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Brans-Dicke Cosmology in 4D from scalar-vacuum in 5D

We show that Brans-Dicke (BD) theory in 5D may explain the present cosmic accelerated expansion without recurring to matter fields in 5D or dark energy in 4D. Without making any assumption on the nature of the extra coordinate or the matter content in 5D, here we demonstrate that the vacuum BD field equations in 5D are equivalent, on every hypersurface orthogonal to the extra dimension, to a BD theory in 4D with a self interacting potential and an effective matter field. The potential is not introduced by hand, instead the reduction procedure provides an expression that determines its shape up to a constant of integration. It also establishes the explicit formulae for the effective matter in 4D. In the context of FRW cosmologies, we show that the reduced BD theory gives rise to models for accelerated expansion of a matter-dominated universe which are consistent with current observations and with a decelerating radiation-dominated epoch.


Introduction
One of the most challenging problems in cosmology today is to explain the observed late-time accelerated expansion of the universe [1]- [13]. Since the gravity of both baryonic (ordinary) matter and radiation is attractive, an accelerated expansion requires the presence of a new form of matter (called dark energy), which could (i) produce gravitational repulsion, i.e., violate the strong energy condition; (ii) account for 70% of the total content of the universe; (iii) remain unclustered on all scales where gravitational clustering of ordinary matter is seen. (For a recent review see Ref. [14]). Possible candidates for dark energy include: a cosmological constant or a time dependent cosmological term [15]- [16]; an evolving scalar field known as quintessence (Q-matter) with a potential giving rise to negative pressure at the present epoch [17]- [20]; dissipative fluids [21]; Chaplygin gas [22]- [23]; K-essence [24]- [27], and other more exotic models [28].
General Relativity (GR) is a well tested theory on solar system scales. In contrast it is poorly tested on cosmic scales [29]. Therefore, the question arises of whether dark energy is not just an observational artifact caused by an inappropriate theory of gravity. This and other puzzles of theoretical and experimental gravity have triggered a huge interest in alternative theories of gravity (See, e.g. [30] and references therein). In recent years there has been a renewed interest in scalar-tensor theories of gravity as viable alternatives to general relativity. In particular, some researchers [31]- [37] have resorted to Brans-Dicke theory (BD) [38] in order to explain the present accelerated expansion of the universe. An attractive feature of BD is that the scalar field is a fundamental element of the theory, as opposed to other models in which the scalar field is postulated separately in an ad hoc fashion. The concept is that the BD scalar field could play the role of Q-matter or K-essence and lead to cosmological acceleration. However, it turns out that this is so only in very particular cases: for values of the coupling parameter ω in the range −2 < ω < −3/2, which not only violate the energy condition on the scalar field but are also inconsistent with a radiation-dominated epoch, unless ω varies with time [32]; when there is a scalar potential, which in turn is added by hand [33]. Further models include the so-called "chameleon fields" that allow the scalar field to interact with matter (see [39] and references therein).
Another attempt to derive the present accelerated expansion from a fundamental concept (or postulate) combines the original BD and Kaluza-Klein (KK) theories with the modern view that our 4D universe can be recovered on a hypersurface orthogonal to the extra dimension. Along these lines, and assuming that the extra coordinate is cyclic (ignorable), it has been shown in [40] that the BD field equations in 5D are equivalent to those of GR in 4D with two scalar fields, viz., the BD scalar field and the γ 55 component of the KK metric (For a recent discussion see Ref. [41]). These two scalar fields may account for the present accelerated expansion of the universe, if one admits the existence of matter in 5D which does not move along the extra dimension and effectively behaves as dust in 4D.
In this work we adhere to the point of view advanced in [40]- [41]. However, we propose that neither the cylindricity condition on the extra coordinate nor the higher dimensional matter hypothesis are necessary. In fact, in this paper we show that, regardless of whether we assume a compact or large extra dimension, BD theory in 5D may explain the present accelerated expansion without introducing matter fields in 5D or dark energy in 4D. We demonstrate that the vacuum BD field equations in 5D are equivalent, on every hypersurface orthogonal to the extra dimension, to a BD theory in 4D with a self interacting potential and an effective matter field. In the context of FRW cosmologies, we show that the reduced BD theory gives rise to models for accelerated expansion of a matter-dominated universe which are consistent with a decelerating radiation-dominated epoch, for the same value of the BD parameter ω.
The paper is organized as follows. In section 2 we perform the dimensional reduction of the scalar-vacuum BD field equations in 5D. Our procedure provides explicit definitions for the effective matter and potential 1 in 4D. In section 3 we study homogeneous and isotropic solutions of the vacuum BD field equations in 5D under the assumption of separation of variables (in general, the metric functions may depend on time and the extra coordinate). We show that there are four classes of separable solutions. We discuss in detail the class of power-law solutions, which in turn consists of five families of solutions. In section 4 we study the 4D cosmological scenarios allowed by the 5D power-law solutions derived in section 3. We find that distinct families of solutions in 5D lead to very much alike scenarios in 4D; they can give accelerating matter-dominated era as well as decelerating radiation-dominated era for the same value of ω in the range −3/2 < ω < −1. In section 5 we give a summary of our results. Finally, in the Appendix we show, by means of explicit integration, that the assumption of separability is consistent with the field equations.

Dimensional reduction of Brans-Dicke theory in 5D
The Brans-Dicke (BD) theory of gravity in 5D is described by the action [40] where R (5) is the curvature scalar associated with the 5D metric γ AB ; γ (5) is the determinant of γ AB ; φ is a scalar field; ω is a dimensionless coupling constant; L (5) m represents the Lagrangian of the matter fields in 5D and does not depend on φ.
The equations for the gravitational field in 5D derived from (1) read where ∇ 2 ≡ ∇ A ∇ A and T AB represents the energy-momentum tensor (EMT) of matter fields in 5D with trace AB . For the sake of generality of the reduction procedure in this section T (5) The field equation for the scalar field φ is determined by (1) as Taking the trace of (2) we find Combining the last two equations we get In this work we use coordinates where the metric in 5D can be written as 2 in such a way that our 4D spacetime can be recovered by going onto a hypersurface Σ y : y = y 0 = constant, which is orthogonal to the 5D unit vectorn along the extra dimension, and g µν can be interpreted as the metric of the spacetime. The effective field equations (FE) in 4D are obtained from dimensional reduction of (2) and (5). To achieve such a reduction we note that where the asterisk denotes partial derivative with respect to the extra coordinate (i.e., ∂/∂y = * ); D α is the covariant derivative on Σ y , which is calculated with g µν , and D 2 ≡ D α D α . Using these expressions, the spacetime components (A = µ, B = ν) of the 5D field equations (2) can be written as To construct the Einstein tensor in 4D we have to express R (5) αβ and R (5) in terms of the corresponding 4D quantities. The Ricci tensor R (4) µν of the metric g µν and the scalar field Φ are related to the Ricci tensor R From (2)-(4) and the second equation in (10) we obtain Substituting this expression into R (5) = γ AB R AB we find where R (4) = g αβ R (4) αβ is the scalar curvature of the spacetime hypersurfaces Σ y . • We are now ready to obtain the effective equations for gravity in 4D. With this aim we substitute the first equation in (10) and (12) into (9) and isolate G where we have introduced the quantity V (φ), which (as we will see bellow) plays the role of an effective or induced scalar potential; S µν is the reduced EMT of the matter fields in 5D and T (BD) µν can be interpreted as an induced EMT for an effective BD theory in 4D. It is given by Since in BD φ acts as the inverse of the Newtonian gravitational constant G, (16) is identical to the induced EMT used in STM (Space-Time-Matter theory) [43]. The second term in (15) depends on the first derivatives of φ with respect to the fifth coordinate and represents the effective EMT in 4D coming from the scalar field. The equation for the scalar potential V is given bellow by (19).
Taking the trace of (13) we obtain a simple relation between R (4) , S = g µν S µν and T (BD) = g µν T (BD) µν , namely (we note that g µν T • To construct the 4D counterpart of (5) we substitute (12) and (17) into (3). After some manipulations we get where This equation, with the r.h.s. evaluated at some Σ y , constitutes a working definition for the potential. We notice that equations (13) and (18) are identical to those of Brans-Dicke theory in 4D derived from the action with m represents the action for matter in 4D. The potential vanishes only in few particular cases. For example when ω = −1, y is a cyclic coordinate and T (5) AB = 0. However, if we are to recover a general version of BD theory in 4D, we must assume ω = −1. Besides, the metric in 5D can depend on y. Therefore, in general V = 0.
• Finally, for the line element (6) we have R The dynamical equation for P αβ is obtained from (2) by setting A = 4, B = µ, viz., (Φφ) P α µ;α = 8πT In the case where T (5) µ4 = 0 and φ = constant, this reduces to P α µ;α = 0. In braneworld theory this quantity is proportional to the EMT of the matter on the brane [44]. If y is a cyclic coordinate, (22) reduces to 0 = 0.
To keep contact with other works in the literature, we note that we can substitute (11) in (14) to obtain an expression where the right hand side of (13) only contains T (5) µν as well as the derivatives of φ, Φ and g µν . In case that there is no y dependence we neatly recover the formulae developed in [40]. However, our equations are completely different from those derived/used in [42].

Scalar-vacuum Brans-Dicke cosmology in 5D
In cosmological applications the 5D metric (6) is commonly taken in the form where k = 0, +1, −1 and (t, r, θ, φ) are the usual coordinates for a spacetime with spherically symmetric spatial sections 3 . For this line element the vacuum (T the spatial components Finally, the wave equation (5) becomes

Separation of variables
In this section, we look for solutions to the above equations under the assumption that the metric coefficients are separable functions of their arguments. In this framework, without loss of generality we can set 4 Then, from (27) we obtain Thus, separability requires * where c 1 , c 2 , c 3 are arbitrary constants and f = f (y) is some function to be determined by the field equations. From (31) we get where N 0 , P 0 and U 0 are constants of integration. Besides, from (30) and (31) it follows that Now, to obtain an equation for f we substitute (29) and (32) into the wave equation (28). The requirement of separability yields a second-order differential equation for f (y) whose first integral is where C s is a separation constant and C is a constant of integration. This equation admits exact integration, in terms of elementary functions, in several cases. For example, setting C = 0 we get Similar power-law and exponential solutions can be obtained for C s = 0 or c 2 = 0.
Next, we substitute the metric functions (29), with (32) and (34), into the FE (24)- (26). We find that these equations have the following structure and W (t) and their derivatives; Inspection of the FE shows that the last two terms in (36) do not cancel out for c 2 = −c 1 = 0 and F ∝ Q. Therefore, the assumption of separability demands either c 1 = 0 for k = 0, or k = 0 for c 1 = 0 as well as the fulfillment of (at least) one of the following conditions: C = 0, η 0 = 0, c 2 = 0.
A detailed analysis of the field equations is provided in the Appendix, where we derive the differential equations to be satisfied by the three functions Q, F and W . Inspection of those equations reveals that they admit power-law and exponential solutions for several choices of the parameters. However, this is clear at the outset if we notice that (30) can also be written as which requiresḞ wherec 1 ,c 2 ,c 3 are arbitrary constants and h = h(t) is some function to be determined by the field equations.
Integrating (38) we get Following the same procedure as above, from the wave equation (28) we find whereC s andC are some new separation and integration constants, respectively. For several choices of the constants we obtain power-law and exponential solutions. For example, it we setC = 0 we get Once again, similar solutions are obtainedC s = 0 orc 2 = 0. In summary, the assumption of separation of variables generates four distinct classes of solutions corresponding to the following choices: where the constants obey the conditions indicated in (35) and (41). When we substitute (42) in the FE (24)-(28) the latter reduce to a system of algebraic equations which provide the appropriate consistency relations for the constants.

Power-law solutions
We now proceed to study in some detail the power-law solutions. We concentrate our attention on the spatially-flat scenario, which seem to be relevant to the present epoch of the universe [45]. We could use the analysis provided in the Appendix and study case by case the differential equations for Q, F and W . However, since we already know the form of the desired solutions it is much easier to start from the family I in (42). For practical reasons it is convenient to adopt another parameterization, namely (c 1 , c 2 , c 3 ) → (α, β, γ) and (c 1 ,c 2 ,c 3 ) → (l, m, s), where the power-law solution takes the form Here A, B, C, D are some constants with the appropriate units; while α, β, γ and l, m, s are parameters that have to satisfy the field equations (24)- (28). Substituting (43) into (28) we obtain The above is satisfied in five different cases We now proceed to study these cases with some detail.

Case 1:
The selection s = 0, γ = 0 corresponds to the usual general relativity in 5D with φ = constant. In this case the power-law cosmological solutions are well-known in the literature [46]. However, to make the paper self-consistent we provide them here.
From (27) we obtain αl + β (m − l) = 0. Thus, assuming l = m we get β = αl/(l − m). In this case the only power-law solution to the field equations is 5 (The field equations cannot be satisfied for s = 0, γ = 0 and l = 1, simultaneously). For l = m the solution is Here, contrary to what we see in (46), the coordinate y is not restricted to be spacelike. Therefore, by virtue of extra symmetry discussed in [47], the transformation of coordinates t = w 3/2ȳ , r =r/ √ w, y = wt with w = ± √ ǫ in (47) generates the "new" solution We should note that although the last two line elements are diffeomorphic in 5D (allowing complex transformations of coordinates in 5D), their interpretation in 4D is quite different 6 .

Case 3:
The We note that this solution is related to (49) by the transformation t ↔ y, β ↔ l, ǫC 2 ↔ A 2 .
The solution for β = 0 is which can formally be obtained from (50) after the transformation t = w 1−α C/A ȳ, r = w −2βr , y = wt with w = ± √ ǫ and replacing α and β by m and l, respectively. In the limit where m → (1 − 3l), for which φ = constant and ω → ∞, we recover the general-relativistic solutions (48) and the second static metric mentioned in footnote 5.
It should be noted that (52) is the general power-law solution for the case where the metric depends on time alone. There is a very particular case, given bellow by (57) with β = 0, for which φ is allowed to depend on y.

Case 4
We now look for solutions to the field equations of the form Substituting these expressions into (24) we get (If γ = 0 then we recover Case 3). Now from (27) we obtain For these quantities we find that the choice β = −γ/3 (s = −3l) produces a simple particular solution, which is given bellow by (68). The general solution to the field equations (24)- (28), in the case under consideration, is generated by a quadratic equation for l. The explicit expression for l is given by Thus, (54)-(56) describe a two-parameter family of solutions. From a mathematical viewpoint, this expression might look cumbersome and/or non-interesting. However, from a physical viewpoint, the fact that l explicitly depends on β and γ means that the expansion rate of the universe, which is determined by l, is manifestly determined by the dynamics along the extra dimension. The above solution can be simplified if we add some physical requirements. The simplest ones arise from the choice of the coordinate/reference system.
It is worthwhile to mention that there are no solutions having simultaneously α = 0 and β = 1/3. Besides, the condition β < 2/3 ensures that l is a real number and ω > −4/3. For completeness, we note that there is a unique solution for which the metric depends only on time, while the scalar field is a function of both t and y. Such a solution is generated by the choice β = 0 in the above expressions.
Gaussian normal coordinate system: A popular choice in the literature is to use the five degrees of freedom to set g 4µ = 0 and g 44 = 1. This is the so-called Gaussian normal coordinate system based on Σ y . The solution in such coordinates can formally be obtained from (57)-(58) by changing t ↔ y, l ↔ β, s ↔ γ; setting ǫC 2 = A 2 ; and allowing the extra coordinate to have one or the other signature. The result is • Let us now go back to the general solution (53). For completeness we should consider the case where the denominator in (55) vanishes. In such a case, from (27) and (55) we find that the parameters β and γ are restricted to be The first set of parameters generates two families of solutions. Namely,
For s = −3l the solution is We note that this is the only solution to the FE, with α = m = 1, which allows the extra coordinate to be timelike (see (46), (67)).
Solution with ω = ω 2 : From the field equation (26) we obtain the condition Since s + 3l = 0 (see (63) and (68)) it follows that It can be verified that when ω = ω 2 the solution to the field equations is given by (67) with the choice of parameters given above. When the denominator in (66) vanishes, the solution is (67) with Thus, the choice ω = ω 2 does not generate new solutions to the FE. It just singles out some particular values for the parameters. Consequently, (67) is the general power-law solution with α = 1, m = 1.

Reduced Brans-Dicke cosmology in 4D
We now proceed to study the effective 4D picture generated by the 5D solutions discussed in the preceding section. For the line element (23), the cosmological metric induced on spacetime hypersurfaces Σ y : y = y 0 = constant is just the spacetime part of (23), and the non-vanishing components of the induced Brans-Dicke energy-momentum tensor (15) are where V = V (φ) should be determined from (19). We note that T where • It turns out that C vanishes identically for all solutions discussed in Cases 3 and 4. Therefore in these cases, integrating (73) and setting the constant of integration equal to zero we find (75) Thus, the effective potential vanishes either in the particular case where ω = −1 or in the Gaussian normal coordinate system (m = 0), i.e., for the solution (59)-(60).
• In Case 5, C = 0 and from (73) we get (76) • The parameter s is equal to zero only in Cases 1 and 2. In Case 1, the general-relativistic 5D solution (46) reduces on every hypersurface Σ y to the spatially flat FRW models of general relativity, viz., where ρ 0 = 3l 2 / 8πt 2 0 A 2 y 2 0 is the energy density measured at some time t = t 0 and n = 2 − 3l 3l . (79) • In Case 2, to evaluate V we substitute (49) into (19). We find that φdV /dφ vanishes identically by virtue of the equation relating β and ω. The same is true for solution (50). Therefore, without loss of generality we can set V = 0. Now, from (72) we find that (49) reduces on Σ y to a radiation dominated universe (ρ = 3p), regardless of ω. Similarly, (50) reduces to empty space (ρ = p = 0) on Σ y .

4D cosmologies induced by the 5D solutions of Cases 3 and 4
Cases 3 and 4, with s = 2, give more general BD cosmological models 9 in 4D. In fact, from (72), using (75) we get where Thus, the effective pressure and density satisfy the barotropic equation of state 9 If s = 2 and m = 0, the logarithmic potential leads to a violation of the weak energy condition ρ ≥ 0. Therefore, to avoid such a violation we should set m = 0 when s = 2, which in turns implies that T (BD) µν = 0 on every Σy.
Matter-dominated universe: Since the present epoch of the universe is matter dominated let us consider the case where n = 0. In this case the weak energy condition is satisfied for any l > 0. On the other hand, the condition ω > −3/2 restricts the range of l to be either l < 2 1 − 2/3 ≈ 0.37 or l > 2 1 + 2/3 ≈ 3.633. The former range requires q > 1 + √ 6 /2 ≈ 1.72, which is inapplicable to the present epoch. However, the latter range leads to q < 1 − √ 6 /2 ≈ −0.72, which is within the current observed measurements q = −0.67 ± 0.25 [50]. We note that the parameter l is a real number only if (1 + ω) ≤ 0.
To avoid misunderstanding, we should mention that the above discussion assumes that the parameters n and ω are independent, which is true in the 4D cosmologies generated by (52) and (53): giving n and ω we obtain l from (88). This in turn defines the rest of the solution (86)-(87). The 5D solutions (51) and (57) only have one independent parameter. Therefore they lead to a more restricted class of BD cosmologies in 4D. In particular, one can show that (51) yields shrinking (instead of expanding) dust models. Regarding the solutions in the synchronous frame (57), they produce 4D models of expanding dust, but do not yield accelerated expansion.
• If we assume V = 0 at the beginning, then we obtain a set of BD cosmologies in 4D which precludes the existence of a dust-filled universe. In fact, in the case where V = 0, (m = 0, ω = −1) instead of (80) we have Thus, a dynamical universe (l = 0) with p = 0 requires ρ = 0.

4D cosmologies induced by the 5D solutions of Case 5
Finally, let us study the effective 4D world generated by the 5D solutions discussed in Case 5. We emphasize that these are the only solutions demanding the extra dimension to be spacelike. Now the effective potential is given by (76). To avoid contradictions with the weak energy condition we disregard the logarithmic potential (see footnote 9). In this context, from (72) we find the effective EMT as where n is the same as in Cases 3 and 4, namely As a consequence, from (65) we find that γ can be expressed as γ = (3ln − 2) [3l (n + 1) − 2] 3l (n + 1) [3l (n + 1) (1 + ω) − 4ω − 3] + 3l (l − 1) + 2 (1 + 2ω) .
Substituting this into (90) we get After some calculations we find that the effective cosmology in 4D is formally identical to (86) except for the fact that now 10 Let us consider the above solution for large values of ω, in accordance with the solar system bound on ω(> 600). From the above expressions it follows that ρ > 0, φ > 0 require l in the range Consequently, as ω → ∞ we recover the usual spatially flat FRW cosmologies of general relativity.
Let us now consider a matter-dominated universe (n = 0). Using that l = 1/(q + 1) we find that the positivity of the effective density requires For the sake of argument, let us consider an accelerated expansion with q = −2/3, which corresponds to l = 3. This requires ω < −71/49 ≈ −1.449. In general, q < 0 requires ω < −1. As q → 1/2, which is the general-relativistic value of q for dust, the theory becomes consistent with increasingly larger values of ω. We note that the solution can give accelerating matter-dominated era as well as decelerating radiation-dominated era (n = 1/3) for the same value of ω in the range −3/2 ≤ ω < −1. For example if we choose ω = −1.45, then a consistent model of decelerating radiation era is obtained for l < 0.88, and accelerated dust era for l > 3, approximately.
• For completeness, let us consider the case where V = 0. Coming back to the original expression for φdV /dφ, we find that V = 0 requires Substituting into (65) we obtain .
Thus, for V = 0 (omitting intermediate calculations) we find Consequently, the effective BD cosmology in 4D can be written as 10 Here φ 0 and V 0 depend on three independent parameters, n, l and ω. If one fixes ω as in (85), then (93) reduces to (87).
In this parameterization, the n = 0 models become identical to those presented in the original BD paper [38]. For n = 0 we recover the type A-I solutions discussed in [37], although in a slightly different notation. From (96) it is easy to see that any ω in the range −2 < ω < −3/2 (1 < l < 2) yields a matter-dominated universe with accelerated expansion. However, this range is inconsistent with a radiation dominated epoch, which demands ω = −3/2 (l = 1/2).

Summary
Brans-Dicke theory in 4D can explain the observed accelerated expansion of the present matter-dominated universe, without invoking the presence of dark energy, if ω is allowed to take some value in the range −2 < ω < −3/2. However, ω in this range does not produce a consistent radiation-dominated epoch with decelerating expansion, as requires the big-bang nucleosynthesis scenario. One way out of this problem is to introduce a self interacting potential [31]. Another way is to consider a modified BD theory with a varying ω [32]. Both approaches are not free of criticism. In one of them the potential is added by hand, while the other creates the necessity of finding the fundamental mechanism driving the variation of ω.
A third way out of this problem is to resort to higher dimensions. In [40] it was shown that the BD field equations in 5D can be reduced, on a hypersurface orthogonal to the extra dimension, to those of GR in 4D coupled to two scalar fields. These fields may account for the late-time accelerated expansion provided several other conditions are met: the extra coordinate is cyclic; T In this work we have presented an approach which is free of the above criticism and conditions. Without making any assumption on the nature of the extra coordinate or the matter content in 5D, we have shown that the BD field equations in 5D are equivalent to those of BD in 4D derived from the action (20) with non-vanishing scalar potential V = V (φ). The potential is not introduced by hand, instead the reduction procedure provides an expression, namely (19), that determines the shape of V (φ) up to a constant of integration [(75)-(76)]. It also establishes the explicit formulae for the effective EMT in 4D [(14)- (16)]. This extends and generalize some previous results recently obtained by the present author [53].
In the context of FRW cosmological models we have integrated the vacuum FE in 5D under the sole assumption of separation of variables. We analyzed in detail the class of power-law solutions (43). We obtained a large family of solutions, namely (67), that has three free parameters; various families with two free parameters [(50), (52), (53)]; one free parameter [(49), (51), (57), (59)], and solutions that only exist for ω = −4/3 [(62), (68)]. Thus, the spectrum of BD power-law solutions in 5D is significantly larger than the one of GR in 5D, which can have at most one free parameter (Case 1).
We discussed the effective 4D world generated by our solutions. We found that the theory yields power-law and logarithmic potentials, except for the solutions derived in Case 2 and those in the Gaussian normal frame where we can set V = 0, without loss of generality. Certainly, one can assume V = 0 at the outset in (19), in which case the theory in 4D would be identical to the original BD [38]. However, such assumption imposes a strong constraint on the parameters of the solutions leading to restricted cosmological models in 4D which have no, or little, relevance to the problem of cosmic acceleration [(89), (96)].
We found that all BD models in 4D can formally be expressed as in (86) with φ 0 and V 0 given by (87) or (93). The models are given in terms of the parameters n, l and ω. When at least two of them are independent, which occurs for a large number of solutions [(50), (52), (53), (67)] we can obtain models that for the same ω can give the present accelerated expansion and a decelerated radiation-dominated epoch as required by primordial nucleosynthesis.
We should mention that as in the conventional 4D Brans-Dicke theory, in our models the (effective) EMT obeys the ordinary conservation law D µ T µ ν (BD) = 0 (the same as in Einstein's theory), which in the cosmological realm yields the usual equation of motionρ + 3ȧ a (ρ + p) = 0.
Although, we do not require the introduction of a self interacting potential in 4D or matter in 5D, the range of ω obtained in our work is consistent with the one obtained by a number of authors in the literature in the context of several cosmological models and different versions of BD theory in 4D and 5D [31]- [41]: In all cases cosmic accelerated expansion requires ω < −1.
There is also evidence in the literature that small values of |ω| are required by inflationary models [51] and structure formation [52] in scalar-tensor theory. Certainly, these theoretical results contradict solar system experiments which impose the constraint ω > 600. However, this solar system limit does not necessarily imply that the evolution of the universe is, at all scales, close to general relativity. The fact is that GR is poorly tested on cosmic scales [29] and no experiment has been done to test BD in cosmological scale yet. Consequently, one cannot discard scalar-tensor scenarios of the sort discussed here and in [31]- [41], [51]- [52] on the basis that in cosmological scales ω does not meet solar system requirements.
Thus, the problem of (in)compatibility between astronomical and cosmological requirements remains open. Our recent work [53] indicates that a variable (homogeneous) ω cannot solve the problem. Perhaps, local inhomogeneities might give rise to high values of ω consistent with astronomical observations [32].

Appendix A: Integrating the field equations
In section 3 we have seen that the scalar-vacuum Brans-Dicke FE for the cosmological metric (23) give five partial differential equations to be solved for four unknowns, n(t, y), a(t, y), Φ(t, y) and φ(t, y). The assumption that the metric is separable increases the number of unknowns to six. The aim of this appendix is to show, by means of explicit integration, the consistency of this assumption with the field equations.
I: First, we assume that c 1 = 0. Then from (33) we get where Q 0 is a constant. Substituting this and (34) into the wave equation (28) we obtain When c 1 = 0 the assumption of separability requires k = 0 and the fulfillment of one of the following conditions (i) C = 0, (ii) η 0 = 0, (iii) c 2 = 0 (cases Ia and Ib bellow).
Ia: Now we substitute the metric functions (29), including (32), (34) and (A-1), into the FE (24)-(26) with k = 0 and either C = 0 or η 0 = 0. We find that (24) is a second-order differential equation for W , while (25) and (26) contain the second derivatives of W and F . Next, we isolateẄ from (A-2) and substitute it into (24) to obtain the following first-order differential equationĉ A similar substitution into (26) yields an expression forF , whose explicit form we omit, asF =F (W, F,Ẇ ,Ḟ ). With theseẄ andF we find that (25) reduces to (A-3). As a result there are two independent equations, viz., (A-2) and (A-3), for the two unknowns W and F .
Ib: For c 1 = 0, c 2 = 0 (C = 0, η 0 = 0), equation (A-2) can be easily integrated. The general solution is F = F 0Ẇ −1 W c3/c1 , where F 0 is a constant of integration. Using this expression, and (A-1) with c 2 = 0, we find that both (24) and (26) generate the same second-order differential equation for W , namely, On the other hand, (25) becomes a third-order differential equation for W , which upon substitution of (A-5) into it reduces to the identity 0 = 0.
• A particular solution to (A-2), for c 2 = 0, is W = constant, which leads to Q = constant. In turn, (24) and (26) become identical to each other, both requiring c 3 = c 1 , in agreement with (A-5). With this requirement (25) yields F ∝ t.

II:
We now consider the case where c 1 = 0 and k = 0 with c 2 = 0 and c 2 = 0 (cases IIa and IIb bellow).
IIa: First we assume c 2 = 0, in which case from (33) we get F = F 0 W −c3/c2 . Substituting this and (34) into the wave equation (28) we obtain 11 When c 2 = 0 separability (36) requires η 0 C = 0. Following the same steps as above, we obtain a first-order differential equation by substitutingẄ from (A-6) into (24). Namely, Taking the time derivative of (A-7) and replacing into it the expression forẄ obtained from (A-6), after some lengthly algebraic manipulations, we derive (26), which now is a second-order differential equation for Q. On the other hand, (25) reduces to (A-7) after we substituteẄ andQ into it. Thus, (A-6)-(A-7) generate all the separable solutions with c 1 = 0, k = 0 and c 2 = 0.
IIb: We now let c 2 = 0 and assume c 3 = 0. In this case one can generate solutions with k = 0 and η 0 C = 0. From (33) it follows that W = constant. Consequently, the wave equation (28) is satisfied identically. From (24) and (26) we obtain the differential equations that govern F and Q. Namely, Taking the time derivative of (A-8) and using (A-9) we obtain an expression forF , which we omit here, asF = Q(F, Q,Ḟ ,Ẇ ). Once again after some manipulations, one can verify that (25) reduces to (A-8) afterF andQ are substituted into it.
Thus, we have obtained the differential equations that generate all distinct cases of separable solutions. They demonstrate, in a constructive way, that the FE are consistent with the assumption of separability. In the case where c 1 = c 2 = c 3 = 0 the metric functions become independent of the extra coordinate. For this case, the solutions to the FE as well as their cosmological applications have recently been discussed by the present author in [53]. 11 In general, when c 2 = 0 from (33) we get F = F 0 W −c 3 /c 2 Q −3c 1 /c 2 and the wave equation (28)