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J. W. Moffat and V. T. Toth submitted recently a comment (arXiv:0903.5291) on our latest paper Modified scalar-tensor-vector gravity theory and the constraint on its parameters [Deng, et al., Phys. Rev. D 79, 044014 (2009); arXiv:0901.3730 ]. We reply to each of their comments and justify our work and conclusions. Especially, their general STVG (MOG) theory has to be modified to fit the modern precision experiments.
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A gravity theory called scalar-tensor-vector gravity (STVG) has been recently developed and succeeded in solar system, astrophysical and cosmological scales without dark matter [J. W. Moffat, J. Cosmol. Astropart. Phys. 03, 004 (2006)]. However, two assumptions have been used: (i) $B(r)=A^{-1}(r)$, where $B(r)$ and $A(r)$ are $g_{00}$ and $g_{rr}$ in the Schwarzschild coordinates (static and spherically symmetric); (ii) scalar field $G=Const.$ in the solar system. These two assumptions actually imply that the standard parametrized post-Newtonian parameter $gamma=1$. In this paper, we relax these two assumptions and study STVG further by using the post-Newtonian (PN) approximation approach. With abandoning the assumptions, we find $gamma eq1$ in general cases of STVG. Then, a version of modified STVG (MSTVG) is proposed through introducing a coupling function of scalar field G: $theta(G)$. We have derived the metric and equations of motion (EOM) in 1PN for general matter without specific equation of state and $N$ point masses firstly. Subsequently, the secular periastron precession $dot{omega}$ of binary pulsars in harmonic coordinates is given. After discussing two PPN parameters ($gamma$ and $beta$) and two Yukawa parameters ($alpha$ and $lambda$), we use $dot{omega}$ of four binary pulsars data (PSR B1913+16, PSR B1534+12, PSR J0737-3039 and PSR B2127+11C) to constrain the Yukawa parameters for MSTVG: $lambda=(3.97pm0.01)times10^{8}$m and $alpha=(2.40pm0.02)times10^{-8}$ if we fix $|2gamma-beta-1|=0$.
The scalar-tensor theory can be formulated in both Jordan and Einstein frames, which are conformally related together with a redefinition of the scalar field. As the solution to the equation of the scalar field in the Jordan frame does not have the one-to-one correspondence with that in the Einstein frame, we give a criterion along with some specific models to check if the scalar field in the Einstein frame is viable or not by confirming whether this field is reversible back to the Jordan frame. We further show that the criterion in the first parameterized post-Newtonian approximation can be determined by the parameters of the osculating approximation of the coupling function in the Einstein frame and can be treated as a viable constraint on any numerical study in the scalar-tensor scenario. We also demonstrate that the Brans-Dicke theory with an infinite constant parameter $omega_{text{BD}}$ is a counterexample of the equivalence between two conformal frames due to the violation of the viable constraint.
The recent data release by the Planck satellite collaboration presents a renewed challenge for modified theories of gravitation. Such theories must be capable of reproducing the observed angular power spectrum of the cosmic microwave background radiation. For modified theories of gravity, an added challenge lies with the fact that standard computational tools do not readily accommodate the features of a theory with a variable gravitational coupling coefficient. An alternative is to use less accurate but more easily modifiable semianalytical approximations to reproduce at least the qualitative features of the angular power spectrum. We extend a calculation that was used previously to demonstrate compatibility between the Scalar-Tensor-Vector-Gravity (STVG) theory, also known by the acronym MOG, and data from the Wilkinson Microwave Anisotropy Probe (WMAP) to show consistency between the theory and the newly released Planck 2018 data. We find that within the limits of this approximation, the theory accurately reproduces the features of the angular power spectrum.
In this paper, we study the properties of gravitational waves in the scalar-tensor-vector gravity theory. The polarizations of the gravitational waves are investigated by analyzing the relative motion of the test particles. It is found that the interaction between the matter and vector field in the theory leads to two additional transverse polarization modes. By making use of the polarization content, the stress-energy pseudo-tensor is calculated by employing the perturbed equation method. Besides, the relaxed field equation for the modified gravity in question is derived by using the Landau-Lifshitz formalism suitable to systems with non-negligible self-gravity.
We present a scalar-tensor theory of gravity on a torsion-free and metric compatible Lyra manifold. This is obtained by generalizing the concept of physical reference frame by considering a scale function defined over the manifold. The choice of a specific frame induces a local base, naturally non-holonomic, whose structure constants give rise to extra terms in the expression of the connection coefficients and in the expression for the covariant derivative. In the Lyra manifold, transformations between reference frames involving both coordinates and scale change the transformation law of tensor fields, when compared to those of the Riemann manifold. From a direct generalization of the Einstein-Hilbert minimal action coupled with a matter term, it was possible to build a Lyra invariant action, which gives rise to the associated Lyra Scalar-Tensor theory of gravity (LyST), with field equations for $g_{mu u}$ and $phi$. These equations have a well-defined Newtonian limit, from which it can be seen that both metric and scale play a role in the description gravitational interaction. We present a spherically symmetric solution for the LyST gravity field equations. It dependent on two parameters $m$ and $r_{L}$, whose physical meaning is carefully investigated. We highlight the properties of LyST spherically symmetric line element and compare it to Schwarzchild solution.
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