Gravity and scalar fields. Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

Transcription

1 Gravity and scalar fields Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste (...soon at the University of Nottingham)

2 µm 1AU 15Mpc Quantum Gravity General Relativity plus unknown corrections? General Relativity General Relativity plus Dark Matter General Relativity plus Dark Matter and Dark Energy [m] 7th Aegean Summer School, Paros, Sep 23rd 2013 Thomas P. Sotiriou

3 GR +?? GR GR + DM + DE [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

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 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

6 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

7 Brans-Dicke theory The action of the theory is S BD = d 4 x g ϕr ω 0 ϕ µ ϕ µ ϕ V (ϕ)+l m (g µν,ψ) and the corresponding field equations are G µν = ω 0 ϕ 2 µ ϕ ν ϕ 1 2 g µν λ ϕ λ ϕ + 1 ϕ ( µ ν ϕ g µν ϕ) V (ϕ) 2ϕ g µν (2ω 0 + 3)ϕ = ϕv 2V Solutions with constant ϕ are admissible and are GR solutions.

8 Brans-Dicke theory However, they are not the only ones. E.g. for V = m 2 (ϕ ϕ 0 ) 2 around static, spherically symmetric stars a nontrivial configuration is necessary and γ h ii i=j h 00 = 2ω 0 +3 exp[ 2ϕ0 2ω 0 +3 mr] 2ω 0 +3+exp[ 2ϕ0 2ω 0 +3 mr] So, hiding the scalar requires, either a very large mass (short range) or a very large Brans-Dicke parameter

9 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.

10 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...

11 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)

12 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

13 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)

14 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!

15 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)

16 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

17 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

18 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!

19 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)

20 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)

21 GR [m] neutron stars stellar and intermediate mass black holes Curvature Cavendish experiment Gravity Probe B double pulsar lunar ranging Mercury s precession [m 2 ] 7th Aegean Summer School, Paros, Sep 23rd 2013 Thomas P. Sotiriou

22 Neutron star physics description requires 1. gravity (Einstein s equations) 2. Magnetohydrodynamics (MHD equations) 3. Microphysics (equation of state) There is too much uncertainty in 3. to start meddling with 1. Better strategy: Assume GR and try to get insight into the microphysics, i.e. Nuclear physics Superfluidity... Neutron Stars as Nuclear Physics labs

23 Black holes as gravity labs Black holes are fully described by just gravity! No matter No microphysics Simple enough yet full GR solutions: perfect! Additional motivation They contain horizons: causal structure probes They contain singularities: this is exactly were GR is supposed to be breaking down!

24 Are black holes different? Not necessarily! Consider scalar-tensor theory (or f(r) gravity) S st = d 4 x g ϕr ω(ϕ) ϕ µ ϕ µ ϕ V (ϕ)+l m (g µν,ψ) Black holes that are Endpoints of collapse, i.e. stationary isolated, i.e. asymptotically flat are GR black holes! T. P. S. and V. Faraoni, Phys. Rev. Lett. 108, (2012) (generalizes the 1972 proof for Brans-Dicke theory) S.W. Hawking, Comm. Math. Phys. 25, 167 (1972)

25 No difference from GR? Actually there is... Perturbations are different! E. Barausse and T.P.S., Phys. Rev. Lett. 101, (2008) They even lead to new effects, e.g. floating orbits V. Cardoso et al., Phys. Rev. Lett. 107, (2011) Cosmic evolution could also lead to scalar hair T. Jacobson, Phys. Rev. Lett. 83, 2699 (1999) M. W. Horbatsch and C. P. Burgess, JCAP 010, 1205 (2012) More general couplings lead to hairy solutions P. Kanti et al., Phys. Rev. D 54, 5049 (1996)

26 Black holes with matter What if there is matter around a black hole? φ U (φ)+a 3 (φ) A (φ)t =0 In general no φ = constant solutions when T = 0unless A (φ 0 )=0 U (φ 0 )=0 Hint that a small amount of matter might perhaps drastically change the solutions Even when these conditions are satisfied Spontaneous scalarization Superradiant instability V. Cardoso, I. P. Carucci, P. Pani and T. P. S., Phys. Rev. Lett. 111, (2013)

27 Conclusions and Perspectives Scalar-tensor theories are simple, interesting examples of modified gravity A scalar field can be traded of for more derivatives or lack of symmetry There are mechanisms to hide the scalar field from local test, yet have it dominate cosmological behaviour Naturalness is still a problem... Strong gravity systems might be able to reveal the existence of scalar fields Even in theories whose black hole solutions are the same as in GR there is new black hole phenomenology!