Modified Gravity and its Consequences for the Solar System, Astrophysics and Cosmology

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Modified Gravity and its Consequences for the
Solar System, Astrophysics and Cosmology
J. W. Moffat
Perimeter Institute For Theoretical
Physics Waterloo, Ontario,
Canada

• Talk given at the Workshop on Alternative Gravity
  Models and Dark Energy
• Edinburgh, Scotland, April 20, 2006

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Contents

• Introduction
• Modified Gravity (MOG)
• Equations of Motion, Weak Fields and Modified
  Acceleration
• Fitting galaxy rotation curves and galaxy
  clusters
• 5. Explaining the Pioneer 10-11 anomalous
  acceleration
• 6. Cosmology
• 7. Conclusions

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1. Introduction

• A fully relativistic modified gravity (MOG)
  called Scalar-Tensor-Vector-Gravity (STVG or
  MSTG) leads to a self-consistent, stable gravity
  theory that can describe solar system,
  astrophysical and cosmological data. The theory
  has an extra degree of freedom, a vector field
  called a phion field whose curl is a skew field
  that couples to matter. The gravitational field
  is described by a symmetric Einstein metric
  tensor.
• The effective classical theory allows the
  gravitational coupling constant G to vary as a
  scalar field with space and time. The effective
  mass of the skew symmetric field and the coupling
  of the field to matter also vary as scalar fields
  with space and time.
• The variation of the constants can be explained
  in a quantum gravity renormalization group (RG)
  flow scenario in which gravity is an
  asymptotically-free theory (Reuter and Weyer,
  JWM). The constants run with momentum k as in
  QCD, and a cutoff procedure leads to space and
  time varying constants. The STVG theory is an
  effective classical description of the RG flow
  scenario. The quantum gravity theory is
  constructed to be non-perturbatively
  renormalizable.

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• We shall show that STVG yields a modified
  Newtonian acceleration law for weak fields that
  can fit a large amount of galaxy rotation curve
  data without non-baryonic dark matter (Brownstein
  and JWM). It also can fit data for X-ray galaxy
  clusters without dark matter. The modified
  acceleration law is consistent with the solar
  system data and can possibly explain the Pioneer
  10-11 anomalous acceleration, and is consistent
  with the binary pulsar PSR 1913 16 data
  (Brownstein and JWM).

• A MOG should explain the following
• The CMB data including the power spectrum data
• The formation of proto-galaxies in the early
  universe and the growth of galaxies
• Gravitational lensing data for galaxies and
  clusters of galaxies
• N-body simulations of galaxy surveys
• The accelerating expansion of the universe.
• We seek a unified description of the
  astrophysical and large-scale cosmological data.

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2. Modified Gravity (MOG)
Our action takes the form
where
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Here, denotes the covariant derivative
with respect to g. Moreover, V denotes a
potential for the fields and
The total energy-momentum tensor is
The field equations are
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The effective gravitational constant G(x)
satisfies the field equations
Similar field equations are obtained for the
scalar fields mu(x) and omega(x).
The field equations for the phion field are
.
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3. Equations of Motion, Weak Fields and the
Modified Acceleration
Let us assume that we are in a distance scale
regime in which the scalar fields, G, and
take their approximate renormalized values
For a static spherically symmetric gravitational
field
If we neglect the mass , then we obtain the
Reissner-Nordstrom static spherically symmetric
solution
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M is a constant of integration and
For large r we obtain the Schwarzschild metric
components
We obtain the equations of motion
where is a constant, and E is a
constant of integration.
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We assume that GM/r ltlt 1 and the slow motion
approximation to give
For weak gravitational fields, the equations of
motion and the Yukawa solution for a static,
source-free spherically symmetric field
are
For the radial acceleration on a test particle we
get
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The acceleration law can be written
The pioneer anomalous acceleration directed
towards the center of the Sun is
We assume the following parametric forms for the
running of the constants and
where and b are constants.
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For material test particles with u1/r we obtain
For photons with ds20
For weak fields and large r we get
For the solar system G(r) G_N and the
perehelion precessions for the planets and the
bending of light by the Sun agree with GR for
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• 4. Fitting Galaxy Rotation Curves
• A fitting routine has been applied to fit a
  large number of galaxy rotation curves (101
  galaxies), using photometric data (58 galaxies)
  and a core model (43 galaxies) (Brownstein and
  JWM, 2005). The fits to the data are remarkably
  good and for the photometric data only one
  parameter, the mass-to-light ratio M/L, is used
  for the fitting once two parameters alpha and
  lambda are universally fixed for galaxies and
  dwarf galaxies. The fits are close to those
  obtained from Milgroms MOND acceleration law
  (Milgrom 1983) in all cases considered. A large
  sample of X-ray mass profile cluster data (106
  clusters) has also been well fitted (Brownstein
  and JWM, 2005). The fitting of the radial
  dependence of the dynamical cluster mass is
  effectively a zero-parameter fit, for the two
  parameters alpha and lambda are fitted to the
  determined bulk mass.

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5. Fitting the Pioneer Anomalous Acceleration
The pioneer anomaly directed towards the center
of the Sun is given by


We use the following parametric representations
of the running of alpha (r) and lambda (r)
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A consequence of a variation of G and GM_sun for
the solar system is a modification of Keplers
third law
For given values of a_pl and T_pl we can
determine G(r)M_sun. We define the standard
semi-major axis value at 1 AU
For a distance varying G(r)M_sun we derive
(Talmadge et al. 1988)
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6. MOG Cosmology

• An important extra-degree of freedom in MOG is a
  light, electrically uncharged vector particle
  called a phion. In the early universe at a
  temperature T lt T_c, where T_c is a critical
  temperature, the phions become a Bose-Einstein
  condensate (BEC) fluid. The phion condensates
  couple weakly with gravitational strength to
  ordinary baryonic matter. This cold fluid has
  zero classical pressure and zero shear viscosity
  and dominates the density of matter at
  cosmological scales and, because of its clumping
  due to gravitational collapse, allows the
  formation of structure and galaxies at
  sub-horizon scales well before recombination.
• We do not postulate the existence of cold dark
  matter in the form of heavy, new stable particles
  such as supersymmetric WIMPS. The phions undergo
  a 2nd-order phase transition through a
  spontaneous symmetry breaking for T lt T_c. The
  non-zero vacuum expectation value ltphigt_0 can
  weakly break Lorentz invariance.

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The correlation function for the temperature
differences across the sky for a given angle
theta takes the form
I use a modified form of the analytic calculation
of C_l given by Mukhanov (2006) to obtain a fit
to the acoustical peaks in the CMB for l gt 100 lt
1200. The adopted density parameters are
Without the dominant BEC phion-matter, the Silk
and finite thickness scales l_s and l_f erase any
peaks above the second peak (baryon drag). The
speed of sound c_s2 14 Omega_b,0 depends on
the baryon density today.
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• For local late-time bound systems such as
  galaxies and clusters of galaxies the symmetry
  breaking is relaxed and the phion Bose-Einstein
  condensates become ultra-light and relativistic.
  For galaxies and clusters of galaxies ordinary
  baryonic matter and neutral hydrogen and helium
  gases now constitute the dominant form of matter.
  
• The phion field and the spatial variation of G
  modify for late-time galaxies Newtons
  acceleration law. The rotational velocity curves
  are flattened, because of the altered dynamics of
  the gravitational field at the outer regions of
  spiral galaxies and not because of the presence
  of a dominant dark matter halo.
• The dual role played by the phion field in
  describing galaxies and the large-scale structure
  of the universe is a generic feature of our MOG
  theory.

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7. Conclusions

• A stable and self-consistent modified gravity
  (MOG) theory is constructed from a
  pseudo-Riemannian geometry and a massive skew
  field obtained from the curl of a massive vector
  field (phion field). The static spherically
  symmetric solution of the field equations yields
  a modified Newtonian acceleration law with a
  scale dependence. The gravitational constant G,
  the effective mass and the coupling strength of
  the skew field run with distance scale r
  according to an infra-red RG flow scenario based
  on an asymptotically free quantum gravity. This
  can be described by an effective classical STVG
  action.
• A fit to 101 galaxy rotations curves is obtained
  and mass profiles of x-ray galaxy clusters are
  also successfully fitted for those clusters that
  are isothermal.
• A possible explanation of the Pioneer 10-11
  anomalous acceleration is obtained from the MOG
  with predictions for the onset of the anomalous
  acceleration at Saturns orbit and for the
  periods of the outer planets.

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• The phion boson field forms Bose-Einstein
  condensates through a spontaneous symmetry
  breaking at large cosmological scales, which can
  explain the formation of proto-galaxies and at
  late times the structure of galaxies and
  clusters.
• A fit to the CMB acoustical power spectrum data
  can be achieved with a 2-component BEC and
  baryon-photon fluid for which the BEC density
  Omega_\phi gt Omega_b.
• For late-time local virialized clusters of
  galaxies and galaxies, the BEC symmetry breaking
  is relaxed and the baryons and neutral gases
  dominate, Omega_b gt Omega_phi, and the MOG
  acceleration law explains galaxy rotation curves
  and mass profiles of clusters.
• Heavy WIMP dark matter particles will not be
  observed in laboratory experiments.
• The MOG theory gives a unified description of
  solar system, astrophysical and cosmological data.

END
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Bibliography

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   Matter, JCAP 0505 (2005) 003, astro-ph/0412195
2. J. W. Moffat, Scalar-Tensor-Vector Gravity
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3. J. R. Brownstein and J. W Moffat, Galaxy
   Rotation Curves without Non-Baryonic Dark Matter,
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