Modified Brans–Dicke Theory in Arbitrary Dimensions
Abstract
Within an algebraic framework, used to construct the induced–matter–theory (IMT) setting, in –dimensional Brans–Dicke (BD) scenario, we obtain a modified BD theory (MBDT) in dimensions. Being more specific, from the –dimensional field equations, a –dimensional BD theory, bearing new features, is extracted by means of a suitable dimensional reduction onto a hypersurface orthogonal to the extra dimension. In particular, the BD scalar field in such –dimensional theory has a self–interacting potential, which can be suitably interpreted as produced by the extra dimension. Subsequently, as an application to cosmology, we consider an extended spatially flat FLRW geometry in a –dimensional space–time. After obtaining the power–law solutions in the bulk, we proceed to construct the corresponding physics, by means of the induced MBDT procedure, on the –dimensional hypersurface. We then contrast the resulted solutions (for different phases of the universe) with those usually extracted from the conventional GR and BD theories in view of current ranges for cosmological parameters. We show that the induced perfect fluid background and the induced scalar potential can be employed, within some limits, for describing different epochs of the universe. Finally, we comment on the observational viability of such a model.
pacs:
04.50.h; 04.50.Kd; 98.80.k; 98.80.JkI Introduction
Models of the universe in more than four dimensions have been widely investigated. KaluzaKlein (KK) type theories Kaluza21 (); Klein26 (); OW97 (), ten–dimensional and eleven–dimensional supergravity FP12.book () as well as string theories DNP86 (); GSW.book () are well–known examples. Multidimensional Brane–World models, space–time–matter or IMT scenarios stm99 (); scalarbook (); 5Dwesson06 (); pav2006 () in five dimensions, seeking the unification of matter and geometry, constitute other settings with additional spatial dimensions.
Unifying electromagnetism with gravity has been investigated by Kaluza by admitting Einstein general relativity (GR) in a five–dimensional space–time under three key assumptions OW97 (): (i) absence of matter in higher dimensional space–time, (ii) defining the geometrical quantities exactly the same as they were in GR and (iii) omitting the derivatives with respect to the extra coordinate (cylinder condition). Since Kaluza’s procedure in 1921, compactified, projective and noncompactified versions have been studied as interesting approaches to higher dimensional unification, in which, at least one of the Kaluza’s key assumptions has been modified. Among these mentioned versions, we will consider the third. In fact, the IMT is one of these developed procedures. In the IMT, the presence of extra dimensions is also elevated to the hypothesis that matter in a four–dimensional space–time has a purely geometric origin. More precisely, it has been proposed stm99 (); 5Dwesson06 (); PW92 (); OW97 () that one large extra dimension is required to obtain a consistent description, at the macroscopic level, of the properties of matter as observed in the four–dimensional space–time of GR. The IMT has been employed in the cosmological context, where concrete scenarios have been appreciated in view of recent cosmological observational data stm99 (); DRJ09 (); RJ10 (). The application of the IMT framework to arbitrary dimensions has been performed in RRT95 (), relating a vacuum –dimensional solution to a –dimensional GR space–time, with induced matter sources.^{1}^{1}1Geometrically generated by means of the dimensional reduction process. This approach has also been employed to obtain lower dimensional gravity from a four–dimensional space–time description.
Another generalization of the IMT, in which the role of GR as a fundamental underlying theory is replaced by the BD theory of gravity, has also been investigated ARB07 (); Ponce1 (); Ponce2 (). The BD theory is an extension of GR, in which the Newton gravitational constant is substituted, in the Jordan frame, by a non–minimally coupled scalar field BD61 (); D62 (). In this latter application of the IMT, it has been shown that five–dimensional BD vacuum^{2}^{2}2From now on, we call “vacuum” to a situation where there is not any other type of ordinary matter, with the BD scalar field being the only formal “source” of gravity. We should also note that in Ponce2 (), the inducing procedure has been started from a very general BD field equations, rather than the vacuum space–time. equations, when reduced to four dimensions, induce a modified four–dimensional BD theory. This feature is of some relevance. In fact, despite of some (“conventional”) versions of a four–dimensional BD setting, where a few assumptions have been advocated in order to obtain an accelerating cosmos,^{3}^{3}3For example, assuming the BD coupling parameter as a function of the time BP01 (), or introducing a time–dependent cosmological term MC07 () and/or adding a particular kind of scalar potential to the Lagrangian (or without considering any scalar potential) by assuming a fluid with dissipative pressure SS01 (); SSS01 (). Further, in SS03 (), the authors derived the accelerating universe in the BD theory by assuming a scalar potential compatible with the power–law expansion of the universe. Also, in BM00 (), it has been shown that the BD setting with a quadratic self–coupling of the BD scalar field and a negative leads to accelerated expansion solutions. the mentioned IMT setup within a BD theory ARB07 (); Ponce1 (); Ponce2 () provides a more appealing perspective based on a fundamental concept. More concretely, in the context of spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology, by employing the Wesson idea stm99 (); 5Dwesson06 (); PW92 (); OW97 () for the BD theory, it has then been shown Ponce1 (); Ponce2 () that the subsequent self–interacting scalar potential (geometrically due to the extra dimension) and the induced matter lead to cosmological acceleration of the matter dominated universe. Furthermore, the generalized Bianchi type I RFS11 () and FLRW BFS11 () models have been studied in this scenario. Our intention in this work will be to generalize the simplest scalar–tensor gravity model, the BD theory, under some of the main assumptions of the IMT, and then critical studying a spatially flat FLRW cosmological model in the extracted gravity model.
In the other reduced BD setting (different from Ponce1 (); Ponce2 ()), that is also based on a fundamental concept, a five–dimensional manifold with a compact and sufficiently small fifth dimension (cylindiricity condition) has been assumed and then the five–dimensional BD equations reduced on a hypersurface orthogonal to the extra dimension qiang2005 (); qiang2009 (). By assuming a few constraints especially on the matter content in fivedimensional space–time, the four–metric in this four–dimensional reduced theory is coupled with two scalar fields, which are responsible for the accelerated expansion of the universe.
In the context conveyed in the previous paragraphs, the objective of our work is to generalize the IMT formulation of the BD setting towards any arbitrary –dimensional space–time. Our work is organized as follows. In Section II, we derive the BD field equations in dimensions and then, by applying a dimensional reduction procedure, within an IMT framework, we construct the MBDT on a hypersurface. Moreover, in this section, the –dimensional field equations lead to a very specific –dimensional^{4}^{4}4By –dimensional, we mean –dimensional space–time. BD theory, where new dynamical ingredients are present, namely, an effective induced self–interacting scalar potential. Subsequently, we investigate cosmological applications. More concretely, in Section III, we discuss exact solutions of BD cosmology in a ()–dimensional vacuum space–time. Then, in Section IV, by means of the MBDT–IMT framework, we study the reduced –dimensional cosmological solutions. We analyze them for different ranges of the equation of state parameter in a four–dimensional spacetime and, subsequently, compare our results with recent constraints on the BD theory LWC13 () based on the new cosmological data (e.g. Planck Planck.XVI ()) as well as obtained results in the context of the standard BD theory, e.g., BP01 ()BM00 (). Finally, we present our conclusions in Section V. In Appendix A, we show that a –dimensional BD theory can be derived from the simplest version of a KK theory in a –dimensional space–time.
Ii –Dimensional Brans–Dicke Theory From Dimensions
The action for the –dimensional BD theory, in the Jordan frame, is written as
(1) 
where is the BD scalar field, is an adjustable dimensionless parameter called the BD coupling parameter,^{5}^{5}5Usually, concerning the possibility of applying a conformal transformation to bring the theory from the Jordan frame to the Einstein frame, the BD coupling parameter is assumed to be for a –dimensional space–time Faraoni.book (). the Latin indices run from zero to , is the curvature scalar associated with the –dimensional space–time metric , is the determinant of the metric and denotes the covariant derivative in –dimensional space–time. The Lagrangian describes ordinary matter in –dimensional space–time, which depends on the metric and other matter fields except on , and we have chosen .
The variation of action (1), with respect to the metric and the scalar field, gives the equations
(2) 
and
(3) 
respectively, where and is the energy–momentum tensor (EMT) of the matter fields in –dimensional space–time. Contraction of the indices in Eq. (2) yields
(4) 
where . By replacing (4) into (3), we further get
(5) 
In the following, by means of the reduction procedure in the context of the BD theory, we relate the –dimensional field equations to the corresponding ones, with geometrically induced sources, on the –dimensional space–time. Let us be more precise. We derive the reduced field equations onto a –dimensional hypersurface by using the BD Eqs. (2) and (5), in a –dimensional space–time described with a line element
(6) 
where the Greek indices run from zero to , is a non–compact coordinate associated to th dimension, which is henceforth labeled with . The indicator allows to choose the extra dimension to be either time–like or space–like, and is a scalar that depends on all coordinates. Choosing the line element (6) is obviously restrictive, but it is also constructive FR04 (), for, as we will convey, it serves our herein purposes. We assume that the whole space–time is foliated by a family of –dimensional hypersurfaces, , defined by fixed values of the extra coordinate. Hence, the intrinsic metric of each hypersurface, e.g. for , is obtained by restricting the line element confined to displacements on it, being orthogonal to the –dimensional unit vector
(7) 
along the extra dimension Ponce1 (); Ponce2 (). Thus, the induced metric on the hypersurface has the form
(8) 
Now, letting and , Eq. (2) gives the –dimensional part of the corresponding –quantity as
where is the covariant derivative on the hypersurface, whose computation employs . Furthermore, the notation denotes the derivative of any quantity with respect to the extra coordinate , and . We also have used the following relations
(10)  
(11)  
(12) 
To obtain the BD effective field equations on the hypersurface, we should construct the Einstein tensor on the hypersurface. Therefore, we relate the and to their corresponding quantities on the –dimensional hypersurface. In this respect, we get
(14) 
Also, by using Eqs. (2), (4), (5) and (14), we obtain
where the relation has been used. By applying relations (14) and (II), we can relate the Ricci scalar in –dimensional space–time to its corresponding one on the hypersurface, as
By using the above expressions, we can eventually obtain the reduced equations onto the –dimensional hypersurface. This will produce our –dimensional MBDT scenario. In what follows, we outline these retrieved equations in three separated steps, providing suitable interpretations.
Firstly, by applying equations (II), (II) and (II), we construct the Einstein equations on the hypersurface as
(18)  
(19) 
The above result conveys the standard BD equations that contain an induced scalar potential, though, there are a few points which we should make clear:

represents the effects of the –dimensional EMT on the hypersurface and is given by
(20) Clearly, if one assumes that the –dimensional space–time is empty of the usual matter fields [i.e., no term in action (2.1)], then will vanish.

The quantity is an induced EMT for a BD theory in dimensions and, in turn, it contains three components, namely,
(21) where
(24) The first part of the induced EMT, i.e. , is the th part of the metric (6) which is geometrically induced on the hypersurface. In fact, as the BD scalar field plays (inversely) the role of the Newton gravitational constant, we can deduce that this part is the modified version of the induced EMT, introduced in the IMT scenario. Whereas, the second part, i.e. , depends on the BD scalar field and its derivatives with respect to the th coordinate, has no analogue in IMT.

The quantity introduced by is the induced scalar potential on the hypersurface, which is derived from the other reduced equation on the hypersurface, see Eq. (II).
Secondly, we obtain the –dimensional counterpart of Eq. (5), the wave equation on the hypersurface. By contracting Eq. (18), we get a relation between , and as
(25) 
Then, by substituting relations (II) and (25) into Eq. (3) and applying relations (11) and (12), we finally achieve
(26) 
where
Hence, in this applied approach, the dimensional reduction procedure provides an expression to obtain the potential, up to a constant of integration, rather than being merely introduced by hand.
Finally, we derive the counterpart equation for a conservation equation introduced within the IMT. For this purpose, by substituting and in Eq. (2), we get
(28) 
where the first equality comes from the metric (6). On the other hand, metric (6) for the mentioned component gives
(29) 
where is given by
(30) 
Therefore, Eqs. (28) and (29) give the dynamical equation for as
(31)  
(32) 
As we conclude this section, let us further clarify a few points about the herein retrieved –dimensional MBDT.

The –dimensional field equations (2) and (5), with a general metric (6), split naturally into four sets of Eqs. (II), (18), (26) and (31). As mentioned, Eqs. (18) and (26) are the BD field equations on a –dimensional space–time, with a geometrically induced energy–momentum source.^{6}^{6}6More precisely, they are retrieved from the action , where specifically . Such a correspondence is guaranteed by the Campbell–Magaard theorem C26 (); M63 (); RTZ95 (); LRTR97 (); SW03 (). Furthermore, it is important to note that Eq. (II) has no standard BD analog, and the set of Eqs. (31) is a generalized conservation law introduced within the IMT.

The induced EMT is covariantly conserved (the same way as in the standard four–dimensional BD theory), i.e. .

In the special case^{7}^{7}7The case corresponds precisely to the value predicted when the BD theory is derived as the low energy limit of some string theories Faraoni.book (); BD12 (). when , is a cyclic coordinate and , the scalar potential, without loss of generality, vanishes. Thus, to reproduce a general version of a –dimensional BD theory by means of the above dimensional reduction procedure, we should notice those requirements.^{8}^{8}8Also, we should notice that when the coupling parameter goes to infinity, with suitable boundary conditions, the approach developed in this section may be viewed as a generalization of the procedure of RRT95 () (but not always, corresponding to the content in BR93 (); BS97 (); Faraoni99 ()).
We would like to close this section by indicating an interesting point regarding how the BD theory can be related to the KK setting. An important benefit of the MBDT is that the induced matter and scalar potential, which depend on the BD scalar field and its derivatives, derived via the BD action (1), should be regarded as fundamental quantities rather than quantities added by hand. However, some questions may be asked: can we accept the BD scalar field in action (1) as a fundamental field? Where does it emerge from?
In Appendix A, by generalizing the approaches of scalarbook (); PS02 (); Faraoni.book (), we show that the –dimensional BD framework (in vacuum) can be derived from a generalized GR (i.e., a simplest version of the KK) theory in a –dimensional space–time by obtaining in which is the number of the compactified extra spatial dimensions. Moreover, in this formalism, the BD scalar field emerges as a geometrical quantity, namely, it is related to the determinant of the metric associated to submanifold of extra dimensions.
Iii exact solutions of BD cosmology in ()–Dimensional vacuum space–time
In this section, we assume an empty^{9}^{9}9Eqs. (2) and (5) are the field equations of the BD theory in –dimensions, though (as described in the Introduction), we propose to employ them in the this section, in terms of “vacuum” cosmological solutions, which are defined as a configuration where there is no other matter source (except the BD scalar field) in –dimensional space–time. In this case, Eq. (3) becomes which means that the –dimensional scalar curvature is generated only by a free scalar field ARB07 (). –dimensional space–time that is described by an extended version of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric and then, in section (IV), by means of the MBDT procedure described in the previous section, we investigate the cosmology reduced on a –dimensional hypersuface. Furthermore, in order to respect the space–time symmetries, we assume that the metric components and the BD scalar field depend only on the comoving time^{10}^{10}10In RFS11 (), it has been shown that, in a five–dimensional space–time, when the usual three scale factors are functions of the cosmic time whereas the scale factor of the extra dimension is a constant (i.e. ), if the BD scalar field is assumed as a function of both and , then, in general, we will encounter inconsistencies in the field equations.. Moreover, we choose the case with the metric
(33) 
where is the cosmic time, and are cosmological scale factors and ’s are the Cartesian coordinates.
The dynamical field equations in vacuum^{11}^{11}11In the present work, we leave a few more extended solutions that can be produced by assuming the following general cases: i) taking the BD scalar field and metric components such that they also depend on the spatial coordinates, specially, the extra coordinate , ii) assuming an ordinary matter in the bulk, iii) and/or iv) considering a more flexible embedding approach Leon06 (); Leon061 (); Leon09 (). Considering such assumptions would make the analysis more realistic. Concerning the second assumption, we should stress that, in this work, we have been studying the BD theory in the Jordan frame in a –dimensional spacetime, in which the is seen as a (scalar) part of the gravitational degrees of freedom rather than a matter degree of freedom (where it could play the role of a –essence field, see, e.g., Kim04 (); Kim05 () in a –dimensional space–time). Moreover, we will assume henceforth that there is no ordinary matter in –dimensional space–time. These assumptions allow to extract the induced matter and the scalar potential as a manifestation of pure geometry in a –dimensional world OW97 (); stm99 (). Within this context, the suggestion is to replace the “base wood” of matter by the “pure marble” of geometry DNP86 (). are given by
(34)  
(35)  
(36)  
(37) 
where equation (34) is obtained from (5), and equations (35), (36) and (37) are associated to the components , and of Eq.(2), respectively, in which we have used equation (34). “ ” denotes the derivative with respect to the cosmic time.
Let us solve Eqs. (34)(37) by using the power–law solutions^{12}^{12}12The power–law solutions, in the conventional BD theory, have resemblance to the inflationary de Sitter attractor in GR. However, in the scalartensor gravity, these solutions have been assumed for investigating the quintessence models Faraoni.book ().
(38) 
where , and are constants determined in an arbitrary fixed time , and , and are parameters, which are not independent, satisfying the field equations. Substituting these solutions in equations (34)–(37), it yields
(39) 
where, by assuming negative values for , we must have
(40) 
Some special solutions, that are of our interest, include:

When tends to zero, then goes to infinity and takes a constant value. In this limit, the field equations are only satisfied^{13}^{13}13We disregard the static case of . for , i.e. . Thus, the –dimensional solution
(41) is obtained, which is the unique solution for the –dimensional metric (33) for the Einstein field equations in vacuum.

In the case where , we cannot set this value of in solutions (39) and then get the values for the exponents , and . However, instead, we should start from the field equations (34)–(37). It is straightforward to show that for , we have a –dimensional de Sitter–like space
(42) where , and are constants.

One of the most well–known class of solutions in standard BD theory is the O’Hanlon and Tupper solution o'hanlontupper72 (). This class corresponds to “vacuum” with a free scalar and the range of the BD parameter in four–dimensional space–time is restricted to , Faraoni.book (). Assuming and at the beginning, thus solutions (38) and (39) are reduced to a generalized O’Hanlon and Tupper solution in a ()–dimensional space–time as
(43) where
(44) and
(45) where and and algebraically are related by constraint^{14}^{14}14For convenience, we will drop the index from the parameters and . . We also assumed . When the cosmic time goes to zero, this solution has a big bang singularity. In a four–dimensional space–time, this solution has been obtained by means of different methods o'hanlontupper72 (); KE95 (); MW95 (); Faraoni.book (). We can easily show that
(46) When tends to zero, then goes to infinity and takes a constant value. In this limit, we have
(47) We should note that in the mentioned limit, the corresponding general relativistic solution is not reproduced; as (47) illustrates, it is not a Minkowski space. Let us check it for, e.g., a four–dimensional space–time (i.e. by setting ). In this special case, the solutions are reduced to
(48) where the scale factor has a decelerated expanding behavior Faraoni.book (); RFK11 ().
In the next section, as an application of the MBDT in cosmology, we proceed to investigate the effective –dimensional picture generated by the exact power–law solutions in –dimensional space–time.
Iv Reduced Brans–Dicke Cosmology in Dimensions
The non–vanishing components of the induced EMT (21) associated to the metric (33) on the hypersurface are
(50) 
where the induced potential will be determined from (II). As the different components of [where with no sum] are equal, thus the induced–matter can be considered as a perfect fluid with an energy density and isotropic pressures .
In order to derive the induced scalar potential, we substitute the power–law solutions (38) into (II) and evaluate it on the hypersurface. Thus, we get
(51) 
where , , , and are given by relations (38) and (39). By integrating this equation, we obtain
where the constants of integration have been set equal to zero. In the special cases, regardless of , where or , the scalar potential will be zero. Also, for the particular case of where the BD scalar field takes constant values, it is straightforward to show that identically vanishes, and thus, without loss of generality, we can set in this case. From now on, we will not investigate the logarithmic potential with , for it leads to some difficulties when the weak energy condition is applied.
By substituting the scalar potential (for ) and also the power–law solutions (38) into relations (IV) and (50), we get the induced quantities in a –dimensional hypersurface as
(52) 
and
(53) 
which, by applying (39), we have
(54) 
and
(55) 
It is straightforward to show that the conservation law for the above induced EMT is satisfied, as expected. Consequently, from relations (54) and (55), the equation of state for the power–law solutions on the –dimensional hypersurface is
(56) 
We proceed to discuss cosmological consequences for different types of matter. Hence, it will be appropriate to express the parameter in terms of the deceleration parameter , namely
(57) 
In order to proceed and analyze the induced quantities on the hypersurface, we should express the exponent associated to the scalar , the scale factor of the th dimension, in terms of , and . Thus, from relation (56), we get
(58) 
where