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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
1. Introduction
2. Modified Gravity (MOG)
3. Equations of Motion, Weak Fields and Modified
Acceleration
4. 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-TensorVector-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 nonperturbatively 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 ReissnerNordstrom 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
integration.
is a constant, and E is a constant of
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We assume that GM/r << 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 u=1/r we obtain
For photons with ds^2=0:
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 Milgrom’s 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 Kepler’s 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 < 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 < T_c. The non-zero vacuum expectation value
<phi>_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 > 100 < 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_s^2 ~ 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 Newton’s 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 Saturn’s 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 > 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 > 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
1. J. W. Moffat, Gravitational Theory, Galaxy Rotation
Curves and Cosmology without Dark Matter, JCAP
0505 (2005) 003, astro-ph/0412195
2. J. W. Moffat, Scalar-Tensor-Vector Gravity Theory,
JCAP 0603 (2006) 004, gr-qc/0506021
3. J. R. Brownstein and J. W Moffat, Galaxy Rotation
Curves without Non-Baryonic Dark Matter,
Astrophys. J. 636 (2006) 721, astro-ph/0506370
4. J. R. Brownstein and J. W Moffat, Galaxy Cluster
Masses Without Non-Baryonic Dark Matter, Mon. Not.
Roy. Astron. Soc. 367 (2006) 527, astro-ph/0507222
5. J. R. Brownstein and J. W Moffat, Gravitational
Solution to the Pioneer 10/11 Anomaly, to be
published in Class. Quant. Grav. 2006, grqc/0511026
6. J. W. Moffat, Spectrum of Cosmic Microwave
Fluctuations and the Formation of Galaxies in a
Modified Gravity Theory, astro-ph/0602607
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