Spherically symmetric gravity with variable G and Lambda

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Spherically symmetric gravity with variable G and
Lambda

• G. Esposito, INFN, Naples (GRG18 Conference,
  Sydney, July 2007), with A. Bonanno, C. Rubano,
  P. Scudellaro

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Main ideas and results

• The asymptotic safety suggested by Weinberg in
  1979 seems to work the running couplings have a
  finite limit at large k, and a new non-Gaussian
  fixed point exists.
• We find that the treatment of cosmological term
  and Newton parameter as dynamical variables,
  jointly with a fixed-point condition, gives
  consistent chances of emulating dark matter in
  long-range gravitational interactions, at least
  on galactic scales.

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Renormalization-group approach

• A scale-dependent effective action is built. If
  this equals the classical action at the UV
  cut-off scale K, one uses the RG equation to
  evaluate Gamma(k) for all k less than K, and then
  sends k to 0 and K to infinity. The continuum
  limit as K tends to infinity should exist after
  ren. finitely many parameters in the action, and
  is taken at a non-Gaussian fixed point of the
  RG-flow.

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A new ultraviolet fixed point
Part of theory space of the Einstein-Hilbert
truncation with its Renormalization Group flow.
The arrows point in the direction of decreasing
values of k. The flow is dominated by a
non-Gaussian fixed point in the first quadrant
and a trivial one at the origin
Lauscher-Reuter in hep-th/0511260
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Action functional
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Spherical symmetry 3-metric of the leaves
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Action from Legendre transform
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Constraints
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Effective Hamiltonian
Hamilton equations
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First-order operator in the Hamilton equations
General form of Hamilton equations
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The U functions are
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The V functions read as
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Fixed-point condition
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Hamiltonian constraint
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All V functions vanish weakly
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All V functions
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Nonlinear equation for G(r)
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Novel form of the Newton parameter
It holds in the singular limit of vanishing
Work is in progress on the generic case
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Approximate linear growth of G(r)

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Large-r behaviour of radial velocities of galaxies
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Convenient parameters
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Equation for G with these parameters
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Vanishing cosmological constant
This is possibly more relevant than the previous
case
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Open problems (among the many)

• 1. Can the solution with vanishing cosmological
  constant, or the previous solution, reproduce the
  flat rotation curves of galaxies?
• 2. Horizon and gravitational collapse