Gravitation modifiée à grande distance & tests dans le système solaire 10 avril 2008

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1 Gravitation modifiée à grande distance et tests dans le système solaire Gilles Esposito-Farèse, GRεCO, IAP et Peter Wolf, LNE-SYRTE 10 avril 2008 Gravitation modifiée à grande distance & tests dans le système solaire 10 avril 2008 Gilles Esposito-Farèse & Peter Wolf

2 Modified gravity at large distances and solar-system tests J.-P. Bruneton and G. Esposito-Farèse GRεCO, Institut d Astrophysique de Paris Phys. Rev. D 76 (2007) Discussions with L. Blanchet, C. Deffayet, B. Fort, G. Mamon, Y. Mellier, M. Milgrom, R. Sanders, J.-P. Uzan, R. Woodard, etc. April 10th, 2008

3 Dark matter and galaxy rotation curves evidences for dark matter: Ω Λ 0.7 (SNIa) and Ω Λ + Ω m 1 (CMB) Ω m 0.3, at least 10 greater than estimates of baryonic matter. Rotation curves of galaxies and clusters: almost rigid bodies v many theoretical candidates for dark matter (e.g. from SUSY) Numerical simulations of structure formation are successful while incorporating (noninteracting, pressureless) dark matter

4 Dark matter and galaxy rotation curves evidences for dark matter: Ω Λ 0.7 (SNIa) and Ω Λ + Ω m 1 (CMB) Ω m 0.3, at least 10 greater than estimates of baryonic matter. Rotation curves of galaxies and clusters: almost rigid bodies v many theoretical candidates for dark matter (e.g. from SUSY) Numerical simulations of structure formation are successful while incorporating (noninteracting, pressureless) dark matter

5 Milgrom s MOND proposal [1983] MOdified Newtonian Dynamics for small accelerations (i.e., at large distances) a = a N = GM r 2 if a > a m.s 2 a = a 0 a N = GMa0 r if a < a 0 Automatically recovers the Tully-Fisher law [1977] v 4 M baryonic Superbly accounts for galaxy rotation curves (but clusters still require some dark matter) [Sanders & McGaugh, Ann. Rev. Astron. Astrophys. 40 (2002) 263]

6 Milgrom s MOND proposal [1983] MOdified Newtonian Dynamics for small accelerations (i.e., at large distances) a = a N = GM r 2 if a > a m.s 2 a = a 0 a N = GMa0 r if a < a 0 Automatically recovers the Tully-Fisher law [1977] v 4 M baryonic Superbly accounts for galaxy rotation curves (but clusters still require some dark matter) [Sanders & McGaugh, Ann. Rev. Astron. Astrophys. 40 (2002) 263] M b v

7 Consistent field theories of MOND? A priori easy to predict a force 1/r : If V(ϕ) = 2a 2 e bϕ, unbounded by below then ϕ = V (ϕ) ϕ = (2/b) ln(abr). Constant coefficient 2/b instead of M. Some papers write actions which depend on the galaxy mass M They are actually using a different theory for each galaxy! Stability Full Hamiltonian should be bounded by below: no tachyon (m 2 0), no ghost (E kinetic 0) Well-posed Cauchy problem Hyperbolic field equations

8 Consistent field theories of MOND? A priori easy to predict a force 1/r : If V(ϕ) = 2a 2 e bϕ, unbounded by below then ϕ = V (ϕ) ϕ = (2/b) ln(abr). Constant coefficient 2/b instead of M. Some papers write actions which depend on the galaxy mass M They are actually using a different theory for each galaxy! Stability Full Hamiltonian should be bounded by below: no tachyon (m 2 0), no ghost (E kinetic 0) Well-posed Cauchy problem Hyperbolic field equations

9 Consistent field theories of MOND? A priori easy to predict a force 1/r : If V(ϕ) = 2a 2 e bϕ, unbounded by below then ϕ = V (ϕ) ϕ = (2/b) ln(abr). Constant coefficient 2/b instead of M. M Some papers write actions which depend on the galaxy mass M They are actually using a different theory for each galaxy! Stability Full Hamiltonian should be bounded by below: no tachyon (m 2 0), no ghost (E kinetic 0) Well-posed Cauchy problem Hyperbolic field equations

10 Consistent field theories of MOND? A priori easy to predict a force 1/r : If V(ϕ) = 2a 2 e bϕ, unbounded by below then ϕ = V (ϕ) ϕ = (2/b) ln(abr). Constant coefficient 2/b instead of M. M Some papers write actions which depend on the galaxy mass M They are actually using a different theory for each galaxy! Stability Full Hamiltonian should be bounded by below: no tachyon (m 2 0), no ghost (E kinetic 0) Well-posed Cauchy problem Hyperbolic field equations

11 Consistent field theories of MOND? A priori easy to predict a force 1/r : If V(ϕ) = 2a 2 e bϕ, unbounded by below then ϕ = V (ϕ) ϕ = (2/b) ln(abr). Constant coefficient 2/b instead of M. M Some papers write actions which depend on the galaxy mass M They are actually using a different theory for each galaxy! Stability Full Hamiltonian should be bounded by below: no tachyon (m 2 0), no ghost (E kinetic 0) Well-posed Cauchy problem Hyperbolic field equations

12 Consistent field theories of MOND? A priori easy to predict a force 1/r : If V(ϕ) = 2a 2 e bϕ, unbounded by below then ϕ = V (ϕ) ϕ = (2/b) ln(abr). Constant coefficient 2/b instead of M. M Some papers write actions which depend on the galaxy mass M They are actually using a different theory for each galaxy! Stability Full Hamiltonian should be bounded by below: no tachyon (m 2 0), no ghost (E kinetic 0) Well-posed Cauchy problem Hyperbolic field equations Causal cone Cauchy surface

13 Most promising framework S = Relativistic AQUAdratic Lagrangians [Bekenstein (TeVeS), Milgrom, Sanders] c 3 d 4 x { } g R 2 f ( µ ϕ µ ϕ) 16πG [ ] +S matter matter ; g µν A 2 (ϕ)g µν + B(ϕ)U µ U ν GMa0 A k-essence kinetic term can yield the MOND force r Matter coupled to the scalar field Disformal term (almost) necessary to predict enough lensing

14 Consistency conditions on f ( µ ϕ µ ϕ) Hyperbolicity of the field equations + Hamiltonian bounded by below x, f (x) > 0 x, 2 x f (x) + f (x) > 0 N.B.: If f (x) > 0, the scalar field propagates faster than gravitons, but still causally no need to impose f (x) 0 These conditions become much more complicated within matter

15 Consistency conditions on f ( µ ϕ µ ϕ) Hyperbolicity of the field equations + Hamiltonian bounded by below x, f (x) > 0 x, 2 x f (x) + f (x) > 0 scalar causal cone graviton causal cone N.B.: If f (x) > 0, the scalar field propagates faster than gravitons, but still causally no need to impose f (x) 0 Cauchy surface These conditions become much more complicated within matter

16 Consistency conditions on f ( µ ϕ µ ϕ) Hyperbolicity of the field equations + Hamiltonian bounded by below x, f (x) > 0 x, 2 x f (x) + f (x) > 0 scalar causal cone graviton causal cone N.B.: If f (x) > 0, the scalar field propagates faster than gravitons, but still causally no need to impose f (x) 0 Cauchy surface These conditions become much more complicated within matter

17 Difficulties of such models Complicated Lagrangians (unnatural) Fine tuning ( fit rather than predictive models): Possible to predict different lensing and rotation curves Discontinuities: can be cured In TeVeS [Bekenstein], gravitons & scalar are slower than photons gravi-cerenkov radiation suppresses high-energy cosmic rays [Moore et al.] Solution: Accept slower photons than gravitons preferred frame (ether) where vector U µ = (1, 0, 0, 0) Maybe not too problematic if U µ is dynamical Vector contribution to Hamiltonian unbounded by below [Clayton] unstable model Post-Newtonian tests very constraining

18 Difficulties of such models Complicated Lagrangians (unnatural) Fine tuning ( fit rather than predictive models): Possible to predict different lensing and rotation curves Discontinuities: can be cured In TeVeS [Bekenstein], gravitons & scalar are slower than photons gravi-cerenkov radiation suppresses high-energy cosmic rays [Moore et al.] Solution: Accept slower photons than gravitons preferred frame (ether) where vector U µ = (1, 0, 0, 0) Maybe not too problematic if U µ is dynamical Vector contribution to Hamiltonian unbounded by below [Clayton] unstable model Post-Newtonian tests very constraining

19 Difficulties of such models Complicated Lagrangians (unnatural) Fine tuning ( fit rather than predictive models): Possible to predict different lensing and rotation curves Discontinuities: can be cured In TeVeS [Bekenstein], gravitons & scalar are slower than photons gravi-cerenkov radiation suppresses high-energy cosmic rays [Moore et al.] Solution: Accept slower photons than gravitons preferred frame (ether) where vector U µ = (1, 0, 0, 0) Maybe not too problematic if U µ is dynamical Vector contribution to Hamiltonian unbounded by below [Clayton] unstable model Post-Newtonian tests very constraining

20 Difficulties of such models Complicated Lagrangians (unnatural) Fine tuning ( fit rather than predictive models): Possible to predict different lensing and rotation curves Discontinuities: can be cured In TeVeS [Bekenstein], gravitons & scalar are slower than photons gravi-cerenkov radiation suppresses high-energy cosmic rays [Moore et al.] Solution: Accept slower photons than gravitons preferred frame (ether) where vector U µ = (1, 0, 0, 0) Maybe not too problematic if U µ is dynamical Vector contribution to Hamiltonian unbounded by below [Clayton] unstable model Post-Newtonian tests very constraining

21 Post-Newtonian constraints Solar-system tests matter a priori weakly coupled to ϕ TeVeS tuned to pass them even for strong matter-scalar coupling Binary-pulsar tests matter must be weakly coupled to ϕ matter α 0 ϕ LLR 10 0 matter-scalar coupling function ln A(ϕ) 10 1 α 0 β 0 < Mercury Cassini β 0 > ALLOWED THEORIES α 0 ϕ β 0 matter ϕ ϕ g µν A 2 (ϕ)g µν general relativity (α 0 = β 0 = 0)

22 Post-Newtonian constraints Solar-system tests matter a priori weakly coupled to ϕ TeVeS tuned to pass them even for strong matter-scalar coupling Binary-pulsar tests matter must be weakly coupled to ϕ matter α 0 ϕ LLR 10 0 matter-scalar coupling function ln A(ϕ) 10 1 α 0 β 0 < Mercury Cassini β 0 > ALLOWED THEORIES α 0 ϕ β 0 matter ϕ ϕ g µν A 2 (ϕ)g µν +B(ϕ)U µ U ν general relativity (α 0 = β 0 = 0)

23 Post-Newtonian constraints Solar-system tests matter a priori weakly coupled to ϕ TeVeS tuned to pass them even for strong matter-scalar coupling Binary-pulsar tests matter must be weakly coupled to ϕ matter α 0 ϕ SEP LLR 10 0 matter-scalar coupling function ln A(ϕ) α J J B B Mercury β 0 < 0 Cassini β 0 > ALLOWED THEORIES α 0 ϕ β 0 matter ϕ ϕ g µν A 2 (ϕ)g µν +B(ϕ)U µ U ν general relativity (α 0 = β 0 = 0)

24 Post-Newtonian constraints Solar-system tests matter a priori weakly coupled to ϕ TeVeS tuned to pass them even for strong matter-scalar coupling Binary-pulsar tests matter must be weakly coupled to ϕ fʼ(x) x x a 0 a GM r AU Too large! GMa 0 r r

25 Post-Newtonian constraints Solar-system tests matter a priori weakly coupled to ϕ TeVeS tuned to pass them even for strong matter-scalar coupling Binary-pulsar tests matter must be weakly coupled to ϕ fʼ(x) a α 2 GM r 2 small enough x a 0 α 2 but MOND GMa 0 effects at too r small distances! x 0 0.1AU r

26 Post-Newtonian constraints Solar-system tests matter a priori weakly coupled to ϕ TeVeS tuned to pass them even for strong matter-scalar coupling Binary-pulsar tests matter must be weakly coupled to ϕ fʼ(x) 1 a 0 a α 2 GM r 2 GMa 0 r x x 0 30AU 7000AU r Quite unnatural! (and not far from being experimentally ruled out)

27 New simpler models? S = Nonminimal metric coupling c 3 d 4 x g R pure G.R. in vacuum 16πG ] + S matter [matter ; g µν f (g µν, R λ µνρ, σ R λ µνρ,... ) Can reproduce MOND, but Ostrogradski [1850] unstable within matter S = Nonminimal scalar-tensor model c 3 d 4 x { } g R 2 µ ϕ µ ϕ Brans-Dicke in vacuum 16πG ] + S matter [matter ; g µν A 2 g µν + B µ ϕ ν ϕ Can reproduce MOND while avoiding Ostrogradskian instability, but field equations not always hyperbolic within outer dilute gas

28 New simpler models? S = Nonminimal metric coupling c 3 d 4 x g R pure G.R. in vacuum 16πG ] + S matter [matter ; g µν f (g µν, R λ µνρ, σ R λ µνρ,... ) Can reproduce MOND, but Ostrogradski [1850] unstable within matter S = Nonminimal scalar-tensor model c 3 d 4 x { } g R 2 µ ϕ µ ϕ Brans-Dicke in vacuum 16πG ] + S matter [matter ; g µν A 2 g µν + B µ ϕ ν ϕ Can reproduce MOND while avoiding Ostrogradskian instability, but field equations not always hyperbolic within outer dilute gas

29 New simpler models? S = Nonminimal metric coupling c 3 d 4 x g R pure G.R. in vacuum 16πG ] + S matter [matter ; g µν f (g µν, R λ µνρ, σ R λ µνρ,... ) Can reproduce MOND, but Ostrogradski [1850] unstable within matter S = Nonminimal scalar-tensor model c 3 d 4 x { } g R 2 µ ϕ µ ϕ Brans-Dicke in vacuum 16πG ] + S matter [matter ; g µν A 2 g µν + B µ ϕ ν ϕ Can reproduce MOND while avoiding Ostrogradskian instability, but field equations not always hyperbolic within outer dilute gas

30 New simpler models? S = Nonminimal metric coupling c 3 d 4 x g R pure G.R. in vacuum 16πG ] + S matter [matter ; g µν f (g µν, R λ µνρ, σ R λ µνρ,... ) Can reproduce MOND, but Ostrogradski [1850] unstable within matter S = Nonminimal scalar-tensor model c 3 d 4 x { } g R 2 µ ϕ µ ϕ Brans-Dicke in vacuum 16πG ] + S matter [matter ; g µν A 2 g µν + B µ ϕ ν ϕ Can reproduce MOND while avoiding Ostrogradskian instability, but field equations not always hyperbolic within outer dilute gas

31 Introduction MOND Consistency RAQUAL Conditions Difficulties 1PN constraints New route Pioneer Conclusions Pioneer 10 & 11 anomaly Extra acceleration m.s 2 towards the Sun between 30 and 70 AU Simpler problemp than galaxy rotation curves (Mdark Mbaryon ), because we do not know how this acceleration is related to M several stable & well-posed solutions Nonminimal scalar-tensor model o n c3 d4 x g R 2 µ ϕ µ ϕ Brans-Dicke in vacuum S = 16πG h µ ϕ ν ϕ i + Smatter matter ; g µν e2αϕ gµν λ ϕ5 Z α2 < 10 5 to pass solar-system & binary-pulsar tests λ α3 (10 4 m)2 to fit Pioneer anomaly Gilles Esposito-Fare se, GRεCO/IAP

32 Introduction MOND Consistency RAQUAL Conditions Difficulties 1PN constraints New route Pioneer Conclusions Pioneer 10 & 11 anomaly Extra acceleration m.s 2 towards the Sun between 30 and 70 AU Simpler problemp than galaxy rotation curves (Mdark Mbaryon ), because we do not know how this acceleration is related to M several stable & well-posed solutions Nonminimal scalar-tensor model o n c3 d4 x g R 2 µ ϕ µ ϕ Brans-Dicke in vacuum S = 16πG h µ ϕ ν ϕ i + Smatter matter ; g µν e2αϕ gµν λ ϕ5 Z α2 < 10 5 to pass solar-system & binary-pulsar tests λ α3 (10 4 m)2 to fit Pioneer anomaly Gilles Esposito-Fare se, GRεCO/IAP

33 Introduction MOND Consistency RAQUAL Conditions Difficulties 1PN constraints New route Pioneer Conclusions Pioneer 10 & 11 anomaly Extra acceleration m.s 2 towards the Sun between 30 and 70 AU Simpler problemp than galaxy rotation curves (Mdark Mbaryon ), because we do not know how this acceleration is related to M several stable & well-posed solutions Nonminimal scalar-tensor model o n c3 d4 x g R 2 µ ϕ µ ϕ Brans-Dicke in vacuum S = 16πG h µ ϕ ν ϕ i + Smatter matter ; g µν e2αϕ gµν λ ϕ5 Z α2 < 10 5 to pass solar-system & binary-pulsar tests λ α3 (10 4 m)2 to fit Pioneer anomaly Gilles Esposito-Fare se, GRεCO/IAP

34 Conclusions A consistent field theory should satisfy different kinds of constraints: Mathematical: stability, well-posedness of the Cauchy problem, no discontinuous nor adynamical field Experimental: solar-system & binary-pulsar tests, galaxy rotation curves, gravitational lensing by dark matter haloes, CMB Esthetical: natural model, rather than fine-tuned fit of data Best present candidate: TeVeS [Bekenstein Sanders], but it has still some mathematical and experimental difficulties simpler models, useful to exhibit the generic difficulties of all MOND-like field theories By-product of our study: a consistent class of models for the Pioneer anomaly (but not natural!) Nonlocal models? [Work in progress with Cédric Deffayet & Richard Woodard]

35 Conclusions A consistent field theory should satisfy different kinds of constraints: Mathematical: stability, well-posedness of the Cauchy problem, no discontinuous nor adynamical field Experimental: solar-system & binary-pulsar tests, galaxy rotation curves, gravitational lensing by dark matter haloes, CMB Esthetical: natural model, rather than fine-tuned fit of data Best present candidate: TeVeS [Bekenstein Sanders], but it has still some mathematical and experimental difficulties simpler models, useful to exhibit the generic difficulties of all MOND-like field theories By-product of our study: a consistent class of models for the Pioneer anomaly (but not natural!) Nonlocal models? [Work in progress with Cédric Deffayet & Richard Woodard]

36 Conclusions A consistent field theory should satisfy different kinds of constraints: Mathematical: stability, well-posedness of the Cauchy problem, no discontinuous nor adynamical field Experimental: solar-system & binary-pulsar tests, galaxy rotation curves, gravitational lensing by dark matter haloes, CMB Esthetical: natural model, rather than fine-tuned fit of data Best present candidate: TeVeS [Bekenstein Sanders], but it has still some mathematical and experimental difficulties simpler models, useful to exhibit the generic difficulties of all MOND-like field theories By-product of our study: a consistent class of models for the Pioneer anomaly (but not natural!) Nonlocal models? [Work in progress with Cédric Deffayet & Richard Woodard]

37 Conclusions A consistent field theory should satisfy different kinds of constraints: Mathematical: stability, well-posedness of the Cauchy problem, no discontinuous nor adynamical field Experimental: solar-system & binary-pulsar tests, galaxy rotation curves, gravitational lensing by dark matter haloes, CMB Esthetical: natural model, rather than fine-tuned fit of data Best present candidate: TeVeS [Bekenstein Sanders], but it has still some mathematical and experimental difficulties simpler models, useful to exhibit the generic difficulties of all MOND-like field theories By-product of our study: a consistent class of models for the Pioneer anomaly (but not natural!) Nonlocal models? [Work in progress with Cédric Deffayet & Richard Woodard]

38 Conclusions A consistent field theory should satisfy different kinds of constraints: Mathematical: stability, well-posedness of the Cauchy problem, no discontinuous nor adynamical field Experimental: solar-system & binary-pulsar tests, galaxy rotation curves, gravitational lensing by dark matter haloes, CMB Esthetical: natural model, rather than fine-tuned fit of data Best present candidate: TeVeS [Bekenstein Sanders], but it has still some mathematical and experimental difficulties simpler models, useful to exhibit the generic difficulties of all MOND-like field theories By-product of our study: a consistent class of models for the Pioneer anomaly (but not natural!) Nonlocal models? [Work in progress with Cédric Deffayet & Richard Woodard]