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Dark matter and cosmic acceleration from Wesson's IMT

In the present work a procedure is build up, that allows obtaining dark matter and cosmic acceleration in our 4 dimensional universe embedded in a 5 dim. manifold. Both, DM and the factor causing cosmic acceleration, as well ordinary matter are induced in the in the 4D space-time by a warped but empty from matter 5D bulk. The procedure is carried out in the framework of the Weyl-Dirac version of Wesson's IMT enriched by Rosen's approach. Making in the given procedure use of present observational data one can develope a model of the vuniverse with conventional matter, DM and cosmic acceleration induced by the 5D bulk.


INTRODUCTION
Two interesting phenomena are for the fast decades in the spotlight of cosmology: dark matter (DM) and the acceleration of the universe at present (cf. e.g. [1][2][3]) These remarkable cosmological events are verified observationally, but the origin of dark matter and of dark energy (DE) causing cosmic acceleration is till unknown.
At present it is believed that DM and DE make up ~ 96% of the mass/energy content of the Universe: DM contributes ~25%, DE -up to 71%. The latter affects the expansion rate of the Universe, which in turn affects the growth of structure and the distances to objects. It was found that the Universe decelerated until 5 . 0 = z and age 9Gyr, when it began accelerating and that the expansion has actually been speeding up for the past years 10 5 5Gyr 9 × ≈ [4][5][6][7] .
There exist a number of theories trying to explain the origin of dark matter, but within the context of the general theory of relativity, cosmic acceleration cannot be explained by any form of conventional matter or energy. It can however be accommodated by a nearly smooth and very elastic ( Thus, what is the nature of DM, that holds the Universe together and of DE that is responsible for the expansion of the Universe to speed up? In the present paper an attempt is made to answer this question. Issuing from Wesson's Induced Matter Theory (MIT) [8] in its Weyl-Dirac (W-D) [9,10] modification and applying Rosen's [11] approach, we have carried out a procedure, that allows getting luminous matter, DM and cosmic acceleration in our Universe, the latter regarded as a 4D brane in the 5D bulk.
Wesson's IMT is based on Einstein's general theory of relativity extended by the idea that our 4D space-time is a surface embedded in a 5-dimensional manifold, the bulk.
As long ago, in 1921, Kaluza proceeding from this idea proposed a unification of electromagnetism and gravitation [12] in the frame of a 4D hypersurface embedded in a as a hypersurface (brane) embedded in a 5D space (bulk) Paul Wesson and collaborators have shown that the matter being in the 4D universe is induced by the geometry of the 5D bulk, the latter being a warped, but empty from matter one. Among works in the framework of the IMT are those dealing with test particles in higher-dimensional models, successful cosmological models, cosmological models with a variable cosmological constant, papers on dark matter, dark energy and induced unified theory of gravitation and electromagnetism.
Some years ago the present writer has shown that in Wesson's IMT, the 5D manifold (bulk) is rather a Weyl space, than a Riemannian one. The Weyl-Dirac [W-D] modification of Wesson's IMT was presented [9,10]. Following the ideas of Weyl [50][51][52] and Dirac [53] in each point of the bulk, in addition to the metric tensor the Weyl connection vector A w and the Dirac gauge function Ω were introduced.
Regarding the 5D Weyl field A w as the prototype of the 4D electromagnetic potential one obtains on the 4D brane both, gravitation and electromagnetism, as well gravitational matter and electric current. As Ω , , A AB w g are integral parts of the W-D geometry describing the bulk one concludes that the latter, being warped but empty, induces on the brane the unified theory. It must be pointed out that a unified theory of gravity and electromagnetism was already obtained in the original IMT [3,30,39] by a K-K procedure. Recently, in the framework of the W-D modification of Wesson's IMT, models of classical (non-quantum) neutral and charged fundamental particles induced by the bulk were presented [54,55].
It turns out that the W-D bulk of Wesson's IMT can create DM and DE on the brane.
As long ago, in 1982, Nathan Rosen [11] in discussing the (4-dimensional) Dirac modification of Weyl's theory pointed out that Dirac had chosen a particular value for a certain parameter appearing in his variational principle and that this value could be taken differently. Rosen showed that the parameter could be chosen so that, instead of an electromagnetic field, one gets a Proca [56] vector field which, from the standpoint of quantum mechanics, can be interpreted as an ensemble of particles having finite mass and spin 1. Later [57][58][59][60] it was suggested by Nathan Rosen and the present writer that these particles, named weylons, interacting gravitationally with ordinary matter, form the CDM in the universe. For the ensemble of weylons in the universe an acceptable cosmological behavior was obtained [58].
In order to get in the framework of Wesson's IMT a 4D brane with induced DM and cosmic acceleration, we will apply Rosen's approach [11] to the 5D W-D bulk.
In the present work following conventions are valid. Upper case Latin indices run from 0 to 4; lowercase Greek indices run from 0 to 3. Partial differentiation is denoted by a comma (,), Riemannian covariant 4D differentiation by a semicolon (;), and Riemannian covariant 5D differentiation by a colon (:). Further, the 5D metric tensor is denoted by AB g , its 4D counterpart by µν h ; sometimes 5D quantities will be marked by a In the original 5D IMT of Wesson, one regards the gravitational 5D bulk as pure geometry (gravitation) without any additional fields. The geometry is described by the metric tensor Thus, the principal phenomenon, which carries information, is a metric perturbation propagating in the form of a gravitational wave. In order to avoid misinterpretations one must assume that all gravitational waves have the same speed.
Therefore, the isotropic 5-dimensional interval 0 2 = dS has to be invariant, whereas an arbitrary line element may vary. The situation resembles the 4D Weyl geometry [11,[50][51][52]60] where the light cone is the principal phenomenon describing the space-time and hence the light-like interval 0 2 = ds is invariant rather than an arbitrary line-element β α αβ dy dy h ds = 2 between two space-time events. Following the ideas of Weyl [50][51][52] and Dirac [53] developed by Nathan Rosen [11] and the present writer [60] the Weyl-Dirac version of Wesson's IMT was proposed recently [9,10]. In that version the 5D manifold {M} (the bulk) is mapped by coordinates { } N x and in every point exists the symmetric metric tensor AB g , as well the Weylian connection vector C w and the Dirac gauge function Ω .
The three fields AB g , C w and Ω are integral parts of the geometric framework, and no additional fields or particles exist in the bulk {M}. In this 5D manifold, field equations, for AB g , C w , and Ω are derived from a geometrically based action.
One considers a 5-dimensional manifold { M } (the "bulk") with a symmetric with Φ being the lapse function.
A 5D quantity (vector, tensor) in the bulk has 4D counterparts located on the brane.
These counterparts may be formed by means of the following system of basis vectors, The brane l Σ is stretched on four ( ) Together with the main basis { } Let us consider a 5D vector in the bulk {M}. Its 4D counterpart on the brane l Σ is given by . ; On the other hand the 5D vector may be written as Actually, (7) is a decomposition of A V into a 4-vector µ V and a part normal to l Σ .

THE ACTION INTEGRAL AND THE FIELD EQUATIONS IN THE BULK
Being guided by Dirac's approach [53] we derive the field equations in the 5D bulk from a variational principle 0 The Lagrangian geom L is formed from a number of suitable, geometrically based terms.
Making use of AB g , C w , Ω and their derivatives we build the following action integral In the action (10) appear the following quantities: the Weyl length curvature tensor , formed from the 5D Christoffel symbols C AB Γ ; Dirac's gauge function Ω and its partial derivatives . The symbol Λ stands for the cosmological constant.
Finally, σ is an arbitrary constant. The dynamical variables in (10) are AB g , D w and Ω .
In the present work we want to get on the 4D brane dark energy causing cosmic acceleration as well dark matter, both induced by the bulk. Therefore we adopt Rosen's [11] idea and assume that 0 ≠ σ ; and that σ can obtain arbitrary values 2 . In this case the Weyl field ( A w ; AB W ) of the bulk is no longer the 5D creator of the 4D Maxwell field.
Considering the variation of (10) with respect to the dynamical variables AB g , D w and Ω , one obtains the field equations in the bulk.
Varying AB g one obtains the "gravitational" equation is the 5D Einstein tensor, and 2 Setting 0 = σ one would obtain from (10) the action integral of the W-D modification considered in [1], so that on the brane one would have a geometrically based unified theory of gravitation and electromagnetism with mass and current induced by the bulk.
(12) AB M can be regarded as the energy-momentum density tensor of the 5D Weyl field.
Let us consider the Ω -equation. On one hand varying Ω in (10) we obtain On the other hand, contracting (11) we have Comparing (14) and (15) It seems that EQ. (16) is a restriction on Ω . However, turning to equation (13) (16). Thus, the − Ω equation (14) is a corollary of (11) and (13) Considering cosmological models we believe that at large scale the universe is isotropic and homogeneous and no cosmic vector functions are present. However, we can adopt the existence of microscopic areas with chaotically oriented vector fields. This standpoint will be applied to the 5D Weyl-Dirac bulk. Thus, on the one hand the bulk has a microstructure caused by chaotically oriented local Weyl (17) By (17), EQ. (13) is satisfied identically and in EQ. (11) are no vector fields.
One can describe both, local and global effects if one writes (18) With the separation (18) between the global and local fields, one rewrites EQ. (11) as and EQ. (13) as ( ) global AB T represent the global effect caused by the local cells of the microstructure.
Turning to the local effect we point out that in a 5D very small micro-region, the Dirac gauge function Ω remains nearly unchanged, so that inside the micro-region one can take the Einstein gauge 1 = Ω . Then EQ. (19) takes the form whereas EQ. (20) may be written as The local field on the brane will be considered in detail in section 5.

EQUATIONS ON THE 4D BRANE
There are two sets of equations in the brane; the first, induced by EQ. (11), gives the equations of gravitation, the second set, being derived from EQ. (13), describes the Weylian field.
The equations of gravitation may be obtained by inserting EQ. (11) into the Gauss equations (cf. [9,46,47]) In (21)  ( ) In EQ. (22) appear the quantities introduced in (21a, b) It is worth noting that 0 ≠ σ , so that in the present framework AB W is not the prototype of a 4D Maxwell field, as it was in previous papers [9,54,55].
As pointed out above, we believe that at large scale the universe is isotropic and homogeneous and no cosmic vector functions are present. However, we allow the existence of small areas with chaotically oriented vector fields. This standpoint was applied to the 5D bulk, and by (6) it is valid in our universe, the 4D brane. According to (6) and (18) the Weyl field on the brane, which is induced by its 5D counterpart, is divided into a cosmic gradient vector field and a chaotically oriented local vector field Omitting the subscribe " loc " we denote from now on by ν w the local chaotically oriented Weyl vector field in microscopic areas.

LOCAL WEYL FIELDS ON THE BRANE
One turns to a very small 4D micro-region, a cell, in which one can take the Einstein gauge 1 = Ω , (cf (19b), (20a)), so that from (22) (25) and from (23) Considering an empty micro-cell, with no induced matter and current (the − ε terms in equations (25,26)), and discarding also the cosmological term in (25) EQ. (27), with the RHS being the energy-momentum density of the Weylian field inside the cell, may be rewritten explicitly as Turning to (28) one obtains a Lorentz-like condition Taking into account the relation In ( It must, however, be noted that 16 years ago in a paper [57] by Nathan Rosen and the present writer, a 4-dimensional Weyl-Dirac scenario was developed in which bosons, named weylons, constitute dark matter in the universe. It was proposed that our universe, being filled with luminous matter, has a chaotic Weylian microstructure, but is described on large scale by the F-R-W metric. The Weylian microstructure creates particlesweylons, that interact with ordinary matter only trough gravitation and not directly, so that weylons are analogous to photons and gravitons but differ in that they are massive.
Two years later the same authors [58] have investigated the behavior of Weylian dark matter in the universe, described either by the non-singular model [61] or by the standard one [62]. It was shown that the ensemble of weylons, being described by the Bose-Einstein statistics, may be considered in the universe as a gas, the thermal behavior of which is given by means of the Boltzmann statistics, provided the weylon mass, . The behavior of this DM was considered in [58] from the time when weylons were created and until the present. For weylons having mass GeV 10 MeV 10 5 W ≤ ≤ m an acceptable cosmological behavior was obtained. It was also shown in [58] that this weylon DM was unimportant in the early stages of the universe, but became important at the time of galaxy formation and may have played a role in this process.
One should distinguish between the DM in and around cosmic inhomogeneities (like galaxies, clusters, etc.) and the DM pervading all of the cosmic space [59]. The first one, known as Cold Dark Matter (CDM), may be presented by the weylon gas considered above. [57], we will adopt the main results and conclusions of [57,58] without repeatedly developing the formalism and proving the formulae.

THE GLOBAL FIELD ON THE BRANE
In the previous section the local Weyl field as creator of cold dark matter (CDM) was considered. Now, let us turn to the gravitational equations of the global field in a FRW universe. Making use of the separation (24) we find that (23)    , and making use of (21a, b) as well of (37), (37a), and denoting, we can write down the gravitational equations (34) explicitly It is worth noting that the terms in the second line of (38) and (39) describe a substance with the equation of state, 0 = + P ρ i.e. prematter (cf. [61]). From these terms it is possible to obtain a repulsing action that may lead to cosmic acceleration, alternatively, , one gets a cosmological constant induced by the bulk (Cf. the paper by Liu and Wesson, [40]).
In EQ.-s (38,39), the gauge function, ) , ( l t Ω is an arbitrary but positive function, are solutions of the gravitational equations.
To simplify the procedure one can turn to ordinary differential EQ.-s, assuming that the dependence of the function, Ω Φ and , , R on t and l may be separated as follows can take one brane on the addition, In Inserting the separation condition (40) into (38, 39) we obtain It is believed that the − β matter (45) will form DE being responsible for the present cosmic acceleration.
The above considered procedure is supplying a plausible framework for constructing cosmological models, which contain induced by the 5D bulk luminous matter, cold dark matter and dark energy, the latter causing the present cosmic acceleration. It is worth emphasizing that all kinds of matter, the luminous, CDM and DE are induced by the 5D bulk in the framework of the W-D version [1,2] of Wesson's geometrically based IMT [8,.

CONCLUSION and DISCUSSION
In this work we have made an attempt to explain the origin of DM and DE causing cosmic acceleration. In our scenario, all forms of matter, luminous, DM and DE are induced by a 5D bulk in the 4D brane, our Universe. The procedure is built up in the framework of the Wesson's IMT [8,, in its W-D version [9,10], where the bulk is characterized rather by the Weyl-Dirac geometry [50][51][52][53], than by Riemann's. In addition an approach proposed by Nathan Rosen [11] is applied in order to get DM and DE..
Up to the observation of DM effects (the flat rotation curves of spiral galaxies and other evidences of "missing mass" in the universe) the widely accepted cosmological model was the standard hot big bang model [62], which describes in detail the evolution of the Universe from a fraction of a second after the beginning, when it was just a hot soup of elementary particles to the present some 9 10 7 . 13 × years later, when it is filled with stars, planets, galaxies, clusters of galaxies and us [4].
There were also many alternative models, among them the Singularity-Free Cosmological Model [61], which describes an oscillatory Universe that underwent a contraction phase before the present expansion. After a prematter period and a transition period this model at sec 10 12 . Weylons created by the chaotically oriented fields in micro cells cannot be detected directly as they interact only gravitationally with conventional matter. But one can perhaps estimate roughly the weylons mass from flat rotation curves of spiral galaxies.
From the rotation curves as well from massive bodies located in the galaxies' center (supposed they are made of weylons) one can estimate the account of weylon CDM in the Universe.
In Sec. 6 we considered the global gravitational field; its total energy-momentum tensor may be split into three parts: the first presenting the induced by the bulk conventional, luminous matter, the second giving the contribution of the CDM formed by weylons, the third part is formed from the 4D Dirac gauge function ( ) t β and, presumably after an appropriate choice of β , it will create DE and global DM.
Finally, in the Appendix it is shown that the bulk is a warped, but empty from matter one.
In order to obtain a cosmological model with luminous matter, DM and DE the following procedure may be proposed. (45) is pervading all of the cosmic space. This is a Global DM effect and one must distinguish between this Global DM and Cold DM, which is invoked by weylons and located in and around cosmic inhomogeneities, like stars, galaxies, clusters etc.
A paper based on the above-considered procedure with recent cosmological data will be presented in the near future.

APPENDIX
Above we have considered a framework of inducing matter by the 5D bulk in the 4D brane. We obtained luminous matter, DM and DE, the latter being regarded as causing the expansion of the Universe to speed up, The procedure is carried out in the framework of Wesson's IMT, in its Weyl-Dirac modification, with Rosen's approach.
We recall that at large scale the bulk was regarded as isotropic and homogeneous, so that no cosmic (global) vector functions can be present. However, we adopted the existence of microscopic areas, cells, with chaotically oriented Weyl vector fields loc A w .
The latter under certain circumstances might create massive particles. As in induced matter theories the bulk has to be a pure geometric one, we show below that in our framework the bulk can be made empty from matter by plausible conditions.
It was found that one can describe both, local and global fields by the separation condition (18) Ω Ω − = + =  We assume that in A w one can separate the dependence on the fifth coordinate from that on the space-time coordinates, so that Conditions (A-8) and (A-10) will be imposed on the local chaotically oriented Weyl vector fields, so that in the bulk can exist gravitational and Weyl waves (A-11), but no matter. The bulk is warped and empty. It must be pointed out that imposing (A-8, A-9, and A-10) does not affect or restrict the discussion and results presented in the present work.