Selected topics MOND-like field theories Jean-Philippe Bruneton Institut d’Astrophysique de Paris Work with Gilles Esposito-Farèse

No Introduction! Causality vs superluminal propagations ; causality in MOND Some difficulties in model building for MOND MOND and the Pioneer anomaly

Causality vs superluminal behavior Let’s recall RAQUAL theory : [a k-essence theory] Let’s recall RAQUAL theory : [a k-essence theory] Quite generically, Quite generically, the  field propagates superluminaly (if f ’’ > 0) Is it a problem for the theory? Is it a problem for the theory? Bekenstein said : « yes » Bekenstein said : « yes » …But the right answer is : it depends

What is causality? First, we need a well-defined notion of time. First, we need a well-defined notion of time. Def : this is known as « causality conditions » [Penrose, Hawking, Geroch, etc…] Then, at a classical level, causality is just : Then, at a classical level, causality is just : « is there a bijective link between initial conditions at t 0 and solutions at t 1 > t 0 » ? Def : this is known as the Cauchy problem.

Causality conditions [Wald, chp 8] Existence of a well-defined notion of time? [ie, no ] Existence of a well-defined notion of time? [ie, no Closed Timelike Curves : CTC] If the spacetime (M,g  ) is globally hyperbolic, then M can be foliated in Cauchy surfaces  and has the topology of  R and there is no CTC If the spacetime (M,g  ) is globally hyperbolic, then M can be foliated in Cauchy surfaces  and has the topology of  R and there is no CTC Powerful theorems prove that spacetime is generically globally hyperbolic in GR if some conditions are satisfied Powerful theorems prove that spacetime is generically globally hyperbolic in GR if some conditions are satisfied [But this is not always the case : eg. Gödel Universe, Kerr Black Hole,…] [But this is not always the case : eg. Gödel Universe, Kerr Black Hole,…]

The Cauchy problem for k-essence theories The equation of the scalar field is The equation of the scalar field isOrWith Theorem [Wald, Chp 10] : in a globally hyperbolic spacetime, Theorem [Wald, Chp 10] : in a globally hyperbolic spacetime, the Cauchy problem is well posed iff the above effective metric C is Lorentzian This condition reads This condition reads (or the opposite, but then it is a ghost)

Causality in k-essence Iff these conditions are fulfilled the theory Iff these conditions are fulfilled the theory is causal, even if there are superluminal propagations The reason is that we simply have two metrics… The reason is that we simply have two metrics… … so that causal structure is preserved! Photons 

So what about RAQUAL/TeVeS, etc… ? We have to check if these conditions hold We have to check if these conditions hold : in RAQUAL/TeVeS we have Generic problem : in RAQUAL/TeVeS we have and therefore the field does not propagate anymore at X=0 The theory is not causally well behaved at the transition between local physics/cosmology M So that : => Existence of an horizon for the  field, around each galaxy :

Causality in MOND Bekenstein was right in worrying about causality, but was finally wrong Bekenstein was right in worrying about causality, but was finally wrong The problem with causality is not superluminal propagations, but the horizon ; and this is The problem with causality is not superluminal propagations, but the horizon ; and this is generic. : consider a modified asymptotic relation Straightforward solution : consider a modified asymptotic relation  = F’(X) ~ X +  => Newton then MOND, then Newton again with Geff=G N /  =>The rotation curves are flat only on a range of r M The rotation curves are flat only on a range of r M <r<r M /  Sanders (86) already study this model, but here we justify it. Sanders (86) already study this model, but here we justify it. Other applications :Other applications : the discontinuity in the F function in TeVeS rises the same problems

Model Building(1) Commun misconceptions : Many papers have v flat 2 =  c 2,where  is a parameter in the Lagrangian Many papers have v flat 2 =  c 2,where  is a parameter in the Lagrangian => this is not Tully-Fischer [but some seem to ignore it] => then they say, let’s take    This a not only « one theory for each galaxy », this is not a theory at all! f(R) theories of gravity : f(R) theories of gravity : equivalent to scalar-tensor theory of gravity, equivalent to scalar-tensor theory of gravity, but with  BD = 0 => excluded by solar system experiments [or maybe ok, but with very fine-tuned potential] Scalar-tensor theories of gravity : Scalar-tensor theories of gravity : if the potential has a minima, the cosmic expansion drives the scalar field to this minima. if the potential has a minima, the cosmic expansion drives the scalar field to this minima. Then V(  )~  2 => Yukawa force => not MOND

Model Building(2) Realize dynamically the k-essence field [phase coupling gravitation, BSTV] Realize dynamically the k-essence field [phase coupling gravitation, BSTV] Perform the transformation : Perform the transformation :<=> If If then in the MOND regime Add a kinetic term for q (=PCG) : Add a kinetic term for q (=PCG) : negative sextic potential = tachyon = unstable

Model Building(3) BSTV : BSTV : Sanders takes a positive quadratic potential => stable? In fact the Hamiltonian is still unbounded. Indeed he has : And J = X and f(q) ~ q^6 ! => GHOST! And J = X and f(q) ~ q^6 ! => GHOST! Summary : k-essence : pb of horizons, PCG-like theories (BSTV) : generically unstable Without k-essence, we need to have access independantly to M and r, in order to construct the MONDian potential : M 1/2 ln(r) Without k-essence, we need to have access independantly to M and r, in order to construct the MONDian potential : M 1/2 ln(r) The only way (?) is to consider higher derivatives of the Newtonian potential : R  2, etc… The only way (?) is to consider higher derivatives of the Newtonian potential : R  2, etc… But higher orders derivatives theories are generically unstable because the Hamiltonian is not bounded from below [cf J.Simon 92, Woodard 2006 ]

Conclusions The issue of causality has been misunderstood. The issue of causality has been misunderstood. For each free function f in TeVeS like theory, one has to check that equations are hyperbolic [it depends on f] For each free function f in TeVeS like theory, one has to check that equations are hyperbolic [it depends on f] Generic pb with an horizon surrounding each galaxies => toward a theory like Newton-MOND-Newton? Generic pb with an horizon surrounding each galaxies => toward a theory like Newton-MOND-Newton? The discontinuities in function f are also problematic The discontinuities in function f are also problematic PCG-like theories seem to be generically unstable, because one has to find a negative sextic potential in a certain regime PCG-like theories seem to be generically unstable, because one has to find a negative sextic potential in a certain regime Without k-essence => theories with higher derivatives => generically unstable Without k-essence => theories with higher derivatives => generically unstable