Towards the explanation of flatness of galaxies rotation curves

We suggest a new explanation of flatness of galaxies rotation curves without invoking dark matter. For this purpose a new gravitational tensor field is introduced in addition to the metric tensor.


Introduction
Velocities of stars and gas rotation around galaxies centers become independent of rotation radii at large enough radii, see e.g.[1] and references therein.This asymptotical flatness of galaxies rotation curves is in obvious contradiction with Newtonian dynamics which demands that velocities should decrease with radii as 1/ √ r.
The contradiction is explained within the dark matter paradigm [2].Within the paradigm a galaxy is placed in a spherical halo of dark matter [3] with mass density of dark matter decreasing as 1/r 2 .Hence the Newtonian gravitational potential becomes logarithmically dependent on r providing a flat rotation curve.But hypothetical constituents of dark matter are still not discovered in direct experiments inspite of numerous searches.
In this situation it is worthwhile to develope explanations of flatness of galaxies rotation curves which do not invoke dark matter.Probably the most known attempt of this type is modified Newtonian dynamics, called MOND [4]- [6] where it is assumed that gravitational forces depend on accelerations of objects participating in interactions.Modified Newtonian dynamics has essential phenomenological successes.In particular it described the known Tully-Fisher relation [7] which establishes correlation between a galaxy luminosity and a corresponding flat rotation speed.But modified Newtonian dynamics does not describe e.g.gravitational lensing by galaxies and galaxies clusters.Modified Newtonian dynamics is not a completely formalized theory although there was an attempt to construct its complete Lagrangian version [8] which was not supported experimentally.
It should be mentioned that the Tully-Fisher relation is more precise in its barionic form which states that a flat rotation speed in a galaxy correlates with its barionic mass i.e. a sum of stars and gas masses, see [9] and references therein.The barionic Tully-Fisher relation states that M bar ∝ V 4 f lat .This tight correlation between visible matter of a galaxy and a corresponding flat rotation speed is also a rather strong motivation to find the explanation of flatness of galaxies rotation curves without dark matter.Also in [10] it is shown that there is a strong correlation between the observed radial star acceleration and the acceleration predicted by the observed distribution of baryons.
It is necessary to point out that, over the years, there have also been other ideas to replace the dark universe paradigm.For example through the so called de Sitter relativity [11]- [14] or through other approaches [15]- [17].
In the present paper we suggest a new explanation of flatness of galaxies rotation curves without invoking dark matter.For this purpose a new tensor gravitational field is introduced in addition to the metric tensor.

Main part
We consider the General Relativity action plus tems with a new gravitational tensor field f µν : here the R-term is the Einstein-Hilbert Lagrangian of General Relativity, the geometrical part of the action; √ −g is as usual the square root of the minus determinant of the metric tensor g µν (x).M 2 P l = 1/(16πG) is the Planck mass squared.
D λ is the standard covariant derivative, for example: where the Christoffell symbols as usual are f µν (x) is the introduced new tensor field additional to the metric tensor g µν (x).The field f µν (x) can be chosen to be the symmetric tensor similar in this sence to the metric tensor g µν (x).It is necessary to ensure that the new gravitational field f µν (x) interacts with light which has traceless energy-momentum tensor T µν light (x).Interaction of the field f µν with light is important in order to produce extra gravitational lensing due to barionic matter as compared to General Relativity, since in the case of absence of dark matter General Relativity alone is not sufficient to describe observed gravitational lensing.In principle, the new dynamical gravitational field f µν can also have asymmetric part in addition to the symmetric one, but this is not essential at the present stage of considerations.
The main new point of the action in eq.( 1) is the non-integer power 3/2 of the factor D λ D λ .This provides the 1/(k 2 ) 3/2 behavior of the propagator of the new dynamical gravitational field f µν in the momentum space, where k 2 = k µ k µ is the square of the four momentum k µ .Due to such a behavior of the propagator one can generate the logarithmic with the distance r gravitational log(r) potential, as it will be shown below, which allows to describe flatness of galaxies rotation curves without invoking dark matter.
G * is the introduced new coupling constant of interaction of the new gravitatonal tensor field f µν with matter fields desribed by energy-momentum tensors T µν i .Of course, it it assumed that the complete action also contains terms describing propogations of matter fields in addition to their interactions with gravitational fields.
m * is the introduced new mass parameter which is necessary to compensate the dimensions of masses m i in the corresponding square rootxs.
The sum i in the action (1) goes over matter objects drscribed by energy-momentum tensors T µν i and having masses m i .Couplings of them with the field f µν depend on m i via 1

√
1+m i /m * ; this property is quite essential and is used below to reproduce the famous barionic Tully-Fisher relation.
Numerical values of new constants G * and m * should be fixed from fitting experiments.It needs an analysis of a huge amount of empirical data for different galaxies and that is why it is a subject for a separate publication.
The Λ-term in eq.( 1) is not essential in perturbation theory which we will consider.
We will use the standard in Quantum Field Theory system of units h = c = 1.
To quantize the theory (1) selfconsistently one should add to the Lagrangian all possible terms quadratic in the Riemann tensor R µνρσ , see [18], [19] where perturbatively reormalizable and unitary model of quantum gravity was for the first time formulated.But these terms are not essential for the present considerations.
As it was already mentioned above we will work within perturbation theory, hence a linearized theory around the flat metric η µν is considered, that is we make the following substitution here the generally accepted in Field Theory convention in four dimensions is chosen η µν = diag(+1, −1, −1, −1).Indexes are raised and lowered in the following by means of the flat metric tensor η µν .Within perturbation theory one makes the standard shift of the metric field h µν → M P l h µν .
Perturbative expansion goes as usual in the inverse powers of M P l or in other words in the powers of the Newton coupling constant G = 1 16πM 2 P l .Let us now get the propagator of the new gravitational tensor field f µν in the momentum space.For this purpose we take the quadratic in the field f µν part of the Lagrangian of the action in the equation ( 1) and perform the Fourier transform to the momentum space: To obtain the propagator D µνρσ of the field f µν one should in the standard way invert the matrix in the brackets of the equation ( 6): Then the propagator has the following form The obtained propagator (8) generates the logarithmic in the distance r gravitational potential.
To derive the logarithmic potential from the propagator (8) one should consider in the standard way the Fourier transform of the propagator in three space dimensions: where the expression in the right hand side is obtained after integrations over angles are done.κ > 0 is a small regularizing parameter, which regularizes the infrared divergency in the integral.k is the lenth of the three vector k.
Then one gets after performing integration in the right hand side of the equation ( 9): where Ci(rκ) is the standard cosine integral function, and in the right hand side the expansion in the limit of small κ is done.Thus one obtains the logarithmic in r potential due to the field f µν .The regularizing parameter κ is inessential when one takes the derivative in r in order to produce the gravitational force from the obtained logarithmic potential.
At large enough distances r, that is at galactic and extragalactic scales, this log r-potential starts to dominate over the Newtonian 1/r-potential which is generated by the metric field h µν of General Relativity for small potentials.
Thus the gravitational field f µν generates a 1/r-force between a point object with a mass M bar having the energy-momentum tensor T µν = δ 0 µ δ 0 ν M bar δ 3 (x) (describing a galaxy with the baarionic mass M bar of stars plus gas ) and an analogous object with a mass M star (describing a star with the mass M star ): At a galaxy mass M bar large compared to the mass parameter m * the square root 1 + M bar /m * becomes approximately just M bar /m * , and the above relation (11) takes the form From the other side according to the second Newton law one gets Equating the expressions ( 12) and ( 13) we obtain the following relation The expression (14) reproduces the barionic Tully-Fisher relation which states that M bar ∝ V 4 f lat .It is interesting to note that the right hand side of the expression (14) has the dependence on the star mass M star .
Thus flat rotation speeds of stars satisfy the barionic Tully-Fisher relation in our model.
We should also mention once more that the introduced tensor field f µν interacts with light and that is why adds additional, as compared to the metric field h µν of General Relativity, gravitational lensing due to barionic matter.

Discussions
The Newtonian theory of gravity could be assumed to be a perfect theory at the galactic and extragalactic distances.But velocities of stars and gas rotation as it is known from the experimental observations are usually essentially larger than velocities generated by visible barionic matter as they are estimated according to the Newtonian dynamics.It is presently commonly accepted to explain this paradox by the presence of the necessary amount of dark matter in galaxies.Also observed gravitational lensing by galaxies and clusters of galaxies is larger then lensng which can be produced within General Relativity due to visible matter only.This is again traditionally explained by the presence of the appropriate amount of dark matter.But constituents of dark matter are still not found in spite of the numerous experimental efforts.In this situation it is worthwhile to develope models alternative to General Relativity although it is clear that approximations of these models for the solar system scales should coincide with General Relativity which is perfectly experimentally tested in the solar system.

Conclusions
We suggested a new explanation of flatness of galaxies rotation curves without invoking dark matter.For this purpose a new tensor gravitational field is introduced in addition to the metric tensor.Flat rotation speeds of stars in our model satisfy the known barionic Tully-Fisher relation.

Acknowledgments
The author is grateful to colleagues from the Theory Division of the Institute for Nuclear Research for helpful discussions.