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

CosmologyCosmology

Talk given at the Workshop on Talk given at the Workshop on Alternative Gravity Models and Dark Alternative Gravity Models and Dark

EnergyEnergyEdinburgh, Scotland, April 20, 2006Edinburgh, Scotland, April 20, 2006

J. W. Moffat Perimeter Institute For Theoretical Physics Waterloo, Ontario, Canada

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Contents

1. Introduction2. Modified Gravity (MOG)3. Equations of Motion, Weak Fields and Modified

Acceleration4. Fitting galaxy rotation curves and galaxy clusters5. Explaining the Pioneer 10-11 anomalous acceleration6. Cosmology7. 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 AccelerationLet 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 << 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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Bibliography1. 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, gr-qc/0511026

6. J. W. Moffat, Spectrum of Cosmic Microwave Fluctuations and the Formation of Galaxies in a Modified Gravity Theory, astro-ph/0602607