Selected topics

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Selected topics
MOND-like field theories
Jean-Philippe Bruneton Institut dAstrophysique
de Paris Work with Gilles Esposito-Farèse
bruneton_at_iap.fr
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• No Introduction!
• Causality vs superluminal propagations
  causality in MOND
• Some difficulties in model building for MOND
• MOND and the Pioneer anomaly

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Causality vs superluminal behavior

• Lets recall RAQUAL theory a k-essence theory
• Quite generically, the f field propagates
  superluminaly
• (if f gt 0)
• Is it a problem for the theory?
• Bekenstein said  yes 
• But the right answer is it depends

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What is causality?

• 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
•  is there a bijective link between initial
  conditions at t0 and solutions at t1gt t0  ?
• Def this is known as the Cauchy problem.

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Causality conditions Wald, chp 8

• Existence of a well-defined notion of time? ie,
  no Closed Timelike Curves CTC
• If the spacetime (M,gmn) is globally hyperbolic,
  then M can be foliated in Cauchy surfaces S and
  has the topology of SR and there is no CTC
• 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,

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The Cauchy problem for k-essence theories

• The equation of the scalar field is
• Or
• With
• 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
• (or the opposite, but then it is a ghost)

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Causality in k-essence

• Iff these conditions are fulfilled the theory is
  causal, even if there are superluminal
  propagations
• The reason is that we simply have two metrics
• so that causal structure is preserved!

Photons
f
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So what about RAQUAL/TeVeS, etc ?

• We have to check if these conditions hold
• Generic problem in RAQUAL/TeVeS we have
• and therefore the field does not propagate
  anymore at X0
• The theory is not causally well behaved at the
  transition between local physics/cosmology

So that

• gt Existence of an horizon for the f field,
  around each galaxy

M
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Causality in MOND

• 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
  generic.
• Straightforward solution consider a modified
  asymptotic relation
• F(X) X e
• gt Newton then MOND, then Newton again with
  GeffGN/e
• gtThe rotation curves are flat only on a range of
  rMltrltrM/e

• Sanders (86) already study this model, but here
  we justify it.
• Other applications the discontinuity in the F
  function in TeVeS rises the same problems

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Model Building(1)

• Commun misconceptions
• Many papers have vflat2 a c2 ,where a is a
  parameter in the Lagrangian
• gt this is not Tully-Fischer but some seem to
  ignore it
• gt then they say, lets take a M1/2
• This a not only  one theory for each galaxy ,
  this is not a theory at all!
• f(R) theories of gravity
• equivalent to scalar-tensor theory of gravity,
  but with wBD 0 gt excluded by solar
  system experiments or maybe ok, but with very
  fine-tuned potential
• Scalar-tensor theories of gravity
• if the potential has a minima, the cosmic
  expansion drives the scalar field to this
  minima. Then V(f) f 2 gt Yukawa force gt not MOND

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Model Building(2)

• Realize dynamically the k-essence field phase
  coupling gravitation, BSTV
• Perform the transformation
• ltgt
• If
• then in the MOND regime
• Add a kinetic term for q (PCG) negative sextic
  potential tachyon unstable

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Model Building(3)

• BSTV
• Sanders takes a positive quadratic potential gt
  stable?
• In fact the Hamiltonian is still unbounded.
  Indeed he has
• And J X and f(q) q6 ! gt 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 M1/2 ln(r)
• The only way (?) is to consider higher
  derivatives of the Newtonian potential Rmnrs2,
  etc
• But higher orders derivatives theories are
  generically unstable
• because the Hamiltonian is not bounded from
  below cf J.Simon 92, Woodard 2006

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Conclusions

• 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
• Generic pb with an horizon surrounding each
  galaxies gt toward a theory like
  Newton-MOND-Newton?
• 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
• Without k-essence gt theories with higher
  derivatives gt generically unstable