Modified Newtonian gravity and field theory

Transcription

1 Modified Newtonian gravity and field theory Gilles Esposito-Farèse GRεCO, Institut d Astrophysique de Paris Based on Phys. Rev. D 76 (2007) 2402 in collaboration with J.-P. Bruneton, on my study of scalar-tensor theories & binary-pulsar tests with T. Damour since 99, on a work in progress with C. Deffayet & R. Woodard about nonlocal models, and discussions with many colleagues, notably L. Blanchet, B. Fort, G. Mamon, Y. Mellier, M. Milgrom, J. Moffat, R. Sanders, J.-P. Uzan, etc. This PDF file is a summary of my lectures at IESC, Cargèse, November 2008 Dark matter and galaxy rotation curves Ω Λ 0.7 (SNIa) and Ω Λ + Ω m (CMB) Ω m 0.3, at least 0 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

2 Milgrom s MOND proposal [983] 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 [977] M b v 4 M baryonic Superbly accounts for galaxy rotation curves v (but clusters still require some dark matter) [Sanders & McGaugh, Ann. Rev. Astron. Astrophys. 40 (2002) 263] Modified gravity or modified inertia? General Relativity c3 d 4 x g R + S matter [matter, g µν ] 6πG } {{ } } {{ } Einstein-Hilbert Metric coupling freely falling elevator Einstein-Hilbert pure spin 2 Metric coupling equivalence principle Example: S point particle = R mc p g µν (x) v µ v ν dt Modified inertia [Milgrom 994, 999]: Keep S Einstein-Hilbert [g µν ], but look for S point particle (x, v, a, ȧ,... ). Galileo invariance nonlocal ( causality?) Modified gravity: Keep metric coupling, but S gravity Einstein-Hilbert ( extra fields) Other possibilities: Both modifications, or none?

3 Mass-dependent models? A priori easy to predict a force /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 Moffat [2004] proposes a consistent field theory (nonsymmetric g µν ) but predicts a = km 2 /r instead of M/r, and assumes then k = M 3/2 In 2005, he introduced a potential in his model to derive k = M 3/2, but the potential depends on M Consistent field theories Field theories All predictions deriving from a single action proposed models in which 2 field equations are inconsistent with each other violation of conservation laws 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

4 Quadratic gravity t Hooft & Veltman [974]: Divergence of needs L = = Counterterm [ g 53 8π 2 (d 4) 90 R2 µνρσ R2 µν + 43 ] 72 R2 [ g 7 8π 2 (d 4) 40 C2 µνρσ + 8 R ] 360 GB C µνρσ : Weyl tensor (fully traceless) GB R 2 µνρσ 4R 2 µν + R 2 : Gauss-Bonnet topological invariant Stelle s thesis [977]: If α 0 and β 0, Quadratic gravity is renormalizable S gravity = d 4 x [ ] g R + αcµνρσ 2 + βr 2 + γgb f (curvature) and ghosts S gravity = But quadratic gravity is unstable d 4 x [ ] g R + αcµνρσ 2 + βr 2 + γgb Intuitive argument: Propagator N.B.: p 2 + α p 4 = p 2 ghost p 2 + α α = m2 of extra d of freedom negative α gives a tachyon, but anyway a ghost Full calculation [Stelle 977; Hindawi, Ovrut, Waldram 996; Tomboulis 996]: R + f (R µν, R µνρσ ) extra massive spin-2 ghost unstable vacuum R + f (R) extra massive spin-0 scalar with E kin > 0 N.B.: Strings predict C 2 µνρσ, but also any higher derivative (nonlocal)

5 f (R) as scalar-tensor theories Introduction of a Lagrange multiplier Φ [Teyssandier & Tourrenc 983] S gravity = d 4 x g f (R) d 4 x g { f (Φ) + (R Φ) f (Φ) } = d 4 x { g f (Φ)R 0 ( µ Φ) 2 [ Φf (Φ) f (Φ) ] } } {{ } } {{ } ω BD =0 potential N.B.: If potential = 0, solar system ω BD > Similarly f (R, R,..., n R) Einstein plus n + scalar fields [Gottlöber, Schmidt, Starobinsky 990; Wands 994] Such scalar fields give generically Yukawa potentials e mr r not MOND (potential ln r) f (R) as scalar-tensor theories (continued) We saw that f (R) theories are equivalent to d 4 x g {f (Φ)R 0 ( µ Φ) 2 [ Φf (Φ) f (Φ) ]} + S matter [matter, g µν ] Let g µν f (Φ)g µν, ϕ 3 ln f (Φ), V(ϕ) Φf (Φ) f (Φ) f 2 (Φ) Standard scalar-tensor theory d 4 x { g R } 2 gµν µ ϕ ν ϕ V(ϕ) graviton + S matter [matter, g µν = e ϕ/ 3 g µν] scalar

6 Relativistic aquadratic Lagrangians RAQUAL models d 4 x { } g R f (g µν µ ϕ ν ϕ) V(ϕ) } {{ } k-essence + S matter [matter, g µν A 2 (ϕ)g µν] The nonlinearity of f ( µ ϕ µ ϕ) now allows us to reproduce a MOND-like potential GMa 0 ln r [Bekenstein & Sanders]: r 2 ( r r 2 f [ ( r ϕ) 2] r ϕ ) T (matter source) f (x) constant for large x : Newtonian limit f (x) x for small x : MOND regime f (x) x x 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

7 Action and reaction principle Replace gravitational field g µν by its value imposed by material sources Fokker Lagrangian L = m A c 2 v2 A c V(m A, m B, r AB, v A, v B ) } {{ } A m A a A = Gm Am B r 2 AB a A = Gm B r 2 AB A B + m A GmB a 0 + r AB GmB a 0 r AB Gm A m B r AB 2m A GmB a 0 ln r AB + m B GmA a 0 r AB + m B Ga0 /m A r AB Divergent acceleration for test masses m A A B C But if self-field effects are treated consistently [Milgrom ], RAQUAL models actually predict in the MOND regime (a < a 0 ) m A a A = 2 Ga0 [ ] (m A + m B ) 3/2 m 3/2 A m 3/2 B 3 r AB m B m A a A = GmB a 0 r AB + O( m A ) Light deflection c 3 6πG RAQUAL models d 4 x { } g R f (g µν µ ϕ ν ϕ) V(ϕ) + S matter [matter, g µν A 2 (ϕ) }{{} } {{ } physical metric matter-scalar coupling g µν ] }{{} spin-2 metric Matter-scalar coupling function ln A() ln A() = " 0 ( 0 ) + # 0 ( 0 ) 2 + matter Effective gravitational constant G eff = G ( ) graviton 2 scalar slope curvature # 0 " 0 0 Scalar field extra attractive force

8 Light deflection (continued) Conformally related metrics g µν A 2 (ϕ)g µν Light rays do not feel the scalar field: ds 2 = g µν dx µ dx ν = 0 g µνdx µ dx ν = 0 photon h µ exists, but photon " = 0 Light deflection angle θ = 4GM bc 2 same as G.R. 4G eff M = bc 2 ( + α0 2) < 4G effm bc 2 } {{ } apparent position star Sun Earth interpreted as smaller than G.R. because G eff > G bare [N.B.: an erroneous theorem (overstatement) by Bekenstein & Sanders about this] Disformal coupling c 3 6πG Stratified theories [Ni, Sanders, Bekenstein (TeVeS)] d 4 x { g R 2 f (g µν µ ϕ ν ϕ) +S matter [ matter ; g µν A 2 (ϕ, U)g µν + B(ϕ, U)U µ U ν ] } U µ is either a new vector field, or µ ϕ itself A 2 > 0 and A 2 + Bg µν U µ U µ > 0 necessary for hyperbolicity (in addition to the previous conditions on f ).

9 Stratified theories Trick to increase light deflection (as dark matter does) Since Schwarzschild is such that g 00 = g rr = ( ) 2GM rc, 2 let us couple ϕ inversely to g 00 and g ij, say g 00 e 2ϕ g 00 and g ij e 2ϕ g ij Covariant rewriting: Let a vector U µ = (, 0, 0, 0) in this preferred frame. Then U µ U ν = behaves as g 00, and `g µν + U µ U ν as g ij Define the physical metric (minimally coupled to matter) as g µν = e 2ϕ ( g µν + U µ U ν ) e 2ϕ U µ U ν = e 2ϕ g µν 2 U µ U ν sinh(2ϕ) N.B.: Other factors depending on ϕ would give a different light deflection this is ad hoc N.B.2: Preferred-frame effects strongly constrained in solar system Bekenstein s TeVeS model [ ] Many years of work complicated S scalar = 2 R d 4 x i g hσ 2 (g αβ U α U β ) α ϕ β ϕ + 2 Gl 2 σ 4 F(kGσ 2 ) Potential F discontinuous F(µ) = F local effects µ(4 + 2µ 4µ 2 + µ 3 ) + 2 ln[( µ) 2 ] µ 2 cosmology µ Sanders defines smoother functions, and promotes σ to a 2nd dynamical scalar field, but this is a tachyon 6 8

10 MOND cosmology? No MOND regime in dense Universe at high redshifts Too small Ω baryon to account for structure formation [Lue & Starkman 2004] Skordis et al. [2006] need Ω ν = 0.7 for CMB (not far from Ω DM = 0.24) dark matter Bekenstein + large " Standard #CDM Bekenstein without " Difficulties of RAQUAL models Action/reaction, light deflection & CMB: solutions 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 = (, 0, 0, 0) Maybe not too problematic if U µ is dynamical Vector contribution to Hamiltonian unbounded by below [Clayton 200] unstable model Post-Newtonian tests very constraining

11 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 0 $ LLR 0 0 matter-scalar coupling function ln A($) 0 # 0 " 0 < 0 0 #2 Mercury Cassini " 0 > 0 0 #3 ALLOWED THEORIES 0 $ 0 #4 #6 #4 # matter $ " 0 $ general relativity ( 0 = " 0 = 0) 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 0 $ LLR 0 0 matter-scalar coupling function ln A($) 0 # 0 " 0 < 0 0 #2 Mercury Cassini " 0 > 0 0 #3 ALLOWED THEORIES 0 $ 0 #4 #6 #4 # matter $ " 0 $ general relativity ( 0 = " 0 = 0)

12 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 $ 0 # SEP LLR 0 0 matter-scalar coupling function ln A(#) 0 J B534+2 $ J B93+6 Mercury " 0 < 0 Cassini " 0 > ALLOWED THEORIES $ 0 # matter # " 0 # general relativity ($ 0 = " 0 = 0) 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 f (x) 0.8 a GM r 2 Too large x a 0 GMa 0 r x AU r

13 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 f (x) 0.8 a 2 GM r 2 small enough x a 0 2 but MOND GMa 0 effects at too r small distances x 0 0.AU r 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 f (x) a 0 a 2 GM r 2 GMa 0 r x 0 0 x 0 30AU 7000AU r Quite unnatural (and not far from being experimentally ruled out)

14 Non-minimal metric coupling Neither modified gravity nor modified inertia c3 d 4 x g 6πG R +S matter [matter, g µν g µν + f (R µνρσ,...)] } {{ } } {{ } Einstein-Hilbert Metric coupling Strictly same spectrum as G.R. in vacuum standard Schwarzschild solution (no extra field, no tachyon nor ghost) Equivalence principle satisfied vertices coupling matter fields to curvature extra degrees of freedom confined within matter (no free propagator) Same structure as finite-size effects in G.R. matter 2 g extra d.o.f. 2 g Non-minimal metric coupling: problems But exhibits all generic problems quite useful toy model to locate hidden assumptions in the literature Near a spherical body, R µνρσ and its covariant derivatives give access to M and r independently One can reproduce the MOND phenomenology, but also any other potential and any light deflection: not predictive MOST IMPORTANTLY, although this model does not involve any tachyon nor ghost, it is anyway unstable: Hamiltonian unbounded by below

15 Generic instability of higher-derivative theories Consider a non-degenerate L(q, q, q): p 2 L q invertible q = f (q, q, p 2) Ostrogradski [850] defines q q q 2 q p L q d dt p 2 L q ( ) L q Then H = p q + p 2 q 2 L(q, q, q) = p q 2 + p 2 f (q, q 2, p 2 ) L (q, q 2, f (q, q 2, p 2 )) is such that q i = H/ p i and ṗ i = H/ q i reproduce the Euler-Lagrange equations derived from L. Nonminimal scalar-tensor model Nonminimal metric coupling (unstable within matter) c 3 d 4 x g 6πG R pure G.R. in vacuum ] + S matter [matter ; g µν f (g µν, R λ µνρ, σr λ µνρ,... ) Nonminimal scalar-tensor model c 3 d 4 x { } g 6πG R 2 g µν Brans-Dicke µ ϕ ν ϕ in vacuum ] + S matter [matter ; g µν A 2 g µν + B µ ϕ ν ϕ Avoids Ostrogradskian instability because g µν depends only on ϕ and ϕ and because S matter only involves g linearly

16 Introduction Model building Kinetic terms Difficulties New routes Conclusions Nonminimal scalar-tensor model (continued) Nonminimal scalar-tensor model, / 0 c3 4 d x g R 2 s Brans-Dicke in vacuum 6πG & ' 2 + Smatter matter ; gµν A gµν + B µ ϕ ν ϕ s gµν µ ϕ ν ϕ A(ϕ, ϕ) eαϕ ϕx α αa0 /4 c s B(ϕ, ϕ) 4 ϕx α s X ln X Reproduces MOND while avoiding Ostrogradskian instability but field equations not always hyperbolic within outer dilute gas Modified Newtonian gravity and field theory IESC, Carge se, November 2008 Introduction Model building Kinetic terms Gilles Esposito-Fare se, GRεCO/IAP Difficulties New routes Conclusions Pioneer 0 & anomaly Extra acceleration m.s 2 towards the Sun between 30 and 70 AU Simpler problem0 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, / 0 c3 Brans-Dicke 4 µν d x g R 2 g µ ϕ ν ϕ 6πG in vacuum & µ ϕ ν ϕ ' 2αϕ + Smatter matter ; gµν e gµν λ ϕ5 α2 < 0 5 to pass solar-system & binary-pulsar tests λ α3 (0 4 m)2 to fit Pioneer anomaly Modified Newtonian gravity and field theory IESC, Carge se, November 2008 Gilles Esposito-Fare se, GRεCO/IAP

17 Other recent ideas Einstein-æther gravity [T. Jacobson, arxiv: (gr-qc)] d 4 x { g R + c ( µ U ν ) 2 + c 2 ( µ U µ ) 2 + c 3 µ U ν ν U µ } +c 4 (U µ µ U ν ) 2 + λ(u µ U µ [ ] + ) + S matter matter ; gµν Nonstandard kinetic term for vector U µ but constant norm. Metric coupling of matter, but T µν (U) 0. H.S. Zhao generalizes this framework to nonconstant coefficients c i (x) to reproduce MOND, but predictions & stability still quite unclear. Special kind of dark matter reproducing MOND predictions? For instance, fluid of gravitational dipoles [L. Blanchet ]. nice relativistic version, although a few difficulties remain. Nonlocal models? [C. Deffayet, GEF, R. Woodard 2008] Actions involving R, or more generally f (R λ µνρ) power f 2 (R λ µνρ). too naive reasoning showing that this should involve a ghost, but this is not always the case. 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) new ideas (æther, nonlocal,... ), but stability still needs to be proven