Eur. Phys. J. C (2017) 77:171DOI 10.1140/epjc/s10052-017-4719-7

Regular Article - Theoretical Physics

Higher-order gravity in higher dimensions: geometrical originsof four-dimensional cosmology?

Antonio Troisia

Dipartimento di Fisica “E.R. Caianiello”, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Salerno, Italy

Received: 9 December 2016 / Accepted: 22 February 2017 / Published online: 18 March 2017© The Author(s) 2017. This article is an open access publication

Abstract Determining the cosmological field equationsis still very much debated and led to a wide discussionaround different theoretical proposals. A suitable concep-tual scheme could be represented by gravity models that nat-urally generalize Einstein theory like higher-order gravitytheories and higher-dimensional ones. Both of these two dif-ferent approaches allow one to define, at the effective level,Einstein field equations equipped with source-like energy-momentum tensors of geometrical origin. In this paper,the possibility is discussed to develop a five-dimensionalfourth-order gravity model whose lower-dimensional reduc-tion could provide an interpretation of cosmological four-dimensional matter–energy components. We describe thebasic concepts of the model, the complete field equationsformalism and the 5-D to 4-D reduction procedure. Five-dimensional f (R) field equations turn out to be equiva-lent, on the four-dimensional hypersurfaces orthogonal tothe extra coordinate, to an Einstein-like cosmological modelwith three matter–energy tensors related with higher deriva-tive and higher-dimensional counter-terms. By consideringthe gravity model with f (R) = f0Rn the possibility is inves-tigated to obtain five-dimensional power law solutions. Theeffective four-dimensional picture and the behaviour of thegeometrically induced sources are finally outlined in corre-spondence to simple cases of such higher-dimensional solu-tions.

1 Introduction

Type Ia supernovae (SNeIa) observations depicted a late-timespeeding up universe [1–7], driven by an unknown compo-nent, whose properties can be ascribed to some sort of exoticfluid. This elusive component has been addressed in termsof a form of dark energy that exhibits an anti-gravitationalnegative equation of state (EoS) [8,9]. The combination

a e-mail: atroisi@unisa.it

of standard matter and dark energy provides an Einstein-like universe where gravitational attraction is counterbal-anced by such an unusual gravitational source, thus repro-ducing cosmological observations. Soon after this discov-ery a plethora of theoretical proposals has been suggested inorder to explain the dark-energy origin. However, as of today,a well endowed and self-consistent physical interpretation isso far unknown [10–13]. The puzzling quest of a satisfactoryexplanation as regards these phenomenological results ledcosmologists to explore several research lines. In particular,two main directions were followed. On the one side, stan-dard Einstein gravity has been reviewed by introducing a newcosmological component: the cosmological constant or anywell-behaved fluid with a negative EoS. Differently, adoptingan unconventional point of view, there have been consideredmodified gravity models that generalize Einstein theory: i.e.scalar-tensor gravity [14–16], f (R) theories [17–20], DGPgravity [21], braneworld scenarios [22–25], induced-mattertheory [26–28], and so on.

Among others, f (R) theories of gravity received con-siderable attention. Such theories, which represent a naturalgeneralization of Einstein gravity, are obtained by relaxingthe hypothesis of linearity contained in the Hilbert–EinsteinLagrangian. Several results, sometimes controversial, havebeen obtained in this framework [29,30]. These models havebeen satisfactory checked with cosmological observations[31–35] and suggest intriguing peculiarities in the low energyand small velocity limit [36] since the gravitational potentialdisplays a Yukawa like correction [37,38]. In order to beviable, f (R) gravity theories have to satisfy some minimalprescriptions. In particular, they have to match cosmologicalobservations avoiding instabilities and ghost-like solutionsand they have to evade solar system tests [29,39]. Constraintsfrom energy conditions represent a further argument to settlea suitable gravity Lagrangian [40]. However, at this stage,a fully satisfactory fourth-order gravity model is far fromhaving been achieved. As a matter of fact, such kinds of

123

171 Page 2 of 18 Eur. Phys. J. C (2017) 77 :171

models have been considered also from other points of view.For example, recently, models with a non-minimal couplingbetween gravity and the matter sector have been taken intoaccount [41–43].

On the other side, historically, attempts for a unificationof gravity with other interactions stimulated the search fortheoretical schemes based on higher dimensions, i.e. beyondour conventional four-dimensional (4-D) spacetime. Nord-strøm [44], who was the first to formulate a unified theorybased on extra dimensions, and Kaluza [45] and Klein [46]developed a five-dimensional (5-D) version of general rela-tivity (GR) in which electrodynamics is derived as a counter-effect of the extra dimension. Successively, a lot of work hasbeen dedicated to such a theoretical proposal both consider-ing fifth-coordinate compactification and large extra dimen-sions, allowing for non-compact mechanisms [27]. The lastapproach led to the so called space–time–matter (STM) orinduced-matter theories (IMT) [26,28] and to braneworldtheories [22]. The basic characteristic of IMT is that 5-Dvacuum field equations can be recast after the reduction pro-cedure as 4-D Einstenian field equations with a source ofgeometrical origin. In such an approach, the matter–energysource of 4-D spacetime represents a manifestation of extradimensions. Generalizations of Kaluza–Klein theory presentthemselves as a suitable scheme to frame modern cosmo-logical observations. In fact, after SNIe observations, severalattempts have been made to match induced-matter theorieswith the dark-energy puzzle [47–52].

In this paper we try to merge the two approaches. Sincevacuum fourth-order gravity theories, as well as IMT, canbe cast as an Einstein model with a matter–energy source ofgeometrical origin [53,54], it seems a significant proposalto confront the two theoretical schemes. The genuine ideaunderlying this work is to investigate 5-D f (R)-gravity fromthe point of view of a completely geometric self-consistentapproach. In this scheme, all matter–energy sources will rep-resent a byproduct of the dimensional reduction related tohigher-order terms and higher-dimensional quantities. Suchan approach determines a completely different conceptualframework with respect to standard five-dimensional fourth-order gravity models. The main purpose is to recover alongthis scheme both dark-matter and dark-energy dynamicaleffects. In the standard realm, in fact, extra dimensions andcurvature counter-terms will only play the role of mendingdark-energy phenomenology in the presence of ordinary mat-ter. Along this orthodox paradigm, 5-D models of fourth-order gravity have been studied in presence of perfect fluidsources [55–57] under peculiar assumptions on the metricpotentials. In a similar fashion, accelerating 4-D cosmolo-gies, induced by generalized 5-D f (R) gravity, have alsobeen studied considering a curvature–matter coupling [58].Furthermore, a vacuum fourth-order Kaluza–Klein theory,defined in terms of the Gauss–Bonnet invariant [59], has

been studied from the point of view of the predictions offield equations. In particular, in the cylinder approximation,the propagation of its electromagnetic degrees of freedom[60] and the particle spectrum in the linear regime [61] havebeen investigated.

Attempts to meet f (R)-gravity and IMT have also beendeveloped in time. For example the possibility has beeninvestigated to obtain information on the space–time–mattertensor starting from the energy-momentum tensor inducedby higher-order curvature counter-terms [62]. On the otherside a new effective coupled F((4)R, ϕ) gravity theory hasbeen proposed as a consequence of a five-dimensional f (R)

model [63]. It has also been demonstrated that the Dolgov–Kawasaki stability criterion of the 5-D f (R) theory remainsthe same as in usual f (R) theories: f ′′(R) > 0, with theprime indicating the derivative with respect to the Ricciscalar.

In our work we try to develop a more general framework;5-D f (R)-gravity is discussed in a complete analytical form.We present general field equations in the f (R)-IMT approachand, by exploiting the reduction procedure from 5-D to the4-D ordinary spacetime (considering a suitable extension ofthe Friedmann–Robertson–Walker metric), we obtain a com-plete set of new Einstein-like field equations with matter–energy sources induced by higher-order derivative terms andhigher dimensions (top–down reduction). In other words, wedevelop a fully geometrical cosmology where ordinary mat-ter and dark components could be addressed, in principle,as the outgrowth of the GR formalism adopted within thetop–down reduction mechanism.

The paper is organized as follows. In Sect. 2 we providea brief review of 4-D f (R)-gravity. Section 3 is devoted tosummarizing induced-matter theory. The 5-D f (R)-gravityformalism is outlined in Sect. 4, and, thereafter, we discussthe 5-D to 4-D reduction procedure of the field equations. InSect. 5 a routine is described to find 5-D power law solutions.Section 6 is dedicated to the analysis of the results and, inparticular, to their interpretation from the point of view ofthe 4-D induced cosmologies. Finally, Sect. 7 presents ourconclusions.

2 The cosmological equations for f (R) gravity

Dark-energy models are based on the underlying assumptionthat Einstein’s general relativity is indeed the correct the-ory of gravity. However, adopting a different point of view,both cosmic speed up and dark matter can be viewed asa breakdown of GR. As a matter of fact, one should con-sider the possibility to generalize the Hilbert–Einstein (H–E) Lagrangian. With these premises in mind, the choice ofthe gravity Lagrangian can be settled by means of data withthe only prescription of adopting an “economic” strategy, i.e.

123

Eur. Phys. J. C (2017) 77 :171 Page 3 of 18 171

only a minimal generalization of the H–E action is taken intoaccount. The so-called f (R)-gravity [17,18,29,30,64,65],which considers an analytic function in terms of the Ricciscalar and leads to fourth-order field equations, is based onthis conceptual scheme. It has to be recalled that higher-ordergravity theories represent the natural effective result of sev-eral theoretical schemes [66,67], i.e. quantum field theorieson curved spacetimes and M-theory. In addition, they havebeen widely studied as inflationary models in the early uni-verse [68,69].

Fourth-order gravity is favoured by the Ostrogradski the-orem. It has in fact been demonstrated [70,71] that f (R)-Lagrangians are the only metric-based, local and potentiallystable modifications of gravity among several that can beconstructed by means of the curvature tensor and possiblyby means of its covariant derivatives.

Let us consider the generic f (R) action; we have

S = 1

2

∫d4x

√−g[f (R) + SM (gμν, ψ)

], (1)

where we have used natural units 8πG = c = h = 1, g isthe determinant of the metric, R is the Ricci scalar and ψ

characterizes the matter fields. Varying with respect to themetric we get the field equations,

f ′(R)Rμν − 1

2f (R)gμν − [∇μ∇ν − gμν�

]f ′(R) = T M

μν,

(2)

where the prime, as said in the introduction, denotes a deriva-tive with respect to R and, as usual,

T Mμν = −2√−g

δSMδgμν

. (3)

Taking the trace of (2) we obtain

f ′(R)R − 2 f (R) + 3� f ′(R) = T M , (4)

where T M = gμνT Mμν , so that the relation between the Ricci

scalar and T M is obtained by means of a differential equa-tion, different from GR where R = − T M . This result sug-gest that f (R)-gravity field equations admit a larger varietyof solutions than Einstein’s theory. In particular, T M = 0solutions will no longer imply Ricci-flat cosmologies andtherefore R = 0. Actually, a suitable property of this modelis that the field equations (2) can be recast in the Einsteinform [17,18,53]:

Gμν = Rμν − 1

2gμνR = T curv

μν + T Mμν/ f

′(R), (5)

with

T curvμν = 1

f ′(R)

{1

2gμν

[f (R) − R f ′(R)

]+

+ f ′(R);αβ(gμαgνβ − gμνgαβ)}, (6)

which represents a curvature stress-energy tensor inducedby higher-order derivative terms (the terms f ′(R);μν renderthe equations of fourth order). The limit f (R) → R reducesEq. (5) to the standard second-order Einstein field equations.

The Einstein-like form of fourth-order field equations (5)suggests that higher-order counter-terms can play the role ofa source-like component within gravity field equations. Inpractice, it is possible to postulate that geometry can playthe role of a mass–energy component when higher than sec-ond order quantities are taken into account. Indeed, the traceequation (4) propagates a scalar-like degree of freedom. Asa matter of fact, in principle, one can imagine to address theuniverse’ dark phenomenology in terms of such a kind ofeffective fluid. From the cosmological point of view a signif-icant role is played by the barotropic factor of the curvature–matter–energy fluid. In the vacuum case this quantity will dis-criminate accelerating solutions from standard matter ones.In the case of a Friedman spacetime, for example, one has[53,54]

ρcurv = 1

f ′(R)

{1

2

[f (R) − R f ′(R)

]− 3H R f ′′(R)

},

(7)

wcurv = −1 + R f ′′(R) + R[R f ′′′(R) − H f ′′(R)

][ f (R) − R f ′(R)] /2 − 3H R f ′′(R)

, (8)

where H = a/a stands for the Hubble parameter and a dotmeans a time derivative.

In the following we will resort to a similar quantity in orderto check the behaviour of the cosmological fluids induced by5-D f (R)-gravity on the 4-D hypersurfaces that slice thehigher dimensional spacetime along the extra coordinate.

In our conventions we will consider Latin indices likeA, B,C, etc. running from 0 to 4, Latin indices like i, j,etc., running from 1 to 3 and Greek indices taking valuesfrom 0 to 3.

3 Induced-matter theory

Modelling a unification theory that contains gravity and par-ticle physics forces, typically, implies the resort to extra-dimensional models [27,72]. Among these 5-D Kaluza–Klein theory [45,46] and its modern revisits induced-matterand membrane theory represent significant approaches; inaddition these kinds of models represent the low energy limit

123

171 Page 4 of 18 Eur. Phys. J. C (2017) 77 :171

of more sophisticated theories (i.e. supergravity) [73–75].The main difference between the seminal approach of Kaluzaand Klein and induced-matter theory is related with the role ofthe extra dimension. In its first conception the fifth dimensionwas “rolled up” to a very small size, answering the question ofwhy we do not “see” the fifth dimension. Modern theories likeIMT postulate that we are constrained to live in a smaller 4-Dhypersurface embedded in a higher-dimensional spacetime.The key point for embedding the 4-D Einstein theory intothe 5-D Kaluza–Klein induced-matter theory is representedby the Campbell–Magaard (CM) theorem. The problem ofembedding a Riemannian manifold in a Ricci-flat space wasstudied by Campbell soon after the discovery of GR [76],and finally it was demonstrated by Magaard in 1963 [77].Further on, Tavakol et al. [78] used these studies to establishmathematical well-endowed bases to the 4-D reinterpreta-tion of the 5-D Kaluza–Klein theory that is dubbed induced-matter theory [79,80]. In such a framework, the 5-D to 4-Dreduction procedure determines on the 4-D hypersurface anEinstein gravity theory plus induced-matter components ofgeometrical origin. The CM theorem can be formalized asfollows:

Any analytic Riemannian space Vn (xμ, t) can be locallyembedded in a Ricci-flat Riemannian space Vn+1

(x A, t

).

Here the “smaller” space has dimensionality n with μ =0, ..., n − 1, while the “host” space has dimensionality N =n + 1 with A running from 0 to n; the extra coordinate canbe both space-like and time-like. We are interested in thecase N = 5. It is important to remark that the CM the-orem is a local embedding theorem. Therefore, more gen-eral issues related with global embedding, i.e. initial-valueproblems, stability or general induced solutions [81,82], can-not be addressed by resorting to this achievement. However,for our purposes, the theorem guarantees the right analyticframework in order to frame 4-D matter phenomenology inrelation to 5-D field equations [80].

Because of the CM theorem it is possible to write downthe 5-D metric as follows:

dS2 = γABdx Adx B = gμν(x, y)dxμdxν + ε�2(x, y)dy2,

(9)

where xμ = (x0, x1, x2, x3) are 4-D coordinates, gμν turnsout to be the spacetime metric and y is the fifth coordinate,ε = ±1 allows for a space-like or a time-like extra dimen-sion. Throughout the paper we adopt the spacetime signa-ture (+,−,−,−). With these premises in mind ordinary 4Dspacetime results as a hypersurface y : y = y0 = constant,orthogonal to the 5D extra-coordinate basis vector

n A = δA4

�, nAn

A = ε. (10)

If one considers vacuum 5-D field equations (Ricci flat):RAB = 0, the CM theorem suggests a natural reduction pro-cess to a 4-D pseudo-Riemannian spacetime. In practice, onecan obtain an Einstein-like 4-D model1 Gμν = Tμν [79,80],once the right hand side of these equations fulfills the defini-tion

T IMTμv = �,μ;v

�− ε

2�2

⎧⎨⎩

∗�

∗gμv

�− ∗∗

gμv + gαβ∗gμα

∗gvβ+

−gαβ∗gαβ

∗gμv

2+ gμv

4

[∗αβ

g∗gαβ +

(gαβ

∗gαβ

)2]⎫⎬⎭ .

(11)

Here, a comma is the ordinary partial derivative, a semi-colon denotes the ordinary 4D covariant and starred quan-tities describe terms derived with respect to the fifth coor-dinate. This quantity defines the so-called induced-mattertensor; the only hypothesis underlying this achievement hasbeen relaxing the cylinder condition within the Kaluza–Kleinscheme (independence on the fifth coordinate).

As observed from the previous result, fourth-order grav-ity theories and Kaluza–Klein IMT models provide the sameconceptual scheme. Both of the approaches allow one toobtain an Einstein-like gravity model where matter–energysources are of geometrical origin. In particular, these effec-tive matter–energy tensors descend, respectively, from thehigher-order derivative contributions and from the higher-dimensional counter-terms. Therefore, it seems that devia-tions from GR can be naturally recast as sources of standardEinstenian models.

In that respect let us notice that IMT and 5-D f (R) grav-ity are built in the framework of a different philosophy withrespect to conventional higher dimensional approaches likebraneworld models. In fact, despite the same working sce-nario: bulk universe with non-trivial dependence on the extra-coordinate, 4-D metric obtained evaluating the backgroundmetric at specific 4-D hypersurfaces and matter fields con-fined in the 4-D spacetime, there are intrinsic conceptualdifferences. 5-D f (R) gravity and induced-matter theoryare based on the hypothesis that standard matter is noth-ing else than a 4-D manifestation of geometrical deviationswith respect to GR. Braneworlds model our universe as afour-dimensional singular hypersurface, the brane, embed-ded in a five-dimensional anti de Sitter spacetime. In such acase the motivation for a non-compactified extra dimensionis to solve the hierarchy problem. The differences in terms ofphysical motivations for large extra dimensions imply alsodifferent technical approaches. Within IMT and 5-D f (R)

gravity one considers a Ricci-flat vacuum bulk and develops

1 We recall that natural units are adopted, therefore 8πG�c4 is setequal to unity.

123

Eur. Phys. J. C (2017) 77 :171 Page 5 of 18 171

the 4-D physics as a byproduct. In braneworlds the oppo-site point of view is assumed. In this case, one deals withsuitable 4-D solutions on the brane for some matter distri-butions and these solutions are matched with an appropriate5-D bulk considering Israel junction conditions. Neverthe-less, despite such conceptual differences it has been shownin time that brane theory, IMT and 5-D f (R) gravity can havea suitable matching. In particular, IMT and braneworlds havebeen put in strict analogy [83]. The key point is to consider theinduced-matter approach in terms of the spacetime extrinsiccurvature. In this case it is possible to write down braneworld-like field equations with a 4-D matter–energy source and abrane tension that are defined in terms of the extrinsic cur-vature. On the same line very recently it has been shownthat Z2 braneworld-like solutions can be obtained as par-ticular maximally symmetric solutions of 5-D f (R) gravitywith matter [63]. In future work one can imagine to deepenthe interconnections that seem to arise among these differenthigher-dimensional approaches.

Such considerations suggest, in principle, the intriguingpossibility that GR experimental shortcomings could be actu-ally referred to the effective property of modified gravitymodels. In the following we will exploit such an approach tostudy a general fourth-order 5-D model, where higher-orderderivative gravity is “merged” with higher dimensions.

4 A 5-D f (R)-gravity model and its 4-D reduction

A f (R) theory of gravity in five dimensions can be describedby the action

S(5) = 1

2

∫d5x√g(5) f (R(5))+ 1

2

∫d5x√g(5)LM (gAB, ψ),

(12)

where R(5) is the 5-D Ricci scalar, Lm(gAB, ψ) is aLagrangian density for matter fields denoted, as in the 4-D case, by ψ and g(5) represents the determinant of the 5-Dmetric tensor gAB . 5-D field equations can be obtained byaction (12) varying with respect to the metric:

f (5),R R(5)

AB − 1

2f (R(5)) gAB +

− [∇A∇B − gAB �(5)] f (5),R = T (5) M

AB , (13)

here T (5) MAB is the energy-momentum tensor for matter

sources, while ∇A is the 5-D covariant derivative, �(5) =gAB∇A∇B is the 5-D d’Alembertian operator. In order tosimplify the notation we have defined f,R(R(5)) ≡ f (5)

,R ,

therefore henceforth by f (5),R is intended to write the 5-D

Ricci derivative of the f (R(5)) gravity Lagrangian.

Together with the field equations (13) one can obtain thetrace

4 �(5) f (5),R + f (5)

,R R(5) − 5

2f (R(5)) = T (5)M (14)

where T (5)M = gAB T (5)MAB .

Actually, 5-D field equations (13) can be naturally recastas a generalization of Einstein 5-D equations in the samemanner of the ordinary 4-D formalism given in Sect. 2. Infact, with some algebra and isolating 5-D Einstein tensor onegets

R(5)AB− 1

2R(5)gAB = 1

f (5),R

{1

2

(f (R(5))− f (5)

,R R(5))gAB+

− [gAB �(5) − ∇A∇B] f (5),R

}+ 1

f (5),R

T (5)MAB , (15)

by fact determining a 5-D new source of geometrical originin the right member of field equations

T (5) CurvAB = 1

f (5),R

{1

2

(f (R(5)) − f (5)

,R R(5))gAB+

− [gAB �(5) − ∇A∇B] f (5),R

}. (16)

This fact, which can resemble only the byproduct of a math-ematical trick, determines significant physical consequencesat 5-D and, as a consequence, also in the ordinary space-time. In fact, as we will see in the following, the 4-D dimen-sional reduction of Eqs. (15) and (16) implies a “new” setof Einstein-like equations with three matter–energy com-ponents all of geometrical origin. One of these quantitiesdescends from the 4-D reduction of the Einstein tensor andthe two terms derivate from the relative 4-D reduction appliedtoT (5) Curv

AB . Since we want to explore how geometric counter-terms can effectively mimic cosmological sources we neglectordinary matter, i.e. from now on we will assume T (5) M

AB = 0,considering gravity equations in vacuum.

Let us now develop the 5-D to 4-D reduction procedureof our higher dimensional framework. At first, by assumingthe metric (9), it is possible to draw the reduction rules forthe differential operators [49]:

∇μ∇ν f(5),R = DμDν f

(5),R + ε

2�2

∗gμν

∗f (5),R , (17)

∇4∇4 f(5),R = ε� (Dα�)

(Dα f (5)

,R

)+

∗∗f (5),R −

∗�

�

∗f (5),R , (18)

123

171 Page 6 of 18 Eur. Phys. J. C (2017) 77 :171

�(5) f (5),R = � f (5)

,R +(Dα�)

(Dα f (5)

,R

)

�+

+ ε

�2

⎡⎣ ∗∗f (5),R +

∗f (5),R

⎛⎝gαβ

∗gαβ

2−

∗�

�

⎞⎠⎤⎦ ;

(19)

here, again, the asterisk denotes the partial derivative withrespect to the extra coordinate (i.e., ∂/∂y =∗); Dα is the four-dimensional covariant derivative defined on the hypersurface y=y0 , calculated with gμν , and the usual d’Alembertian � isreferred to 4-D quantities. In the same way, all the quantitiesthat are not labelled in terms of the fifth coordinate will beintended to refer to ordinary spacetime. On this basis, it ispossible to rewrite the 5-D field equations (15) by separatingthe spacetime part (μ = 0, .., 3; ν = 0, .., 3) from the extra-coordinate (y) one:

G(5)μν = 1

f (5),R

{1

2

(f (R(5))− f (5)

,R R(5)

)gμν+

−⎡⎣� f (5)

,R +(Dα�)

(Dα f (5)

,R

)

�+

+ ε

�2

⎛⎝ ∗∗

f (5),R +

∗f (5),R

⎛⎝gαβ

∗gαβ

2−

∗�

�

⎞⎠⎞⎠⎤⎦ gμν+

+ ∇μ∇ν f (5),R + ε

2�2

∗gμν

∗f (5),R

}, (20)

G(5)44 = 1

f (5),R

⎧⎨⎩

1

2

(f (R(5))− f (5)

,R R(5))g44+

−⎡⎣� f (5)

,R +(Dα�)

(Dα f (5)

,R

)

�+

+ ε

�2

⎛⎝ ∗∗

f (5),R +

∗f (5),R

⎛⎝gαβ

∗gαβ

2−

∗�

�

⎞⎠⎞⎠⎤⎦ g44+

+ε� (Dα�)(Dα f (5)

,R

)+

∗∗f (5),R −

∗�

�

∗f (5),R

⎫⎬⎭. (21)

Our purpose is now to disentangle the extra-coordinatedependence from the Einstein tensor. In this way it will bepossible to isolate the 4-D part of Einstein tensor on the leftmember and move on the r.h.s. all quantities that depend onhigher derivative terms and on the extra coordinate. All thehigher-order and higher-dimensional counter-terms will playthe role of effective source terms on the 4-D hypersurface.It is evident that since the fourth-order Lagrangian f (R(5))

depends on the 5-D Ricci scalar, this dependence cannot be

completely untwined until the Lagrangian dependence is notspecified.

The reduction rules for the Einstein tensor [80] give

R(5)μν = Rμν − DμDν�

�+ ε

2�2

⎛⎝

∗�

∗gμν

�− ∗∗

gμν +

+ gλα∗gμλ

∗gνα −gαβ

∗gαβ

∗gμν

2

⎞⎠ , (22)

R(5)44 = −ε��� −

∗αβ

g∗gαβ

4− gαβ

∗∗g αβ

2+

∗� gαβ

∗gαβ

2�. (23)

The next step is to calculate the 5-D Ricci scalar R(5) =gμνR(5)

μν + g44R(5)44 . After the substitution of Eq. (23) into

Eq. (21) and some tedious algebra one gets the relation

��

�= ε

�2

⎛⎝

∗�

�

gαβg∗αβ

2− g

∗αβ g∗

αβ

4− gαβg∗∗

αβ

2

⎞⎠+

− 1

2

f(R(5)

)f,

(5)R

+ �(5) f,(5)R

f,(5)R

− (Dα�)(Dα f,(5)R )

� f,(5)R

+

− ε

�2

⎛⎜⎝

∗∗(5)

f,Rf,

(5)R

−∗�

�

f,∗

(5)R

f,(5)R

⎞⎟⎠ , (24)

which reminds one of an analogous expression given in [49]for the 4-D reduction of 5-D Brans–Dicke theory. Furthermanipulations of (24) allow one to obtain a relatively simpleexpression:

��

�= ε

�2

⎛⎜⎝

∗�

�

gαβ ∗gαβ

2−

∗gαβ ∗

gαβ

4− gαβ ∗∗

gαβ

2

⎞⎟⎠+

− 1

2

f (R(5))

f (5),R

+ � f (5),R

f (5),R

− ε

�2

gαβ ∗gαβ

2

∗f (5),R

f (5),R

, (25)

which finally can be used in the reduction of Ricci scalar tocollect some terms. Therefore, by using also the trace equa-tion, one has

R(5) = R − 2��

�+⎛⎝2

∗�

�gαβ ∗

gαβ +

− 2gαβ ∗∗gαβ −3

2

∗gαβ ∗

gαβ − (gαβ ∗gαβ)2

2

). (26)

123

Eur. Phys. J. C (2017) 77 :171 Page 7 of 18 171

It is easy to check that this result reproduces the analogousrelation of 5-D GR [27] when f (R(5)) → R(5). We remem-ber that in such a case the vacuum Einstein field equationsreduce to the system R(5)

AB = 0 and R(5) vanishes.Once having developed the several aspects of the 5-D to

4-D reduction procedure it is possible to obtain the induced4-D field equations. By inserting (22), (23) and (26) into Eqs.(20), (21) the 4-D spacetime equations get the form2:

Gμν = T IMTμν + T curv

μν + TMixμν . (27)

Here

T IMTμν = �,μ;ν

�− ε

2�2

⎧⎨⎩

∗�

∗gμν

�− ∗∗

gμν +gαβ∗gμα

∗gνβ +

−gαβ∗gαβ

∗gμν

2+ gμν

4

[∗g

αβ ∗gαβ +

(gαβ

∗gαβ

)2]⎫⎬⎭

is the usual IMT tensor, derived from the 5-D to 4-D reductionof the Einstein tensor, and

TCurvμν = 1

f (5),R

{1

2

(f (R(5)) − f (5)

,R R)gμν+

− [gμν � − ∇μ∇ν] f (5),R

}(28)

is the curvature tensor written preserving the 4-D form. Insuch a case extra-dimension contributions are still hidden inthe scalar curvature nested within the definition of the gravityLagrangian and of its derivatives. Finally,

TMixμν = ε

2�2

∗gμν

∗f (5),R

f (5),R

− (Dα�)(Dα f (5),R )

� f (5),R

gμν+

− ε

�2

⎡⎢⎣

∗∗f (5),R

f (5),R

+∗f (5),R

f (5),R

⎛⎝gαβ ∗

gαβ

2−

∗�

�

⎞⎠⎤⎥⎦ gμν+

+ ε

2�2

⎡⎢⎣

∗gαβ ∗

gαβ

4+ (gαβ ∗

gαβ)2

4

⎤⎥⎦ gμν (29)

represents a mixed tensor containing terms that dependexplicitly on the fifth coordinate and on the derivative off (5),R with respect to y.

2 The equations presented throughout this section partially reproduceprevious results obtained in [63]. In fact, besides the different conceptualscheme, there are also some analytic differences that make the twoschemes non-completely indistinguishable.

Therefore, we have obtained a framework where 4-D grav-ity is fully geometrized. In fact, within such a scheme matter–energy sources are related only with geometrical counter-effects deriving either from higher-dimensional metric quan-tities or from higher-order derivative ones.

In order to complete our discussion one should take intoaccount also the off-diagonal equation. By considering that

R(5)4α = �Pβ

α ;β with Pαβ = 1

2�(

∗gαβ −gσρ

∗gσρ gαβ) [80],

one obtains

�Pβ

α ;β =∗

f (5),R ,α

f (5),R

− ∗gασ gβσ

f (5),R ,β

f (5),R

− �,α

�

∗f (5),R

f (5),R

, (30)

which coincides with similar expressions obtained elsewhere[49,63] and in the limit f (R(5)) → R(5) gives back, in theabsence of ordinary matter, the conservation law Pβ

α ;β = 0[27]. Equation (30) resembles a more general conservationequation which relates the spacetime derivatives of Pβ

α and∗f (5),R . It has been conjectured [84] that the spacetime compo-

nents of the field equations relate geometry with the macro-scopic properties of matter, while the extra-coordinate part(α4 ) and (4

4) might describe their microscopic ones. With thesehypotheses, within fourth-order gravity Kaluza–Klein mod-els, one would find that microscopic properties of matter areinfluenced by spacetime derivatives of the scalar degree offreedom intrinsic in f (R) gravity. Furthermore, in our case,the Pβ

α dynamics depends also on �, which, in turn (see Eq.(25)), shows an evolution driven by the higher-order grav-ity counter-terms. It seems that there is a strict interconnec-tion between the extra-dimension properties and the intrin-sic f (R) scalar degree of freedom. For example, observingEq. (25), it seems that f ′(R) guarantees a sort of Machianeffect for this kind of models. However, all these consid-erations represent, at this stage, nothing more than specu-lations and, therefore, we do not discuss this issue furtherhere.

5 A solving algorithm for 5-D f(R) gravity

5.1 5-D f(R) gravity cosmological solutions

Let us now verify what kind of solutions can be derived forour 5-D f (R)-gravity model. Once general cosmologicalsolutions have been obtained, it is be possible to concludeto 5-D effects on y=y0 hypersurfaces, deriving the effective4-D picture. In particular, one can determine the spacetimeeffective matter–energy behaviour of the different geometri-cal components described in the previous section. In order tosearch for field equation solutions we assume the 5-D metric:

123

171 Page 8 of 18 Eur. Phys. J. C (2017) 77 :171

dS2 = n2(t, y)dt2 − a2(t, y)

[dr2

1 − kr2 +

+ r2(

dθ2 + sin2 θdϕ2) ]

+ ε�2(t, y)dy2, (31)

where k = 0,+1,−1 is referred to the 3-D spacetime cur-vature and (t, r, θ, φ) are the usual coordinates for spheri-cally symmetric spatial sections. It is important to notice that,because of the metric choice, our f (R(5)) Lagrangian and itsderivatives do not depend on the 4-D spatial coordinates. Infact, the 5-D Ricci scalar calculated over the metric (31) is afunction of the (t, y) coordinates alone. If the metric (31) isintroduced in Eqs. (20)–(21), we obtain the system of equa-tions, respectively, for the components: 0

0, ii (i = 1, 2, 3), 44

and 04:

− f (R(5))

2 f,(5)R

+ 3 ε∗a

∗f,

(5)R

�2a f,(5)R

− 3 ε∗a

∗n

�2an− ε

∗f,

(5)R

∗�

f,(5)R �3

+

+ε∗n

∗�

n�3 + ε

∗∗f,

(5)R

�2 f,(5)R

− ε∗∗n

�2n+

+ 3a˙

f,(5)R

a f,(5)R n2

+ 3an

an3 +˙

f,(5)R �

n2 f,(5)R �

+ n�

n3�− 3a

n2a− �

n2�= 0,

(32)

− f (R(5))

2 f,(5)R

− 2 ε∗a

2

�2a2 + 2 ε∗a

∗f,

(5)R

�2a f,(5)R

− ε∗a

∗n

�2an+

+ ε

∗f,

(5)R

∗n

�2 f,(5)R n

+ ε∗a

∗�

a�3 − ε f,(5)R

∗�

�3 f,(5)R

− ε∗∗a

�2a+ ε

∗∗f,

(5)R

�2 f,(5)R

+

− 2a2

a2n2 + 2a˙

f,(5)R

a f,(5)R n2

+ an

an3 −˙

f,(5)R n

f,(5)R n3

+

− a�

an2�+

˙f,

(5)R �

f,(5)R n2�

− a

an2 +¨

f,(5)R

f,(5)R n2

= 0, (33)

− f (R(5))

2 f (5),R

+ 3ε∗a

∗f (5),R

�2a f (5),R

+ ε

∗f (5),R

∗n

�2 f (5),R n

+ 3ε∗a

∗�

a�3 + ε∗n

∗�

n�3 +

−3ε∗∗a

�2a− ε

∗n

�2n+ 3a

˙f (5),R

a f (5),R n2

+

−˙f (5),R n

f (5),R n3

− 3a�

an2�+ n�

n3�+

¨f (5),R

f (5),R n2

− �

n2�= 0, (34)

−3a∗n

an+

∗n ˙f,

(5)R

n f,(5)R

+ 3∗a �

a�−

∗f,

(5)R �

f,(5)R �

−˙

3∗a

a−

∗f,

(5)R

f,(5)R

= 0. (35)

Equation (35) reminds one, again, of the result obtained in[49] for 5-D Brans–Dicke vacuum solutions. Therefore itseems a suitable choice to pursue the same approach per-formed in this work to get some particular solutions of oursystem of equations (32)–(35). If one considers the metriccoefficients as separable functions of their arguments,

n(t, y) = N (y), a(t, y) = P(y)Q(t),

�(t, y) = F(t), f (5),R (t, y) = U (y)W (t), (36)

from Eq. (35) one gets

3F∗P

FP+ 3

∗N Q

NQ− 3

∗P Q

PQ+ F

∗U

FU+

∗N W

NW−

∗U W

UW= 0.

(37)

This equation suggests the possibility to look for power lawcosmological solutions. In the following section we will dis-cuss some examples in this sense.

5.2 Preliminary power law solutions

In order to look for 5-D f (R) gravity power law solutions weconsider a set of metric potentials defined as follows [49]:

n(t, y) = N0yδ, a(t, y) = A0t

α yβ, �(t, y) = �0tγ yσ ,

(38)

with N0, A0,�0 some constants with appropriate units andα, β, γ, δ, σ that represent the unknown parameters requiredto satisfy 5-D field equations.

Up to now we have developed a completely generalscheme, without any assumption on the gravity Lagrangian.However, to completely define the solving algorithm one hasto make a choice about the Ricci scalar function entering thegravity action. To remain conservative with the solution pro-cedure, we consider a power law function of the Ricci scalarf (R) = f0Rn . Such a model has been extensively studied inthe literature at four dimensions both in cosmology (curva-ture quintessence) [17,18,53,54] and in the low energy limit[36,85]. In addition, the 5-D phenomenology of a power lawfourth-order gravity has been investigated in the standardapproach in the presence of a perfect fluid matter source[56,58]. In particular, in such a case, a compact fifth dimen-sion is assumed and it is supposed that the homogeneouslydistributed fluid does not travel along the fifth dimension.As a matter of fact the matter–energy density and the pres-sure experienced by a four-dimensional observer have to beintegrated throughout the compact extra-dimensional ring. Inour approach we have overcome this point of view, we relaxthe hypotheses on the extra coordinate and, above all, wediscard ordinary matter in favour of a completely geometricself-consistent approach. In accordance with cosmological

123

Eur. Phys. J. C (2017) 77 :171 Page 9 of 18 171

observations [86–90], we will study field equations consid-ering a flat spacetime geometry (k = 0).

A suitable recipe to study Eqs. (32)–(35) is to plug the met-ric functions (38) into the last of these relations and to searchfor its solutions. This approach allows one to obtain, withsome effort, a number of constraints on the model parameters.Then one can try to satisfy also the other more complicatedfield equations (32)–(34). By inserting the metric potentialsinto Eq. 0

4, one obtains a rather cumbersome expression:

�(α, β, γ, δ, σ, n, t, y)

ϒ(α, β, γ, δ, σ, n, t, y)= 0 (39)

where

�(α, β, γ, δ, σ, n, t, y) = y−1−2δ(

3(βγ + α(−β + δ))

× (N 20 t

2y2δ(6β2 + 3β(−1 + δ − σ) + δ(−1 + δ − σ)

) ++t2γ y2+2σ

(6α2 + 3α(−1 + γ ) + (−1 + γ )γ

)ε�2

0

)2 −+2(−1 + n)

(N 4

0 t4y4δγ (δ + (−1 + 2n)(1 + σ))

× (−6β2 + 3β(1 − δ + σ) + δ(1 − δ + σ))2 +

+N 20 t

2+2γ y2(1+δ+σ)(6α2 + 3α(−1 + γ ) + (−1 + γ )γ

)× ε

(6β2 + 3β(−1 + δ − σ)+

+δ(−1 + δ − σ)) (3δ + 2(−2 + n)(1 + σ)++γ (3 − 2δ + 2δn + 3σ))�2

0++t4γ y4+4σ

(6α2 + 3α(−1 + γ ) + (−1 + γ )γ

)2×δε2(−1 + γ + 2n)�4

0

) )

and

ϒ(α, β, γ, δ, σ, n, t, y) = N 20 t(N 2

0 t2y2δ

×(

6β2 + 3β(−1 + δ − σ) + δ(−1 + δ − σ))

+

+t2γ y2+2σ(

6α2 + 3α(−1 + γ ) + (−1 + γ )γ)

ε�20

)2.

This result, evidently, suggests the possibility to obtain, inprinciple, much more solutions than the Brans–Dicke casestudied in [49]. To simplify our search, as a preliminary step,we make some trivial hypotheses about the model parame-ters, leaving a complete analysis of the field equations solu-tions to a future dedicated work.

5.2.1 Kaluza–Klein GR limit

As a first step we want to verify that the Kaluza–Klein GRlimit is recovered. To perform the standard KK limit wehave to settle n = 1 and, in addition, following customaryapproaches to the model, we assume σ = 0.

In such a case Eq. (39) becomes very simple and can beverified if βγ + α(−β + δ) = 0. In particular, if γ �= α

one obtains β → αδ

α − γ(a result that is in agreement with

[49]). Looking to the other equations, together some trivialnon-evolving solutions (α = 03), one obtains the solving setsof parameters:

δ = 1, γ = 1, α = − 1, β = 1

and N 20 = 4�2

0 with ε = −1, (40)

δ = 0, γ = −1/2, α =1/2, β = 0, (41)

with ε = ±1 (no restrictions on the y coordinate), and

δ = 1, γ = 1, α = −1, β = α

α − 1

and �20 = N 2

0

(α − 1)2 with ε = −1. (42)

All these solutions are well known in the literature [49,91],therefore, our model completely reproduces standard GRKaluza–Klein models in the f (R(5)) → R(5) limit.

5.2.2 Standard 4-D f (R) gravity limit

What about standard 4-D fourth-order gravity? One shouldexpect that this framework has to represent a natural subcaseof 5-D f (R) gravity. In order to get this limit one has toassign a solutions set with δ = 0, β = 0, σ = 0 and γ = 0,which means there is no dependence on y and no dynamicson the fifth coordinate. Starting with this assumptions and byconsidering equations (32), (33) and (35), one obtains

α = −1 + 3n − 2n2

−2 + n, (43)

which exactly matches the well-known power law f (R)grav-ity solution in the case of the vacuum field equations [17,18].However, in the 5-D case, to fully satisfy the theory we haveto fulfill one more equation, Eq. (34). This requirement, infact, settles the power law indexn. Therefore the only allowedcombination is δ = 0, β = 0, σ = 0, γ = 0, n = 5/4, andα = 1/2.

5.2.3 Cylinder-type solutions

We can now relax some hypotheses in order to look for gener-alizations of 5-D GR. To search for such a kind of solutionswe start, as a first simple case, from the assumption of nodependence on the extra coordinate. This means to study ourtheory, looking for solutions of Eqs. (32)–(35), in the limit ofthe cylinder condition. Different from the standard case, in

3 It is possible to find some trivial solutions with α = 0, β = 0,δ = (0, 1), γ = 1.

123

171 Page 10 of 18 Eur. Phys. J. C (2017) 77 :171

the f (R) generalization of induced-matter theory this con-dition, as we will see, does not imply radiation as the onlypossible kind of induced matter.

At first, let us observe Eq. (25). In the case of 5-D f (R)

gravity, independence on the fifth coordinate does not meanthat one has a massless Klein–Gordon equation for the extra-coordinate potential �. Higher-order gravity counter-termswill play the role of a mass term and � will play a completelynew role in the dynamics:

�� = −[

1

2

f (R(5))

f (5),R

+ � f (5),R

f (5),R

]�. (44)

In addition, as a consequence, T IMTμν will no more have zero

trace [27], therefore the induced matter tensor can span moregeneral kinds of matter in relation to the underlying cosmo-logical solution.

From the field equations point of view, power law solu-tions of cylinder-type are obtained when the model param-eters in (38) are settled as δ = 0, β = 0, σ = 0 with freen. Neglecting again static solutions4 it is possible to find aset of implicit solutions that can be expressed in terms of thef (R) Lagrangian power index n:

α1 = 1

12(−3 + 6n − f (n)) ,

γ1 = −17 + 28n2 + 3 f (n) − 2n (24 + f (n))

2 (−5 + 2n + f (n)); (45a)

α2 = 1

12(−3 + 6n + f (n)) ,

γ2 = 17 + 28n2 − 3 f (n) + 2n (−24 + f (n))

2 (5 − 2n + f (n)); (45b)

α3 = γ3 = 2 − 6n + 4n2

5 − 2n, (45c)

with f (n) = √−39 + 108n − 60n2. Equations (45a)–(45c)represent an interesting achievement. In the following wewill show that these solutions provide, after the 5-D to 4-D reduction, cosmological significant behaviours of the 4-Dmatter–energy tensors induced by 5-D geometry.

6 4-D induced f (R) gravity: a top–downgeometrization of matter

The effective 4-D picture induced by higher dimensions canbe obtained once cosmological solutions obtained in Sect. 5are plugged into the 5-D to 4-D reduction framework previ-ously outlined. In particular the 4-D setting is obtained, as

4 There are also in this case trivial solutions i.e. n = 1, α = 0, γ =(0, 1).

already said, by considering hypersurfaces y=y0 that slicethe 5-D universe along the fifth coordinate. In this scheme,the extra-coordinate effects on 4-D physical quantities areevaluated taking y = y0 = const. The resulting cosmo-logical model is a 4-D induced-matter f (R) theory, wherefield equations are given in the form of Einstein equationsequipped with three different matter–energy sources of geo-metrical origin. Deviations from standard GR provide, in theeffective 4-D spacetime description, matter–energy sourcesand one could wonder if these contributions are related withthe elusive nature of dark energy and dark matter. Actu-ally, we study the 4-D induced cosmologies in correspon-dence with the solutions given in Sect. 5. In particular, weare interested in investigating the behaviour of the three dif-ferent induced cosmological fluids (11), (28), (29) on they = y0 = const hypersurfaces. One can notice that for allof these quantities we have T 1 K

1 = T 2 K2 = T 3 K

3 (with K =IMT, Curv, Mix); therefore it is possible to describe suchmatter–energy tensors as perfect fluid sources with ρK =T 0 K

0 and the isotropic pressure defined as pK = −T 1 K1 .

The cosmological nature of each component can be eval-uated by means of the respective equation of state (EoS)ωK = −T 1 K

1 /T 0 K0 .

6.1 4-D GR limit

If one considers solutions obtained along the standard KKGR limit, the related 4-D effective dynamics is describedby well-known cosmological models. In particular, the solu-tion (41) represents the standard radiation universe withp = 1/3ρ. In such a case the three matter–energy tensorscollapse into one matter–energy source and, in particular, thecurvature quintessence tensor and the mixed tensor, respec-tively, vanish. On the other side, the solution (42) repre-sents a spatially flat FRW metric once the extra coordinate isfixed:

ds2| y=N 2

0 y20 dt2−A2

0y2α

α−10 t2α

[dr2 + r2(dθ2 + sin2 dϕ)

].

(46)

The induced-matter tensor plays, again, the role of the onlyone effective cosmological fluid. In fact, the other two quan-tities combine to give a cancelling result. The cosmologi-

cal energy density scales as ρ = ρ0t−2, with ρ0 = 3α2

N 20 y

20

,

while the barotropic factor is ω = 2 − 3α

3α. This result is

in complete agreement with previous achievements on non-compactified KK gravity [49].

123

Eur. Phys. J. C (2017) 77 :171 Page 11 of 18 171

6.2 4-D cosmology from standard f (R) gravity limit

We have seen that the attempt to get the standard 4-D f (R)

gravity limit is frustrated by the higher equations number ofour model. In fact, it is possible to get the right 4-D power lawsolution only if the extra-coordinate equation is neglected.When this equation enters the game the power law indexn gets fixed. The behaviour of the three matter–energy ten-sors describing the top–down effect of higher dimensionsand higher derivative terms on the 4-D gravity hypersurfaceconfirm such a result. In fact, if we evaluate ωcurv on the 4-Df (R) solution (43) we obtain

ωcurv = 1 + 7n + 6n2

3 − 9n + 6n2 , (47)

which, again, matches the already known result for this kindof model [17,18]. However, since the fifth equation selectsthe allowed values forn, we know that the admissible solutionis indeed δ = 0, β = 0, σ = 0, γ = 0, n = 5/4 andα = 1/2. The effective 4-D consequent cosmology showsρIMT = ρMix = 0, ρcurv = 3

4N20 t

2 and ωcurv = 1/3. As

a matter of fact, we find a new radiation like universe withn = 5/4. The radiation component derives from the higherderivatives counter-terms, while the induced-matter tensorvanishes.

6.3 4-D Cylinder models

Cosmologies induced by cylinder solutions are quite interest-ing and allow different dynamics. In particular, the matter–energy sector, characterized by means of the three matter–energy sources of geometrical origin, assumes a variety ofsignificant behaviours. A relevant aspect of these solutions isthat the mixed tensor TMix

μν provides a cosmological constant-like source. In fact, as is possible to observe from the defini-tion (29), in such a case (no extra-coordinate dependence) it isT 0 Mix

0 = T i Mixi , therefore ωMix = −1. Since for each solu-

tion, ρMix = T 0 Mix0 ∼ Xi (n)N0t−2, with Xi (n) depending

on the solution we consider, we obtain an inverse square lawcosmological constant. This behaviour is favoured in stringcosmologies [92] and in time-varying � theories that accom-modate a large � for early times and a negligible � for latetimes [93]. On the other side, the effective barotropic fac-tors of the other two matter–energy sources depend on theparameter n in relation to the solutions (45a)–(45c). In orderto have a schematic portrait of these results we have plottedthe behaviour of each ωK with respect to n in Fig. 1. It ispossible to observe that the value of the gravity Lagrangianpower index n determines the possibility to have differentcosmological components. Both the standard matter sectorand the dark energy one can be mimicked by the geometri-cally induced fluids according with previous results in this

sense obtained in the framework of IMT [79,94–96] andfourth-order gravity [31]. Very interesting is the coexistenceof different regimes with the same value of n.

Together the behaviour of ωK we display, for all the com-ponents (see Fig. 2), the energy density value ρK and theoverall sum of these quantities.5 It is evident that energyconditions are violated for certain values of n. For exampleit is easy to observe that the weak energy condition, ρi ≥ 0,for the first solution, is not satisfied by ρIMT and alterna-tively by the other two energy densities ρCurv and ρMix. Forthe second solution we have again no n intervals where allcomponents satisfy this energy prescription, since ρIMT isalways quite well behaved but ρCurv and ρMix are alterna-tively negative. Finally, for the third solution we have a smallregion around n ≈ 1.5 where all energy densities are pos-itive, while the WEC is violated elsewhere. Although sin-gle matter–energy sources can violate the energy conditions,the total geometrized matter (dotted line in each panel ofFig. 2) satisfies the WEC for all the ranges of the parameterspace in which cylinder solutions are defined. In addition,it has been recently showed that cosmological models withnegative energy components can be suitably matched withobservations if these “disturbing” quantities are coupled withgreater magnitude of positive energy forms [97]. Therefore,the total top–down geometrized matter is physical and pre-serves conceptions about energy conditions in 4-D. Such aresult is quite customary when dealing with induced mattertheory models [27].

Finally, let us discuss the cosmological evolution of thefour-dimensional sections determined by each of the solu-tions (45a)–(45c) and the behaviour of the extra dimension.Figure 3 allows one to acquire significant information asregards this behaviour describing the dependence of a(t, y)and �(t, y) with respect to n in the different cases.

In order to display accelerating dynamics, a power lawscale factor requires that the power index has to be negativeor greater than 1. In particular, universes expanding at anaccelerating rate need α > 1 if we consider the definitiongiven in Eq. (38).

Actually, since our approach in the 4-D limit determines amulti-fluid model, the acceleration condition for the cosmo-logical dynamics turns out to be characterized by the generalrelation for the scale factor,

a

a= −4πG

3(ρtot + 3ptot).

Therefore, barotropic fluids like ones outlined by our model,with ρtot = ∑

K ρK and ptot = ∑K pK , imply that such a

relation can be written as

5 In order to achieve these plots we settle the quantity N 20 t

2 = 1.

123

171 Page 12 of 18 Eur. Phys. J. C (2017) 77 :171

Fig. 1 The behaviour of thebarotropic factors related tothree matter–energy tensors thatarise after the 5-D to 4-Dreduction. The three panels referto the EoS comportment withrespect to n along each of thecylinder solutions (45a)–(45c)(from top to down)

0.6 0.8 1.0 1.210

5

0

5

10

n

Mix

Curv

IMT

0.6 0.8 1.0 1.22

1

0

1

2

n

6 4 2 0 2 4 63

2

1

0

1

2

3

n

Mix

Curv

IMT

Mix

Curv

IMT

123

Eur. Phys. J. C (2017) 77 :171 Page 13 of 18 171

Fig. 2 Energy densities for thedifferent species ofmatter–energy obtained in a 5Df(R)-gravity model whenreduced to the standard 4-Dspacetime. Curves labelled withρtot describe the behaviour ofthe whole matter–energycosmological budget; naturalunits are adopted. The upperpanel refers to solution (45a),the middle one to (45b), finallythe last one refers to the solution(45c)

0.4 0.6 0.8 1.0 1.2 1.41.0

0.5

0.0

0.5

1.0

n

TOT

Mix

Curv

IMT

0.4 0.6 0.8 1.0 1.2 1.41.0

0.5

0.0

0.5

1.0

n

2 1 0 1 2 33

2

1

0

1

2

3

n

TOT

Mix

Curv

IMT

TOT

Mix

Curv

IMT

123

171 Page 14 of 18 Eur. Phys. J. C (2017) 77 :171

a

a= −4πG

3

∑K

(1 + 3ωK ) ρK , (48)

and the condition for accelerating cosmologies becomesϒ = ∑

K (1 + 3ωK ) ρK < 0. As one can see in Fig. 4for our model the regions of the parameters space that fulfillthis requirement exactly determine accelerating expansions(the deceleration parameter q = − aa

a2 is negative). Let usremember that the observations suggest q ≈ −0.5 [98,99],albeit models considering power law solutions can also allowfor higher limits, i.e. q ≈ −0.3 [100].

A peculiar aspect of our model is represented by the pos-sibility that single energy densities can get negative definedvalues. Of course, as said above, the overall evolution is stillphysical since the effective energy density budget is alwayspositive definite. This behaviour, however, can affect theeffective cosmological evolution. For example, it can preventthe possibility to obtain models with speeding up evolutionthat accord with cosmological observations. In fact, it is pos-sible to observe (cf. Figs. 1, 4) that there are regions of theparameter space where the induced cosmological fluids allassume negative barotropic factors, while the cosmologicaldynamics is still decelerating (q ≥ 0). On the other side, thispeculiarity can also determine suitable conditions to framemodels that experience a cosmic speeding up.

It is evident that, at this stage, our results can only beconsidered as a toy model achievement. Power law solutionscan only describe limited stages of universe evolution. Moregeneral analyses are required in order to understand the realpredictions of the model. In particular, the possibility to havea coexistence of negative defined equations of state with bothdecelerating dynamics and accelerating ones can represent anintriguing perspective to deepen in the framework of moregeneral cosmological solutions.

Let us discuss in some detail the results contained in Figs.1, 2, 3, 4.

– Solution (α1, γ1)–(45a) This solution is allowed in theinterval 1

2 < n < 1310 . Values of n < 1 give three neg-

ative EoS fluids within the cosmic pie. The GR limit(n = 1) implies a trivial behaviour with vanishing ρK .Forn ∼ 1.3 we have two standard matter components andthe cosmological constant-like source driven by TMix

μν .In such a case ρIMT, ρCurv > 0 while ρMix < 0, thewhole cosmic mass–energy budget is positively definedas already said above.Looking to the cosmological dynamics one can noticethat this solution implies a negative power law index forthe scale factor within the interval 1

2 < n < 1 while0 ≤ α < 1 for 1 < n < 13

10 . In accordance with Figs.3 and 4 this behaviour indicates accelerated contract-ing universes in the first interval and standard matter-like dominated cosmologies in the second one. The extra

dimension is always decreasing, since 0 < γ < 1. It isinteresting to notice that the ordinary matter-like regimeis achieved: in the interval 1 < n < 1.25 with two of thethree fluids that behave as sources with a negative definedEoS (the other one is a stiffed fluid with ωIMT > 1) andin the range 1.25 < n < 13

10 with two standard matter-like sources (0 < ωIMT < 1 and ωCurv > 1) together thecosmological constant contributes. The divergence of thebarotropic factor ωCurv for n = 1.25 is regulated by thecorrespondent vanishing of ρCurv.

– Solution (α2, γ2)–(45b) We have again 12 < n < 13

10 .For this solution, the scale factor shows a standard mat-ter rate since always 0 < α < 1

2 , there is no accel-eration (q(α2, γ2) > 0). The induced-matter compo-nent ρIMT is always a well-behaved thermodynamicalfluid with 0 ≤ ωIMT ≤ 2/3. The curvature inducedsources can have a negative EoS or a positive one with−5/6 ≤ ωCurv ≤ 3/2. For values n � 1.5 we havetwo standard matter components and a cosmologicalconstant-like source. Energy densities can be defined pos-itive or negative. Around n ∼ 1.2 we have two standardmatter-like sources with positive ρi , while ρMix is neg-ative, ρtot is always ≥ 0. For n = 1, the two curvatureinduced sources collapse to a zero-valued energy den-sity, while the induced-matter tensor plays the role ofa radiation fluid with ωIMT = 1/3 as in usual KK-GRcylinder models. The fifth dimension can experience areduction when 1

2 < n < 54 and an increasing evolution

for 54 < n < 13

10 , the case is static when γ = 12 , 5

4 .– Solution (α3, γ3)–(45c) Admissible values for n ∈ R −

{5/2}. This solution explicitly resembles the standard 4-D f (R)-gravity solution given in [53]. The scale fac-tor and the extra dimension show the same dynamicssince α3 = γ3. There are two interesting regions aroundn ∼ 0.2 and n ∼ 1.6. In the first case we have twostandard matter components, dust-like and radiation-like,with ωCurv ≈ 0, ωIMT ≈ 0.26 together a cosmologicalconstant-like one with ωMix = −1. The correspondentscale factor is, however, non-accelerating since q is pos-itive. The second region shows a dust-like source withωCurv ≈ 0 and two dark-energy-like components, one ofthem phantom-like with ωIMT ≈ −2.6. In such a casethe 4-D section shows an accelerating expansion sinceq < 0 (q0 = −0.5 for n ≈ 1.69) as well as an increasingextra dimension. The zeros of ωIMT (dust-like behaviourof this source) are trivial since the corresponding valueof the energy density is vanishing. Actually, this solutionallows for a plethora of different behaviours. In particu-lar, expanding solutions with no acceleration are obtainedfor − 1

2 ≤ n < 12 and 1 < n ≤ 3

2 with two standardmatter-like sources plus the effective cosmological con-stant contribute given by TMix

μν ; on the other side acceler-

123

Eur. Phys. J. C (2017) 77 :171 Page 15 of 18 171

Fig. 3 The scale factor and theextra coordinate power lawindices vs. n for the threecylinder solutions obtainedalong the 5-D to 4-D reductionof higher-order gravity powerlaw models. Both plots contain adetail that enlightens thebehaviour in the region n ≈ 1

10 5 0 5 1020

10

0

10

20

n

1.0 0.5 0.0 0.5 1.0 1.5 2.01.0

0.5

0.0

0.5

1.0

n

3

2

1

10 5 0 5 1020

10

0

10

20

n

0.5 0.0 0.5 1.0 1.5 2.02

1

0

1

2

n

3

2

1

ated expansions are given by models with n < − 12 and

32 < n < 5

2 . In these intervals TCurvμν plays the role of

a baryonic component and the other two matter–energysources act as dark-energy-like sources. The energy den-sities have, again, both negative and positive values withρtot ≥ 0. For n > 1.5 we have all positive ρK withωK ∼ −1, in such a case the three induced componentsbehave all as cosmological constant-like sources.

As a matter of fact, fourth-order gravity provides, inthe cylindric case, a completely different framework fromthe standard induced-matter theory. The standard schemeimplies, in the 4-D limit, that induced matter behaves asa radiation-like fluid (p = ρ/3) [27] according with theresult obtained with an initial condition of cylindricity byKaluza [45]. In the case of higher-order Kaluza–Klein mod-

els the energy-momentum tensors entering the 4-D inducedfield equations can describe more general matter sources.Therefore one can obtain dynamically significant cosmolo-gies also in the presence of a Killing symmetry for the extracoordinate. Of course, the significance of such solutionshas to be corroborated with cosmological and astrophysicaldata.

7 Conclusions

We discussed the possibility to develop a higher-order andhigher dimensional gravity model. Within such an approach,after the 5-D to 4-D reduction procedure, it has been possibleto suggest a new interpretive scheme for the 4-D cosmologi-cal phenomenology. Specifically, we have considered a vac-

123

171 Page 16 of 18 Eur. Phys. J. C (2017) 77 :171

Fig. 4 The decelerationparameter and the ϒ term,which enters the scale factoracceleration equation, vs. n.Again the three plot are relatedto the cylinder solutions(45a)–(45c)

3 2 1 0 1 2 330

20

10

0

10

20

30

n

q

3 2 1 0 1 2 34

2

0

2

4

n

uum 5-D f (R)-gravity model. The 4-D effective reductionprovides, in this case, a GR-like cosmology characterizedby means of three induced-matter–energy tensors of geo-metrical origin. Within this matter geometrization paradigmthe cosmological fluid sources can be obtained consider-ing the standard induced matter tensor of non-compactifiedKK theory T IMT

μν , the curvature quintessence tensor TCurvμν

coming from higher-order derivative terms of f (R)-gravityand TMix

μν , a mixed quantity that arises from both higher-dimensional and higher-order curvature counter-terms. Thislast quantity embodies the 5-D to 4-D reduction of the twocombined approaches. We have obtained the complete fieldequations formalism and in order to check the predictionof the model we have worked out a routine to search forpower law solutions in the case f (R) = f0Rn . The whole

scheme provides the right KK-GR limit once f (R) → R.In such a case, on shell, radiation dominated cosmologies(p = 1/3ρ) are obtained if no dependence on the extracoordinate is taken into account. More generally, the KK-GR limit furnishes solutions that agree with similar resultsalready present in the literature. In particular, the curvatureinduced-matter–energy tensors vanish, while the induced-matter tensor, which collects all metric terms depending onextra coordinate, turns out to be the only source of the 4-Dfield equation.

Relaxing the hypotheses on n, i.e. by assuming a genericpower law fourth-order gravity model, it is possible to searchfor more general solutions. As a first preliminary step, cylin-dricity has been taken into account. A more general studyhas been left to a dedicated forthcoming publication. Dif-

123

Eur. Phys. J. C (2017) 77 :171 Page 17 of 18 171

ferent from the standard KK-GR theory, the combination offourth-order gravity with higher dimensions implies, in sucha case, non-trivial solutions that can give rise to interestingcosmologies. We have obtained a set of solutions, parame-terized by the f (R) power index n, that determine a varietyof possible barotropic fluids in the 4-D mass–energy sec-tor. In relation to the value of n one can have both standardmatter components and dark-energy-like ones. The cylin-der condition implies that TMix

μν , according to its definition,behaves as an effective cosmological constant-like term withωMix = −1. Actually, in such a case, the energy densityρMix depends on time with an inverse square law. This factsuggests the intriguing possibility to have a time varyingcosmological constant. The cosmological behaviour of the4-D sections shows that it is possible to have both accel-erating and non-accelerating expansions. In particular, non-accelerated expansion can be obtained also in the presenceof subdominant dark-energy-like components. On the otherside, a dark-energy-dominated speeding up universe can eas-ily be framed generalizing results obtained in the standard 4-D f (R)-gravity. The interesting aspect of this last solution isthat geometry can provide both standard matter-like sourcesand dark-energy-like ones, and one of these can mimic aphantom fluid. The extra dimension follows an independentevolution, that, in the third solution, is equivalent to one ofthe scale factors. Therefore, in such a case, expanding accel-erated evolutions are characterized also by a growing fifthcoordinate.

The price to pay for geometrizing matter–energy sourcesis that cosmological energy densities can assume negativevalues violating the WEC. We have shown that although thisis true for some values of n, also in interesting regions ofthe parameters space, the whole mass–energy budget ρtot

is always positive definite and, therefore, total top–downgeometrized matter is physical.

The fact that geometrically induced-matter–energy ten-sors could, alternatively, play the role of standard mattercomponents and of dark-energy-like sources can representan intriguing perspective in order to interpret cosmologicalobservations. In particular, this achievement seems to deservemore studies in the direction of dark-energy and dark-matterinterpretation. Of course, more insights are necessary in orderto evaluate such a theoretical scheme. In particular, the mat-ter sector (i.e. matter-like components), which has been satis-factory checked within standard IMT [94,95,101], requires acareful investigation since matter properties should be com-pletely induced by gravity. In addition, model predictionsalso have to be checked against cosmological and astrophys-ical data.

Furthermore, if f (R) Kaluza–Klein gravity determines, inthe 5-D to 4-D reduction of cylinder solutions, cosmologicalmodels with a suitable top–down matter geometrization, inprinciple, more general solutions could allow one to develop

cosmological models with a wider phenomenology. On theother side, the fifth dimension role requires a careful analysiswhen the extra-coordinate Killing symmetry is discarded. Infact, if the cylinder condition is relaxed one naturally gets intothe game of a length scale related with the extra coordinate. Inthis sense, it has been shown that 5-D f (R) gravity suggeststhat, eventually, there is a strict interconnection between thefifth dimension dynamics and the higher derivative counter-terms.

Acknowledgements The author is thankful with Prof. S. De Pasqualeand Prof. S. Pace for their help and support. The author is grateful tothe CENTRA-IST of Lisbon for the hospitality during the developmentof this work.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.

References

1. A.G. Riess et al., AJ 116, 1009 (1998)2. S. Perlmutter et al., APJ 517, 565 (1999)3. P.M. Garnavich et al., APJ 74, 509 (1998)4. R. Rebolo et al., Mon. Not. R. Astron. Soc. 353, 747 (2004)5. A.C. Pope et al., Astrophys. J. 607, 655 (2004)6. M. Tegmark et al. (SDSS Collaboration), Phys. Rev. D 74, 123507

(2006)7. W.J. Percival, S. Cole, D.J. Eisenstein, R.C. Nichol, J.A. Peacock,

A.C. Pope, A.S. Szalay, Mon. Not. R. Astron. Soc. 381, 1053(2007)

8. V. Sahni, A.A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000)9. P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75, 559 (2003)

10. C. P. Burgess, The Cosmological Constant Problem: Why it’s hardto get Dark Energy from Micro-physics. doi:10.1093/acprof:oso/9780198728856.003.0004. arXiv:1309.4133 [hep-th] (2013)

11. E.J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D 15,1753 (2006)

12. J. Yoo, Y. Watanabe, Int. J. Mod. Phys. D 21, 1230002 (2012)13. A. Joyce, L. Lombriser, F. Schmidt, Ann. Rev. Nucl. Part. Sci. 66,

95 (2016)14. C. Brans, R.H. Dicke, Phys. Rev. 124, 124 (1961)15. N. Bartolo, M. Pietroni, Phys. Rev. D 61(2), 023518 (1999)16. V. Faraoni, Cosmology in scalar–tensor gravity (Kluwer Aca-

demic, Dordrecht, 2004)17. S. Capozziello, S. Carloni, A. Troisi, Res. Dev. Astron. Astrophys.

1, 625 (2003)18. S.M. Carroll, V. Duvvuri, M. Trodden, M.S. Turner, Phys. Rev. D

70, 043528 (2004)19. S. Capozziello, V. Faraoni, Beyond Einstein Gravity (Springer,

Dordrecht, 2011)20. S. Nojiri, S.D. Odintsov, Phys. Rep. 505, 59–144 (2011)21. G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485, 208 (2000)22. P. Brax, C. van de Bruck, Class. Quant. Gravit. 20, R201 (2003)23. R. Maartens, K. Koyama, Living Rev. Relat. 13, 5 (2010)24. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999)25. L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999)

123

171 Page 18 of 18 Eur. Phys. J. C (2017) 77 :171

26. P.S. Wesson, Space–Time–Matter (World Scientific, Singapore,1999)

27. J.M. Overduin, P.S. Wesson, Phys. Rep. 283, 302 (1997)28. P.S. Wesson, Five-Dimensional Physics (World Scientific, Singa-

pore, 2006)29. T.P. Sotiriou, V. Faraoni, Rev. Mod. Phys. 82, 451 (2010)30. T. Clifton, P.G. Ferreira, A. Padilla, C. Skordis, Phys. Rep. 513,

1 (2012)31. S. Capozziello, V.F. Cardone, A. Troisi, Phys. Rev. D 71, 043503

(2005)32. S. Nojiri, S.D. Odintsov, Phys. Rev. D 74, 086005 (2006)33. W. Hu, I. Sawicki, Phys. Rev. D 76, 064004 (2007)34. A.A. Starobinsky, JETP Lett. 86(3), 157 (2007)35. A. de la Cruz-Dombriz, P.K.S. Dunsby, S. Kandhai, D. Saez-

Gomez, Phys. Rev. D 93(8), 084016 (2016)36. S. Capozziello, A. Stabile, A. Troisi, Phys. Rev. D 76(10), 10401

(2007)37. K.S. Stelle, Gen. Relat. Gravit. 9, 353 (1978)38. H.J. Schmidt, Int. J. Geom. Methods Mod. Phys. 04, 209 (2007)39. A. De Felice, S. Tsujikawa, Living Rev. Relat. 13, 3 (2010)40. J. Santos, J.S. Alcaniz, M.J. Rebouças, F.C. Carvalho, Phys. Rev.

D 76, 083513 (2007)41. O. Bertolami, C.G. Bohmer, T. Harko, F.S.N. Lobo, Phys. Rev. D

75, 104016 (2007)42. V. Faraoni, Phys. Rev. D 76, 127501 (2007)43. T. Harko, F.S.N. Lobo, S. Nojiri, S.D. Odintsov, Phys. Rev. D 84,

024020 (2011)44. G. Nordstrøm, Phys. Z. 15, 504 (1914)45. T. Kaluza, Sitz. Preuss. Akad. Wiss. 33, 966 (1921)46. O. Klein, Z. Phys. 37, 895 (1926)47. J.E.M. Aguilar, C. Romero, A. Barros, Gen. Relat. Gravit. 40, 117

(2008)48. J. Ponce de Leon, Class. Quant. Gravit. 27, 095002 (2010)49. J. Ponce de Leon, JCAP 03, 030 (2010)50. S.M.M. Rasouli, M. Farhoudi, H.R. Sepangi, Class. Quant. Gravit.

28, 155004 (2011)51. A.F. Bahrehbakhsh, M. Farhoudi, H. Shojaie, Gen. Relat. Gravit.

43, 847 (2011)52. A.F. Bahrehbakhsh, M. Farhoudi, H. Vakili, Int. J. Mod. Phys. D

22, 1350070 (2013)53. S. Capozziello, V.F. Cardone, S. Carloni, A. Troisi, Int. J. Mod.

Phys. D 12, 1969 (2003)54. S. Capozziello, V.F. Cardone, A. Troisi, J. Cosmol. Astropart.

Phys. 08, 001 (2006)55. A. Aghmohammadi, M.R. Kh Saaidi, A.Vajdi Abolhassani, Phys.

Scripta 80, 065008 (2009)56. B. Huang, S. Li, Y. Ma, Phys. Rev. D 81, 064003 (2010)57. K.A. Bronnikov, R.V. Konoplich, S.G. Rubin, Class. Quant. Grav.

24, 1261 (2007)58. Y.B. Wu et al., Eur. Phys. J. C 74, 2791 (2014)59. S. Nojiri, S.D. Odintsov, Gen. Rel. Grav. 37, 1419 (2005)60. R. Schimming, M. Abdel-Megied, F. Ibrahim, Chaos Solitons

Fractals 14(8), 1255 (2002)61. R. Schimming, M. Abdel-Megied, F. Ibrahim, Chaos Solitons

Fractals 15(1), 57 (2003)62. K. Atazadeh, F. Darabi, H.R. Sepangi, Int. J. Mod. Phys. D 18,

1049 (2009)

63. J.E.M. Aguilar, Eur. Phys. J. C 75(12), 575 (2015)64. S. Carloni, A. Troisi, P.K.S. Dunsby, Gen. Rel. Grav. 41, 1757

(2009)65. H.A. Buchdahl, Mon. Not. R. Astron. Soc. 150, 1 (1970)66. R. Utiyama et al., J. Math. Phys. 3, 608 (1962)67. I.L. Buchbinder, S.D. Odintsov, I.L. Shapiro., Effective action in

quantum gravity, (IOP, Bristol, 1992), p. 41368. A.A. Starobinsky, Phys. Lett. B 91, 99 (1980)69. R. Kerner, Gen. Relat. Gravit. 14, 453 (1982)70. R.P. Woodard, Lect. Notes. Phys. 720, 403 (2007)71. R.P. Woodard, arXiv:1506.02210 [hep-th]72. S. Carlip, Rep. Progr. Phys. 64(8), 885 (2001)73. M. Kaku, Introduction to Superstrings and M-Theory (Springer,

Berlin, 1999)74. Y. Choquet-Bruhat, Supergravities and Kaluza–Klein Theories.

Topological Properties and Global Structure of Space–Time, pp.31–48. Springer US (1986)

75. O. Aharony et al., Phys. Rep. 323(3), 183 (2000)76. J.E. Campbell, A Course of Differential Geometry (Clarendon

Press, Oxford, 1926)77. L. Magaard, Zur Einbettung Riemannscher Räume in Einstein-

Räume und konform-euklidische Räume? PhD Thesis, Universityof Kiel (1963)

78. C. Romero, R. Tavakol, R. Zalaletdinov, Gen. Relat. Gravit. 28(3),365 (1996)

79. P.S. Wesson, Astrophys. J. 397, L91 (1992)80. P.S. Wesson, J. Ponce de Leon, J. Math. Phys. 33, 3883 (1992)81. R. Kerner, Gen. Relat. Gravit. 9(3), 257 (1978)82. R. Kerner, S. Vitale, Int. J. Mod. Phys. A 24, 1465 (2009)83. J. Ponce de Leon, Mod. Phys. Lett. A 16, 2291 (2001)84. P.S. Wesson, J. Ponce de Leon, Gen. Relat. Gravit. 26, 555 (1994)85. S. Capozziello, V.F. Cardone, A. Troisi, Mon. Not. R. Astron. Soc.

375, 1423 (2007)86. C.L. Bennett et al., Astrophys. J. Lett. 464, L1 (1996)87. P. de Bernardis et al., Nature 404(6781), 955 (2000)88. G. Hinshaw et al., Astrophys. J. Suppl. 170, 288 (2007)89. Planck Collaboration: P. A. R. Ade, et al., arXiv:1502.0158990. Planck Collaboration: P. A. R. Ade, et al., arXiv:1502.0159391. J. Ponce de Leon, Gen. Relat. Gravit. 20, 539 (1988)92. J.L. Lopez, D.V. Nanopoulos, Mod. Phys. Lett. A 11, 1 (1996)93. J.M. Overduin, F.I. Cooperstock, Phys. Rev. D 58, 043506 (1998)94. H.Y. Liu, P.S. Wesson, J. Math. Phys. 33, 3888 (1992)95. P.S. Wesson, H.Y. Liu, Phys. Lett. B 432, 266 (1998)96. J. Socorro, V.M. Villanueva, L.O. Pimentel, Int. J. Mod. Phys. A

11, 5495 (1996)97. R.J. Nemiroff, R. Joshi, B.R. Patla, J. Cosmol. Astrop. Phys. 1506,

6 (2015)98. R. Giostri, M. Vargas dos Santos, I. Waga et al., JCAP 03, 027

(2012)99. M. Vargas dos Santosa, R.R.R. Reisa, I. Waga, JCAP 02, 066

(2016)100. S. Rani, A. Altaibayeva, M. Shahalam, J.K. Singha, R. Myrza-

kulov, JCAP 03, 031 (2015)101. A.P. Billyard, W.N. Sajko, Gen. Relat. Gravit. 33, 1929 (2001)

123