantize Horava gravity But seen as a covariant theory, it would look severely fine tuned! T. P. Sotiriou, M. Visser and S. Weinfurtner, Phys. Rev. D 83, (2011)

18 Constraints Classical stability and positivity of energy (no ghosts) λ < 1 3 λ > 1, 2ξ > η > 0 Avoidance of vacuum Cherenkov radiation by matter = superluminal excitations in gravity sector Agreement with Solar system experiments (vanishing preferred frame parameters) α 1 = α 1 (λ, ξ, η) < 10 4, α 2 = α 1 (λ, ξ, η) < 10 7 D. Blas, O. Pujolas and S. Sibiryakov, JHEP 1104, 018 (2011)

19 Strong coupling The low energy action exhibits strong coupling at energy M sc = f( λ 1, η)m pl Can be a large energy scale, but problem with renormalizability! A. Papazoglou and T. P. S., Phys. Lett. B 685, 197 (2010) I. Kimpton and A. Padilla, JHEP 1007, 014 (2010) Strong coupling problem can be circumvented if M sc >M D. Blas, O. Pujolas and S. Sibiryakov, Phys.Lett. B 688, 350 (2010) But then potential tension with observations! GeV >M >M obs A. Papazoglou and T. P. Sotiriou, Phys. Lett. B 685, 197 (2010)

20 Percolation of LV But what about the matter sector and lower order operators? Different speeds for different fields in the IR, with logarithmic running! Possible ways out: R. Iengo, J. G. Russo and M. Serone, JHEP 0911, 020 (2009) Some extra symmetry, e.g. supersymmetry S. Groot Nibbelink and M. Pospelov, Phys. Rev. Lett. 94, (2005) Assume Lorentz symmetry in matter and let the weak coupling to gravity (the Lorentz-violating sector) do the rest M. Pospelov and Y. Shang, Phys. Rev. D 85, (2012)

21 Hierarchy of scales Consider the dispersion relation E 2 = m 2 + p 2 + η 4 p 4 M 2 LV + O( p6 M 4 LV ) Assume that there is a universal LV scale, so M LV M Constraint from synchrotron radiation from the Crab Nebula: M obs > GeV M cannot be a universal scale! S. Liberati, L. Maccione and T. P. S., Phys. Rev. Lett. 109, (2012)

22 Summary Higher order equation means extra degrees of freedom Less symmetry means extra degrees of freedom There are innovative ways to hide these degrees of freedom (at least for scalars) It is hard to guess fundamental symmetries from an effective theory! Combined constraints can be very, very efficient!