Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology?
aa r X i v : . [ g r- q c ] F e b EPJ manuscript No. (will be inserted by the editor)
Higher-order gravity in higher dimensions: Geometrical origins offour-dimensional cosmology?
Antonio Troisi Dipartimento di Fisica “E.R. Caianiello”, Universit`a degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084, Salerno, Italy.February 24, 2017
Abstract.
Determining cosmological field equations represents a still very debated matter and implies awide discussion around different theoretical proposals. A suitable conceptual scheme could be representedby gravity models that naturally generalize Einstein Theory like higher order gravity theories and higherdimensional ones. Both of these two different approaches allow to define, at the effective level, Einstein fieldequations equipped with source-like energy momentum tensors of geometrical origin. In this paper, it isdiscussed the possibility to develop a five dimensional fourth order gravity model whose lower dimensionalreduction could provide an interpretation of cosmological four dimensional matter-energy components. Wedescribe the basic concepts of the model, the complete field equations formalism and the 5-Dim to 4-Dimreduction procedure. Five dimensional f ( R ) field equations turn out to be equivalent, on the four dimen-sional hypersurfaces orthogonal to the extra-coordinate, to an Einstein like cosmological model with threematter-energy tensors related with higher derivative and higher dimensional counter-terms. By consider-ing a gravity model f ( R ) = f R n it is investigated the possibility to obtain five dimensional power lawsolutions. The effective four dimensional picture and the behaviour of the geometrically induced sourcesare finally outlined in correspondence to simple cases of such higher dimensional solutions. Type Ia supernovae (SNeIa) observations depicted a late-time speeding up universe [1,2,3], driven by an unknowncomponent, whose properties can be ascribed to some sortof exotic fluid. This elusive component has been addressedto a form of dark energy that exhibits an anti-gravitationalnegative equation of state (EoS) [4]. The combination ofstandard matter and dark energy provides an Einstein-like Universe where gravitational attraction is counterbal-anced by such an unusual gravitational source thereforereproducing cosmological observations. Soon after this dis-cover a plethora of theoretical proposals has been sug-gested in order to explain dark energy origin. However, attoday, a well endowed and self consistent physical inter-pretation is so far unknown [5,6,7]. The puzzling quest ofa satisfactory explanation about these phenomenologicalresults led cosmologists to explore several research lines.In particular, people followed two main directions. Fromone side, standard Einstein gravity has been reviewed byintroducing a new cosmological component: the cosmo-logical constant or a whatsoever well behaved fluid witha negative EoS. Differently, adopting an unconventionalpoint of view, they have been considered modified gravitymodels that generalize Einstein theory: i.e. scalar-tensorgravity [8,9,10], f ( R ) theories [11,12,13], DGP gravity ⋆ e-mail: [email protected] [14], braneworld scenarios [15,16,17], induced matter the-ory [18,19,20] and so on.Among others f ( R ) theories of gravity received a con-siderable attention. Such theories, which represent a nat-ural generalization of Einstein gravity, are obtained by re-laxing the hypothesis of linearity contained in the Hilbert-Einstein Lagrangian. Several results, sometimes contro-versial, have been obtained in this framework [21]. Thesemodels have been satisfactory checked with cosmologicalobservations [22,23,24] and suggest intriguing peculiari-ties in the low energy and small velocity limit [25] sincethe gravitational potential displays a Yukawa like correc-tion [26]. In order to be viable f ( R ) gravity theories haveto satisfy some minimal prescriptions. In particular, theyhave to match cosmological observations avoiding insta-bilities and ghost-like solutions and they have to evadesolar system tests [27]. Constraints from energy conditionsrepresent a further argument to settle a suitable gravityLagrangian [28]. However, at this stage, a fully satisfac-tory fourth order gravity model is far to be achieved. Asa matter of fact, such kind of models have been consid-ered also from other points of view. For example, recently,models with a non-minimal coupling between gravity andthe matter sector have been taken into account [29,30].On the other side, historically, attempts for a unifi-cation of gravity with other interactions stimulated thesearch for theoretical schemes based on higher dimen-sions, i.e. beyond our conventional four dimensional (4-D) ntonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology? 3 spacetime. Nordstrøm [31], who was the first to formu-late a unified theory based on extra dimensions, Kaluza[32] and Klein [33] developed a five-dimensional (5-D) ver-sion of General Relativity (GR) in which electrodynamicsderivates as a counter-effect of the extra-dimension. Suc-cessively, a lot of work has been dedicated to such the-oretical proposal both considering fifth-coordinate com-pactification and large extra-dimensions, allowing for noncompact mechanisms [19]. This last approach led to the socalled Space-Time-Matter (STM) or Induced-Matter The-ories (IMT) [18,20] and to braneworld theories [15]. Thebasic characteristic of IMT is that 5-D vacuum field equa-tions can be recast after the reduction procedure as 4-DEinstenian field equations with a source of geometrical ori-gin. In such an approach, the matter-energy source of 4-Dspacetime represents a manifestation of extra dimensions.Generalizations of Kaluza-Klein theory represent them-selves a suitable scheme to frame modern cosmologicalobservations. In fact, after SNIe observations, several at-tempts have been pursued to match Induced Matter The-ories with the dark energy puzzle [34,35,36,37].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 sourceis of geometrical origin [38,39], it seems a significant pro-posal to confront the two theoretical schemes. The gen-uine idea underlying this work is to investigate 5-D f ( R )-gravity from the point of view of a completely geomet-ric self-consistent approach. In this scheme, all matter-energy sources will represent a byproduct of the dimen-sional reduction related to higher order terms and higherdimensional quantities. Such an approach determines acompletely different conceptual framework with respectto standard five dimensional fourth order gravity mod-els. The main purpose is to recover along this schemeboth dark matter and dark energy dynamical effects. Inthe standard realm, in fact, extra dimensions and cur-vature counter-terms will only play the role of mendingdark energy phenomenology in presence of ordinary mat-ter. Along this orthodox paradigm, 5-D models of fourthorder gravity have been studied in presence of perfect fluidsources [42,43] under peculiar assumptions on the metricpotentials. In a similar fashion, accelerating 4-D cosmolo-gies, induced by generalized 5-D f ( R ) gravity, have beenalso studied considering a curvature-matter coupling [44].Furthermore, a vacuum fourth order Kaluza-Klein theory,defined in term of the Gauss-Bonnet invariant, has beenstudied from the point of view field equations predictions.In particular, it has been investigated, in the cylinder ap-proximation, the propagation of its electromagnetic de-grees of freedom [40] and the particle spectrum in thelinear regime [41].Attempts to meet f ( R )-gravity and IMT have been alsodeveloped in time. For example it has been investigatedthe possibility to draw information on the Space-Time-Matter tensor starting from the energy-momentum tensorinduced by higher order curvature counter-terms [45]. Onthe other side a new effective coupled F ( (4) R, ϕ ) gravitytheory has been proposed as a consequence of a five di- mensional f ( R ) model [46]. Within such a paper it hasbeen also demonstrated that the Dolgov-Kawasaki stabil-ity criterion of the 5-D f ( R ) theory remains the same asin usual f ( R ) theories: f ′′ ( R ) >
0, with prime that indi-cates the derivative with respect to the Ricci scalar.In our work we try to develop a more general framework,at first 5-D f ( R )-gravity is discussed in a complete analyt-ical form. We provide general field equations in the f ( R )-IMT approach and, by exploiting the reduction procedurefrom 5-D to the 4-D ordinary spacetime (considering asuitable extension of Friedmann-Robertson-Walker met-ric), we obtain a complete set of new Einstein-like fieldequations with matter-energy sources induced by higherorder derivative terms and higher dimensions (top-downreduction). In other words, we develop a fully geometri-cal cosmology where ordinary matter and dark compo-nents could be addressed, in principle, as the outgrowthof the GR formalism adopted within the top-down reduc-tion mechanism.The paper is organized as follows. In section II we providea brief review of 4-D f ( R )-gravity. Section III is devoted tosummarize Induced Matter Theory. The 5-D f ( R )-gravityformalism is outlined in section IV, and, thereafter, we dis-cuss the 5-D to 4-D reduction procedure of field equations.In section V it is described a routine to find 5-D powerlaw solutions. Section VI is dedicated to the analysis ofthe results and, in particular, to their interpretation fromthe point of view of the 4-D induced cosmologies. Finally,Section VII is left to conclusions. f ( R ) Gravity
Dark energy models are based on the underlying assump-tion that Einstein’s General Relativity is indeed the cor-rect theory of gravity. However, adopting a different pointof view, both cosmic speed up and dark matter can beaddressed to a breakdown of GR. As a matter of fact, oneshould consider the possibility to generalize the Hilbert-Einstein (H-E) Lagrangian. With these premises in mind,the choice of the gravity Lagrangian can be settled bymeans of data with the only prescription of adopting an“economic” strategy, i.e. only minimal generalization ofH-E action are taken into account. The so called f ( R )-gravity [11,21,47], which consider an analytic function interm of Ricci scalar and provides fourth order field equa-tions, is based on this conceptual scheme. It has to bereminded that higher order gravity theories represent thenatural effective result of several theoretical schemes [48],i.e. quantum field theories on curved spacetimes and M-theory. In addition, they have been widely studied as in-flationary models in the early universe [49,50].Fourth order gravity is favored by Ostrogradski the-orem. It has been in fact demonstrated [51] that f ( R )-Lagrangians are the only metric-based, local and poten-tially stable modifications of gravity among the severalthat can be constructed by means of the curvature tensorand possibly by means of its covariant derivatives. Antonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology?
Let us consider the generic f ( R ) action, we have: S = 12 Z d x √− g [ f ( R ) + S M ( g µν , ψ )] , (1)where we have used natural units 8 πG = c = ~ = 1, g isthe determinant of the metric and R is the Ricci scalar and ψ characterizes the matter fields. Varying with respect tothe metric we get field equations f ′ ( R ) R µν − f ( R ) g µν − [ ∇ µ ∇ ν − g µν (cid:3) ] f ′ ( R ) = T Mµν , (2)where, the prime, as said in the introduction, denotesderivative with respect to R and, as usual, T Mµν = − √− g δS M δg µν . (3)Tracing (2) we obtain f ′ ( R ) R − f ( R ) + 3 (cid:3) f ′ ( R ) = T M , (4)where T M = g µν T Mµν , so that the relation between theRicci scalar and T M is obtained by means a differentialequation, differently than GR where R = − T M . Thisresult suggest that f ( R )-gravity field equations admit alarger variety of solutions than Einstein’s theory. In par-ticular, T M = 0 solutions will no longer imply Ricci flatcosmologies and therefore that R = 0. Actually, a suitableproperty of this model is that field equations (2) can berecast in the Einstein form [11,38]: G µν = R µν − g µν R = T curvµν + T Mµν /f ′ ( R ) , (5)with T curvµν = 1 f ′ ( R ) n g µν [ f ( R ) − Rf ′ ( R )] + f ′ ( R ) ; αβ ( g µα g νβ − g µν g αβ ) o , (6)that represents a curvature stress - energy tensor inducedby higher order derivative terms (the terms f ′ ( R ) ; µν ren-der the equations of fourth order). The limit f ( R ) → R reduces Eqs.(5) to the standard second-order Einstein fieldequations.The Einstein-like form of fourth order field equations(5) suggests that higher order counter-terms can play therole of a source-like component within gravity field equa-tions. In practice, it is possible to postulate that geometrycan play the role of a mass-energy component when higherthan second order quantities are taken into account. In-deed, the trace equation (4) propagates a scalar-like de-gree of freedom. As a matter of fact, in principle, onecan imagine to address Universe dark phenomenology tosuch a kind of effective fluid. From the cosmological pointof view a relevant role is played by the barotropic factorof the curvature matter-energy fluid. In the vacuum casethis quantity will discriminate accelerating solutions from standard matter ones. In the case of a Friedman space-time, for example, one has [38,39]: ρ curv = 1 f ′ ( R ) (cid:26)
12 [ f ( R ) − Rf ′ ( R )] − H ˙ Rf ′′ ( R ) (cid:27) , (7) w curv = − Rf ′′ ( R ) + ˙ R h ˙ Rf ′′′ ( R ) − Hf ′′ ( R ) i [ f ( R ) − Rf ′ ( R )] / − H ˙ Rf ′′ ( R ) , (8)where H = ˙ a/a stands for the Hubble parameter and dotmeans a time derivative.In the following we will resort to a similar quantity inorder to check the behaviour of the cosmological fluids in-duced by 5-D f ( R )-gravity on the 4-D hypersurfaces thatslice the higher dimensional spacetime along the extra-coordinate.In our conventions we will consider latin indices like A, B, C, etc. running from 0 to 4, latin indices like i, j, etc., run-ning from 1 to 3 and greek indices taking values from 0 to3.
Modeling a unification theory that contains gravity andparticle physics forces, typically, implies the resort to extra-dimensional models [19,52]. Among these 5-D Kaluza-Klein theory [32,33] and its modern revisitations Induced-Matter and membrane theory represent significant ap-proaches; in addition these kind of models represent thelow energy limit of more sophisticate theories (i.e. super-gravity) [53,54]. The main difference between the seminalapproach of Kaluza and Klein and Induced Matter Theoryis related with the role of the extra-dimension. In its firstconception the fifth dimension was “rolled up” to a verysmall size, answering the question of why we do not “see”the fifth dimension. Modern theories like IMT postulatethat we are constrained to live in a smaller 4-D hyper-surface embedded in a higher dimensional spacetime. Thekey-point for embedding the 4-D Einstein theory into the5-D Kaluza-Klein Induced-Matter-Theory is representedby Campbell-Magaard (CM) theorem. The problem of em-bedding a Riemannian manifold in a Ricci-flat space wasstudied by Campbell soon after GR discovery [55], and fi-nally demonstrated by Magaard in 1963 [56]. Further on,Tavakol and coworkers [57] used these studies to establishmathematical well endowed bases to the 4-D reinterpreta-tion of 5-D Kaluza-Klein theory that is dubbed Induced-Matter Theory [58,60]. In such a framework, the 5-D to4-D reduction procedure determines on the 4-D hypersur-face an Einstein gravity theory plus induced matter com-ponents of geometrical origin. The CM theorem can beformalized as follows:
Any analytic Riemannian space V n ( x µ , t ) can be lo-cally embedded in a Ricci-flat Riemannian space V n +1 (cid:0) x A , t (cid:1) . ntonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology? 5 Here the “smaller” space has dimensionality n with µ = 0 , ..., n −
1, while the “host” space has dimensional-ity N = n + 1 with A running from 0 to n , the extra-coordinate can be both space-like and time-like. We areinterested to the case N = 5. It is important to remarkthat CM theorem is a local embedding theorem. Therefore,more general issues related with global embedding, i.e.initial-value problems, stability or general induced solu-tions [59], cannot be resort to this achievement. However,for our purposes, the theorem guarantees the right ana-lytic framework in order to frame 4-D matter phenomenol-ogy in relation to 5-D field equations [60].Because of CM theorem it is possible to write downthe 5-D metric as follows: dS = γ AB dx A dx B = g µν ( x, y ) dx µ dx ν + ǫΦ ( x, y ) dy , (9)where x µ = ( x , x , x , x ) are 4-D coordinates, g µν turnsout to be the spacetime metric and y is the fifth coordi-nate, ǫ = ± , − , − , − ). With these premises in mind ordinary 4 D spacetime result as a hypersurface Σ y : y = y = con-stant, orthogonal to the 5 D extra-coordinate basis vectorˆ n A = δ A Φ , n A n A = ǫ. (10)If one considers vacuum 5-D field equations (Ricci flat): R AB = 0, the CM theorem, suggests a natural reductionprocess to a 4-D pseudo-Riemannian spacetime. In prac-tice, one can obtain an Einstein-like 4-D model G µν = T µν [58,60], once the right hand side of these equationsfulfills the definition T IMTµν = Φ ,µ ; ν Φ − ǫ Φ ( ∗ Φ ∗ g µν Φ − ∗∗ g µν + g αβ ∗ g µα ∗ g νβ + − g αβ ∗ g αβ ∗ g µν g µν (cid:20) ∗ g αβ ∗ g αβ + (cid:16) g αβ ∗ g αβ (cid:17) (cid:21)) . (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 Mat-ter Tensor; the only hypothesis underlying this achieve-ment has been relaxing the cylinder condition within theKaluza-Klein scheme (independence on the fifth coordi-nate).As it is possible to observe from the previous result,fourth order gravity theories and Kaluza-Klein IMT mod-els provide the same conceptual scheme. Both of the ap-proaches allow to obtain an Einstein-like gravity modelwhere matter-energy sources are of geometrical origin. In We recall that natural units are adopted, therefore 8 πG (cid:30) c is set equal to unity. particular, these effective matter-energy tensors descend,respectively, from the higher order derivative contribu-tions and from the higher dimensional counter-terms. There-fore, it seems that deviations from GR can be naturallyrecast as sources of standard Einstenian models.In that respect let us notice that IMT and 5-D f ( R ) grav-ity are built in a different philosophy with respect to con-ventional higher dimensional approaches like braneworldmodels. In fact, despite the same working scenario: bulkuniverse with non trivial dependence on the extra-coordinate,4-D metric obtained evaluating the background metric atspecific 4-D hypersurfaces, matter fields confined in the4-D spacetime, there are intrinsic conceptual differences.5-D f ( R ) gravity and Induced Matter Theory are based onthe hypothesis that standard matter is nothing else than a4-D manifestation of geometrical deviations with respectto GR. Braneworlds model our universe as a four dimen-sional singular hypersurface, the brane, embedded in afive dimensional anti De Sitter spacetime. In such a casethe motivation for a non compactified extra-dimension isto solve the hierarchy problem. The differences in termof physical motivations for large extra-dimensions implyalso different technical approaches. Within IMT and 5-D f ( R ) gravity one considers a Ricci flat vacuum bulk anddevelops the 4-D physics as a byproduct. In braneworldsit is assumed the opposite point of view. In this case, onedeals with suitable 4-D solutions on the brane for somematter distributions and these solutions are matched withan appropriate 5-D bulk considering Israel junction con-ditions. Nevertheless, despite such conceptual differencesit has been showed in time that brane theory, IMT and5-D f ( R ) gravity can have a suitable matching. In particu-lar, IMT and braneworlds have been put in strict analogy[61]. The key point is to consider the induced matter ap-proach in term of the spacetime extrinsic curvature. Inthis case it is possible to write down braneworld-like fieldequations with a 4-D matter-energy source and a branetension that are defined in term of the extrinsic curva-ture. On the same line very recently it has been shown that Z braneworld-like solutions can be obtained as particularmaximally symmetric solutions of 5-D f ( R ) gravity withmatter [46]. In future works one can imagine to deepen theinterconnections that seem to arise among these differenthigher dimensional approaches.Such considerations suggest, in principle, the intriguingpossibility that GR experimental shortcomings could beactually addressed to the effective property of modifiedgravity models. In the following we will exploit such ap-proach to study a general fourth order 5-D model, wherehigher order derivative gravity is “merged” with higherdimensions. Antonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology? f ( R ) -gravity model and its 4-Dreduction A f ( R ) theory of gravity in five dimensions can be de-scribed by the action S (5) = 12 Z d x p g (5) f ( R (5) ) + 12 Z d x p g (5) L M ( g AB , ψ ) , (12)where R (5) is the 5-D Ricci scalar, L m ( g AB , ψ ) is a La-grangian density for matter fields denoted, as in the 4-Dcase, by ψ and g (5) represents the determinant of the 5-Dmetric tensor g AB . 5-D field equations can be obtained byaction (12) varying with respect to the metric: f (5) ,R R (5) AB − f ( R (5) ) g AB + (13) − [ ∇ A ∇ B − g AB (cid:3) (5) ] f (5) ,R = T (5) MAB , here T (5) MAB is the energy-momentum tensor for mattersources, while ∇ A is the 5D covariant derivative, (cid:3) (5) = g AB ∇ A ∇ B is the 5D D’Alambertian operator. In order tosimplify the notation we have defined f ,R ( R (5) ) ≡ f (5) ,R ,therefore henceforth with f (5) ,R it is intended the 5-D Ricciderivative of the f ( R (5) ) gravity Lagrangian.Together the field equations (13) one can obtain thetrace 4 (cid:3) (5) f (5) ,R + f (5) ,R R (5) − f ( R (5) ) = T (5) M (14)where T (5) M = g AB 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 section2. In fact, with some algebra and isolating 5-D Einsteintensor one gets : R (5) AB − R (5) g AB = 1 f (5) ,R (cid:26) (cid:16) f ( R (5) ) − f (5) ,R R (5) (cid:17) g AB − [ g AB (cid:3) (5) − ∇ A ∇ B ] f (5) ,R o + 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 (cid:26) (cid:16) f ( R (5) ) − f (5) ,R R (5) (cid:17) g AB + − [ g AB (cid:3) (5) − ∇ A ∇ B ] f (5) ,R o . (16)This fact, which can resemble only the byproduct of 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 tensorand the two terms derivate from the relative 4-D reduc-tion applied to T (5) CurvAB . Since we want to explore howgeometric counter-terms can effectively mimic cosmolog-ical sources we neglect ordinary matter, i.e. here on wewill assume T (5) MAB = 0, considering gravity equation invacuum.Let us now develop the 5-D to 4-D reduction procedureof our higher dimensional framework. At first, by assumingthe metric (31), it is possible to draw the reduction rulesfor the differential operators [36] : ∇ µ ∇ ν f (5) ,R = D µ D ν f (5) ,R + ǫ Φ ∗ g µν ∗ f (5) ,R , (17) ∇ ∇ f (5) ,R = ǫΦ ( D α Φ ) (cid:16) D α f (5) ,R (cid:17) + ∗∗ f (5) ,R − ∗ ΦΦ ∗ f (5) ,R , (18) (cid:3) (5) f (5) ,R = (cid:3) f (5) ,R + ( D α Φ ) (cid:16) D α f (5) ,R (cid:17) Φ ++ ǫΦ " ∗∗ f (5) ,R + ∗ f (5) ,R g αβ ∗ g αβ − ∗ ΦΦ ! ; (19)here, again, the asterisk denotes partial derivative withrespect to the extra coordinate (i.e., ∂/∂y = ∗ ); D α isthe four-dimensional covariant derivative defined on thehypersurface Σ y = y , calculated with g µν , and the usualD’Alambertian (cid:3) is referred to 4-D quantities. In the sameshape, all the quantities that are not labelled in term ofthe 5 th coordinate will be intended to refer to ordinaryspacetime. On this basis, it is possible to rewrite the 5-D field equations (15) by separating the spacetime part( µ = 0 , .., ν = 0 , ..,
3) from the extra-coordinate ( y )one : ntonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology? 7 G (5) µν = 1 f (5) ,R (cid:16) f ( R (5) ) − f (5) ,R R (5) (cid:17) g µν − (cid:3) f (5) ,R + ( D α Φ ) (cid:16) D α f (5) ,R (cid:17) Φ ++ ǫΦ ∗∗ f (5) ,R + ∗ f (5) ,R g αβ ∗ g αβ − ∗ ΦΦ !! g µν + ∇ µ ∇ ν f (5) ,R + ǫ Φ ∗ g µν ∗ f (5) ,R ) , (20) G (5)44 = 1 f (5) ,R (cid:16) f ( R (5) ) − f (5) ,R R (5) (cid:17) g − (cid:3) f (5) ,R + ( D α Φ ) (cid:16) D α f (5) ,R (cid:17) Φ ++ ǫΦ ∗∗ f (5) ,R + ∗ f (5) ,R g αβ ∗ g αβ − ∗ ΦΦ !! g + ǫΦ ( D α Φ ) (cid:16) D α f (5) ,R (cid:17) + ∗∗ f (5) ,R − ∗ ΦΦ ∗ 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 theleft member and move on the r.h.s all quantities that de-pend on higher derivative terms and on the extra-coordinate.All the higher order and higher dimensional counter-termswill play the role of effective source terms on the 4-D hy-persurface. It is evident that since the fourth order La-grangian f ( R (5) ) depends on the 5-D Ricci scalar, thisdependence cannot be completely untwined until the La-grangian dependence is not specified.The reduction rules for the Einstein tensor [60] give R (5) µν = R µν − D µ D ν ΦΦ + ǫ Φ ∗ Φ ∗ g µν Φ − ∗∗ g µν ++ g λα ∗ g µλ ∗ g να − g αβ ∗ g αβ ∗ g µν ! , (22) R (5)44 = − ǫΦ (cid:3) Φ − ∗ g αβ ∗ g αβ − g αβ ∗∗ g αβ ∗ Φ g αβ ∗ g αβ Φ . (23)The next step is to calculate the 5-D Ricci scalar R (5) = g µν R (5) µν + g R (5)44 . After the substitution of the expression(23) within Eq.(21) and some tedious algebra one gets therelation (cid:3) ΦΦ = ǫΦ ∗ ΦΦ g αβ ∗ g αβ − ∗ g αβ ∗ g αβ − g αβ ∗∗ g αβ + − f ( R (5) ) f (5) ,R + (cid:3) (5) f (5) ,R f (5) ,R − ( D α Φ )( D α f (5) ,R ) Φf (5) ,R + (24) − ǫΦ ∗∗ f (5) ,R f (5) ,R − ∗ ΦΦ ∗ f (5) ,R f (5) ,R , that reminds an analogous expression given in [36] for the4-D reduction of 5-D Brans-Dicke theory. Further manip-ulations of (24) allow to obtain a relatively simpler expres-sion : (cid:3) ΦΦ = ǫΦ ∗ ΦΦ g αβ ∗ g αβ − ∗ g αβ ∗ g αβ − g αβ ∗∗ g αβ + − f ( R (5) ) f (5) ,R + (cid:3) f (5) ,R f (5) ,R − ǫΦ g αβ ∗ g αβ ∗ f (5) ,R f (5) ,R , (25)that can be, finally, used in the reduction of Ricci scalarto collect some terms. Therefore, by using also the traceequation, one has : R (5) = R − (cid:3) ΦΦ + ∗ ΦΦ g αβ ∗ g αβ − g αβ ∗∗ g αβ − ∗ g αβ ∗ g αβ − ( g αβ ∗ g αβ ) ! . (26)It is easy to check that this result reproduces the analo-gous relation of 5-D GR [19] when f ( R (5) ) → R (5) . Weremember that in such a case vacuum Einstein field equa-tions reduce to the system R (5) AB = 0 and R (5) vanishes.Once developed the several aspects of the 5-D to 4-Dreduction procedure it is possible to obtain the induced 4-D field equations. By inserting (22), (23) and (26) withinEqs. (20)-(21) 4-D spacetime equations get the form : G µν = T IMTµν + T curvµν + T Mixµν . (27) The equations provided along this section partially repro-duce previous results obtained in [46]. In fact, besides the dif-ferent conceptual scheme, there are also some analytic differ-ences that make the two schemes non completely indistinguish-able. Antonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology?
Here: T IMTµν = Φ ,µ ; ν Φ − ǫ Φ ( ∗ Φ ∗ g µν Φ − ∗∗ g µν + g αβ ∗ g µα ∗ g νβ + − g αβ ∗ g αβ ∗ g µν g µν (cid:20) ∗ g αβ ∗ g αβ + (cid:16) g αβ ∗ g αβ (cid:17) (cid:21)) . is the usual IMT tensor, deriving from the 5-D to 4-Dreduction of Einstein tensor, T Curvµν = 1 f (5) ,R (cid:26) (cid:16) f ( R (5) ) − f (5) ,R R (cid:17) g µν + − [ g µν (cid:3) − ∇ µ ∇ ν ] f (5) ,R o . (28)is the curvature tensor written preserving the 4-D form. Insuch a case extra-dimension contributions are still hiddenin the scalar curvature nested within the definition of thegravity Lagrangian and of its derivatives. Finally, T Mixµν = ǫ Φ ∗ g µν ∗ f (5) ,R f (5) ,R − ( D α Φ )( D α f (5) ,R ) Φf (5) ,R g µν + − ǫΦ ∗∗ f (5) ,R f (5) ,R + ∗ f (5) ,R f (5) ,R g αβ ∗ g αβ − ∗ ΦΦ ! g µν +(29)+ ǫ Φ ∗ g αβ ∗ g αβ g αβ ∗ g αβ ) g µν represents a mixed tensor containing terms that dependexplicitly on the fifth coordinate and on the derivative of f (5) ,R with respect to y .Therefore, we have obtained a framework where 4-Dgravity is fully geometrized. In fact, within such a schemematter-energy sources are related only with geometricalcounter-effects deriving either from higher dimensional met-ric quantities or from higher order derivative ones.In order to complete our discussion one should takeinto account also the off-diagonal equation. By consideringthat R (5)4 α = ΦP βα ; β with P αβ = 12 Φ ( ∗ g αβ − g σρ ∗ g σρ g αβ )[60], one obtains ΦP βα ; β = ∗ f (5) ,R ,α f (5) ,R − ∗ g ασ g βσ f (5) ,R ,β f (5) ,R − Φ ,α Φ ∗ f (5) ,R f (5) ,R (30) that coincides with similar expressions achieved elsewhere[36,46] and in the limit f ( R (5) ) → R (5) gives back, in ab-sence of ordinary matter, the conservation law P βα ; β = 0[19]. Equation (30) resembles a more general conservationequation which relates the spacetime derivatives of P βα and ∗ f (5) ,R . It has been conjectured [65] that the spacetimecomponents of the field equations relate geometry withthe macroscopic properties of matter, while the extra-coordinate part ( α ) and ( ) might describe their micro-scopic ones. With these hypotheses, within fourth ordergravity Kaluza Klein models, one would have that micro-scopic properties of matter are influenced by spacetimederivatives of the scalar degree of freedom intrinsic in f ( R )gravity. Furthermore, in our case, P βα dynamics dependsalso on Φ , that, in turn (see Eq.(25)), shows an evolu-tion driven by the higher order gravity counter-terms. Itseems that there is a strict interconnection between theextra-dimension properties and the intrinsic f ( R ) scalardegree of freedom. For example, looking to equation (25),it seems that f ′ ( R ) guarantees a sort of Machian effectfor this kind of models. However, all these considerationsrepresent, at this stage, nothing more than speculationsand, therefore, we do not discuss this issue further here. Let us now verify what kind of solutions can be derivedfor our 5-D f ( R )-gravity model. Once general cosmologi-cal solutions have been achieved, it is be possible to draw5-D effects on Σ y = y hypersurfaces, deriving the effective4-D picture. In particular, one can determine the space-time effectual matter-energy behaviour of the different ge-ometrical components described in the previous section. Inorder to search for field equations solutions we assume the5-D metric: dS = n ( t, y ) dt − a ( t, y ) (cid:20) dr − kr ++ r (cid:0) dθ + sin θdϕ (cid:1)(cid:3) + ǫΦ ( t, y ) dy , (31)where k = 0 , +1 , − t, r, θ, φ ) are the usual coordinates for spheri-cally symmetric spatial sections. It is important to noticethat, since the metric choice, our f ( R (5) ) Lagrangian andits derivatives do not depend on the 4-D spatial coordi-nates. In fact, the 5-D Ricci scalar calculated over themetric (31) is a function of the ( t, y ) coordinates alone.If metric (31) is introduced in Eqs.(20)-(21), we obtainthe equations system, respectively for the components: , ii ( i = 1 , , and : ntonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology? 9 − f ( R (5) )2 f (5) ,R + 3 ǫ ∗ a ∗ f (5) ,R Φ af (5) ,R − ǫ ∗ a ∗ nΦ an − ǫ ∗ f (5) ,R ∗ Φf (5) ,R Φ + ǫ ∗ n ∗ ΦnΦ + ǫ ∗∗ f (5) ,R Φ f (5) ,R − ǫ ∗∗ nΦ n ++ 3 ˙ a ˙ f (5) ,R af (5) ,R n + 3 ˙ a ˙ nan + ˙ f (5) ,R ˙ Φn f (5) ,R Φ + ˙ n ˙ Φn Φ − an a − ¨ Φn Φ = 0 , (32) − f ( R (5) )2 f (5) ,R − ǫ ∗ a Φ a + 2 ǫ ∗ a ∗ f (5) ,R Φ af (5) ,R − ǫ ∗ a ∗ nΦ an + ǫ ∗ f (5) ,R ∗ nΦ f (5) ,R n + ǫ ∗ a ∗ ΦaΦ − ǫf (5) ,R ∗ ΦΦ f (5) ,R − ǫ ∗∗ aΦ a + ǫ ∗∗ f (5) ,R Φ f (5) ,R + − a a n + 2 ˙ a ˙ f (5) ,R af (5) ,R n + ˙ a ˙ nan − ˙ f (5) ,R ˙ nf (5) ,R n − ˙ a ˙ Φan Φ + ˙ f (5) ,R ˙ Φf (5) ,R n Φ − ¨ aan + ¨ f (5) ,R f (5) ,R n = 0 , (33) − f ( R (5) )2 f (5) ,R + 3 ǫ ∗ a ∗ f (5) ,R Φ af (5) ,R + ǫ ∗ f (5) ,R ∗ nΦ f (5) ,R n + 3 ǫ ∗ a ∗ ΦaΦ + ǫ ∗ n ∗ ΦnΦ − ǫ ∗∗ aΦ a − ǫ ∗ nΦ n ++ 3 ˙ a ˙ f (5) ,R af (5) ,R n − ˙ f (5) ,R ˙ nf (5) ,R n − a ˙ Φan Φ + ˙ n ˙ Φn Φ + ¨ f (5) ,R f (5) ,R n − ¨ Φn Φ = 0 , (34)3 ˙ a ∗ nan + ∗ n ˙ f (5) ,R nf (5) ,R + 3 ∗ a ˙ ΦaΦ + ∗ f (5) ,R ˙ Φf (5) ,R Φ − ∗ aa − ˙ ∗ f (5) ,R f (5) ,R = 0 . (35)The last equation (35) recalls, again, the result ob-tained in [36] for 5-D Brans-Dicke vacuum solutions. There-fore it seems a suitable choice to pursue the same approachperformed in this work to get some particular solutions ofour equations system (32)-(35). If we consider the metriccoefficients as separable functions of their arguments : n ( t, y ) = N ( y ) , a ( t, y ) = P ( y ) Q ( t ) ,Φ ( t, y ) = F ( t ) , f (5) ,R ( t, y ) = U ( y ) W ( t ) , (36)from Eq. one gets :3 ˙ F ∗ PF P + 3 ∗ N ˙ QN Q − ∗ P ˙ QP Q + ˙ F ∗ UF U + ∗ N ˙ WN W − ∗ U ˙ WU W = 0 . (37)This equation suggests the possibility to look for powerlaw cosmological solutions. In the following section we willdiscuss some examples in this sense. In order to look for 5-D f ( R ) gravity power law solutionswe consider a set of metric potentials defined as follows[36]: n ( t, y ) = N y δ , a ( t, y ) = A t α y β , Φ ( t, y ) = Φ t γ y σ , (38)with N , A , Φ some constants with appropriate units and α, β, γ, δ, σ that represent the unknown parameters askedto satisfy 5-D field equations.Up to now we have developed a completely general scheme,without any assumption on the gravity Lagrangian. How-ever, to completely define the solving algorithm one has tomake a choice about the Ricci scalar function entering thegravity action. To remain conservative with the solutionprocedure, we consider a power law function of the Ricciscalar f ( R ) = f R n . Such a model has been extensivelystudied in literature at four dimensions both in cosmol-ogy (curvature quintessence) [11,38,39] and in the low en-ergy limit [25,64]. In addition, the 5-D phenomenology ofa power law fourth order gravity has been investigated inthe standard approach in presence of a perfect fluid mattersource [43,44]. In particular, in such a case, it is assumed acompact fifth dimension and it is supposed that the homo-geneously distributed fluid does not travel along the fifthdimension. As a matter of fact the matter energy densityand the pressure experienced by a four-dimensional ob-server have to be an integrated throughout the compactextra-dimensional ring. In our approach we have overcome this point of view, we relax the hypotheses on the extra-coordinate and, above all, we discard ordinary matter infavor of a completely geometric self-consistent approach.According with cosmological observations [62,63], we willstudy field equations considering a flat spacetime geome-try ( k = 0).A suitable recipe to study Eqs. (32)-(35), is to plugthe metric functions (38) within the last of these relationsand to search for its solutions. This approach allows to obtain, with some efforts, a number of constraints on themodel parameters. Then, one can try to satisfy also theother more complicated field equations (32)-(34). By in-serting metric potentials within Eq. , one obtains a rathercumbersome expression : Ξ ( α, β, γ, δ, σ, n, t, y ) Υ ( α, β, γ, δ, σ, n, t, y ) = 0 (39)where Ξ ( α, β, γ, δ, σ, n, t, y ) = y − − δ (cid:16) βγ + α ( − β + δ )) (cid:0) N t y δ (cid:0) β + 3 β ( − δ − σ ) + δ ( − δ − σ ) (cid:1) ++ t γ y σ (cid:0) α + 3 α ( − γ ) + ( − γ ) γ (cid:1) ǫΦ (cid:1) − +2( − n ) (cid:16) N t y δ γ ( δ + ( − n )(1 + σ )) (cid:0) − β + 3 β (1 − δ + σ ) + δ (1 − δ + σ ) (cid:1) ++ N t γ y δ + σ ) (cid:0) α + 3 α ( − γ ) + ( − γ ) γ (cid:1) ǫ (cid:0) β + 3 β ( − δ − σ )++ δ ( − δ − σ )) (3 δ + 2( − n )(1 + σ ) + γ (3 − δ + 2 δn + 3 σ )) Φ ++ t γ y σ (cid:0) α + 3 α ( − γ ) + ( − γ ) γ (cid:1) δǫ ( − γ + 2 n ) Φ (cid:17) (cid:17) and Υ ( α, β, γ, δ, σ, n, t, y ) = N t (cid:0) N t y δ (cid:0) β + 3 β ( − δ − σ ) + δ ( − δ − σ ) (cid:1) + t γ y σ (cid:0) α + 3 α ( − γ ) + ( − γ ) γ (cid:1) ǫΦ (cid:1) . This result, evidently, suggests the possibility to ob-tain, in principle, much more solutions than the Brans-Dicke case studied in [36]. To simplify our search, as apreliminary step, we make some trivial hypotheses aboutmodel parameters, leaving a complete analysis of fieldequations solutions to a future dedicated work.
As a first step we want to verify that the Kaluza-KleinGR limit is recovered. To perform 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 canbe verified if βγ + α ( − β + δ ) = 0. In particular, if γ = α one obtains β → αδα − γ (a result that is in agreement with[36]). Looking to the other equations, together some trivialnon evolving solutions ( α = 0 ), one obtains the solvingsets of parameters : It is possible to find some trivial solutions with, α = 0, β = 0, δ = (0 , γ = 1 δ = 1 , γ = 1 , α = − , β = 1and N = 4 Φ with ǫ = − , (40) δ = 0 , γ = − / , α =1 / , β = 0 , (41)with ǫ = ± δ = 1 , γ = 1 , α = − , β = αα − Φ = N ( α − with ǫ = − . (42)All these solutions are well known in literature [36,66],therefore, our model completely reproduces standard GRKaluza-Klein models in the f ( R (5) ) → R (5) limit. f ( R ) gravity limit What about standard 4-D fourth order gravity? One shouldexpect that this framework has to represent a natural sub-case of 5-D f ( R ) gravity. In order to get this limit one has ntonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology? 11 to assign a solutions set with δ = 0 , β = 0 , σ = 0 and γ = 0, that means, no dependence on y and no dynamicson the fifth coordinate. Starting with this assumptions andby considering equations (32)-(33) and (35), one obtains α = − n − n − n , (43)that exactly matches the well known power law f ( R ) grav-ity solution in the case of vacuum field equations [11].However, in the 5-D case, to fully satisfy the theory wehave to fulfill an equation more, Eq. (34). This request, byfact, settles the power law index n . Therefore the only al-lowed combination is δ = 0 , β = 0 , σ = 0 , γ = 0 , n = 5 / α = 1 / We can now relax some hypotheses in order to look forgeneralizations of 5-D GR. To search for such a kind of so-lutions we start, as a first simple case, from the assumptionof no dependence on the extra-coordinate. This means tostudy our theory, looking for solutions of Eqs.(32)-(35), inthe limit of cylinder condition. Differently than the stan-dard case, in the f ( R ) generalization of Induced Matter Theory this condition, as we will see, does not imply ra-diation as the only possible kind of induced matter.At first, let us gibe a look to Eq.(25). In the case of5-D f ( R ) gravity, independence on the fifth coordinatedoes not mean to have a massless Klein-Gordon equationfor the extra-coordinate potential Φ . Higher order gravitycounter-terms will play the role of a mass term and Φ willplay a completely new role into the dynamics : (cid:3) Φ = − " f ( R (5) ) f (5) ,R + (cid:3) f (5) ,R f (5) ,R Φ. (44)In addition, as a consequence, T IMTµν will be no moretraced to zero [19], therefore the Induced Matter Tensorcan span more general kinds of matter in relation to theunderlying cosmological solution.From the field equations point of view, power law so-lutions of cylinder-type are achieved when the model pa-rameters in (38) are settled as: δ = 0 , β = 0 , σ = 0 withfree n . Neglecting again static solutions it is possible tofind a set of implicit solutions that can be expressed interm of the f ( R ) Lagrangian power index n : α = 112 ( − n − f ( n )) , γ = −
17 + 28 n + 3 f ( n ) − n (24 + f ( n ))2 ( − n + f ( n )) ; (45a) α = 112 ( − n + f ( n )) , γ = 17 + 28 n − f ( n ) + 2 n ( −
24 + f ( n ))2 (5 − n + f ( n )) ; (45b) α = γ = 2 − n + 4 n − n ; (45c)with f ( n ) = √−
39 + 108 n − n . Eqs.(45a)-(45c) repre-sent an interesting achievement. In the following we willshow that these solutions provide, after the 5-D to 4-Dreduction, cosmological significant behaviours of the 4-Dmatter-energy tensors induced by 5-D geometry. f ( R ) gravity: a top-downgeometrization of matter The effective 4-D picture induced by higher dimensionscan be achieved once cosmological solutions obtained inSec.5 are plugged within the 5-D to 4-D reduction frame-work previously outlined. In particular the 4-D settingis achieved, as already said, by considering hypersurfaces Σ y = y that slice the 5-D universe along the fifth coordi-nate. In this scheme, the extra-coordinate effects on 4-D There are also in this case trivial solutions i.e. n = 1, α = 0, γ = (0 , physical quantities are evaluated taking y = y = const .The resulting cosmological model is a 4-D induced matter f ( R ) theory, where field equations are given in the form ofEinstein equations equipped with three different matter-energy sources of geometrical origin. Deviations from stan-dard GR provide, in the effective 4-D spacetime descrip-tion, matter-energy sources and one could wonder if thesecontributions are related with the elusive nature of DarkEnergy and Dark Matter. Actually, we study the 4-D in-duced cosmologies in correspondence of the solutions givenin Sec.5. In particular, we are interested to investigate thebehaviour of the three different induced cosmological flu-ids (11), (28), (29) on the y = y = const hypersurfaces.One can notice that for all of these quantities it holds T K = T K = T K (with K = IM T , Curv , M ix ), there-fore it is possible to describe such matter-energy tensorsas perfect fluid sources with ρ K = T K and the isotropicpressure defined as p K = − T K . The cosmological na- ture of each component can be evaluated by means of therespective equation of state (EoS) ω K = − T K /T K . If one considers solutions obtained along the standard KKGR limit, the related 4-D effective dynamics are describedby well known cosmological models. In particular, the so-lution (41) represents the standard radiation universe with p = 1 / ρ . In such a case the three matter-energy tensorscollapse into one matter-energy source, in particular thecurvature quintessence tensor and the mixed tensor re-spectively vanish. On the other side, the solution (42)represents a spatially flat FRW metric once the extra-coordinate is fixed : ds | Σ y = N y dt − A y αα − t α (cid:2) dr + r ( dθ + sin dϕ ) (cid:3) . (46)The induced matter tensor plays, again, the role of theonly one effective cosmological fluid. In fact, the other twoquantities combine to give a cancelling result. The cos-mological energy density scales as ρ = ρ t − , with ρ =3 α N y , while the barotropic factor is ω = 2 − α α . This re-sult is in complete agreement with previous achievementson non compactified KK gravity [36]. 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 equationsnumber of our model. In fact, it is possible to get the right4-D power law solution only if the extra-coordinate equa-tion is neglected. When this equations enters the game thepower law index n gets fixed. The behaviour of the threematter-energy tensors describing the top-down effect ofhigher dimensions and higher derivative terms on the 4-Dgravity hypersurface confirmate such a result. In fact, ifwe evaluate ω curv on the 4-D f ( R ) solution (43) we obtain ω curv = 1 + 7 n + 6 n − n + 6 n , (47)that, again, matches the already known result for this kindof model [11]. However, since the fifth equation select theallowed values for n , we know that the admissible solu-tion is indeed δ = 0 , β = 0 , σ = 0 , γ = 0 , n = 5 / α = 1 /
2. The effective, 4-D, consequent cosmologyshows ρ IMT = ρ Mix = 0 , ρ curv = N t and ω curv = 1 / n = 5 /
4. The radiation component derivates from thehigher derivatives counter-terms while the Induced MatterTensor vanishes.
Cosmologies induced by cylinder solutions are quite in-teresting and allow different dynamics. In particular, thematter-energy sector, characterized by means of the threematter-energy sources of geometrical origin, assumes a va-riety of significant behaviours. A relevant aspect of thesesolutions is that the mixed tensor T Mixµν provides a cos-mological constant-like source. In fact, as it is possible toobserve from the definition (29), in such a case (no extra-coordinate dependence) it is T Mix = T i Mixi , therefore ω Mix = −
1. Since for each solution, ρ Mix = T Mix ∼ X i ( n ) N t − , with X i ( n ) depending on the solution weconsider, we obtain an inverse square law cosmologicalconstant. This behaviour is favored in string cosmologies[67] and in time-varying Λ theories that accommodate alarge Λ for early times and a negligible Λ for late times[68]. On the other side, the effective barotropic factors ofthe other two matter-energy sources depend on the pa-rameter n in relation to the solutions (45a)-(45c). In or-der to have a schematic portrait of these result we haveplotted the behaviour of each ω K with respect to n inFig.1. It is possible to observe that the value of the grav-ity Lagrangian power index n determines the possibilityto have different cosmological components. Both the stan-dard matter sector and the dark energy one can be mim-icked by the geometrically induced fluids according withprevious results in this sense obtained in the frameworkof IMT [58,69,70] and fourth order gravity [22]. Very in-teresting is the coexistence of different regimes with thesame value of n .Together the behaviour of ω K we display, for all thecomponents (see Fig.2), the energy density value ρ K andthe overall sum of these quantities . It is evident that en-ergy conditions are violated for certain values of n . Forexample it is easy to observe that the Weak Energy Con-dition, ρ i ≥
0, for the first solution, is not satisfied by ρ IMT and alternatively by the other two energy densities ρ Curv and ρ Mix . For the second solution we have againno n intervals where all components satisfy this energyprescription, since ρ IMT is quite always well behaved but ρ Curv and ρ Mix are alternatively negative. Finally, for thethird solution we have a small region around n ≈ . In order to achieve these plots we settle the quantity N t = 1.ntonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology? 13 - - Ω Ω Mix Ω Curv Ω IMT - - Ω Ω Mix Ω Curv Ω IMT - - - - - - Ω Ω Mix Ω Curv Ω IMT
Fig. 1.
The behaviour of the barotropic factors related to three matter-energy tensors that arise after the 5-D to 4-D reduction.The three panel refer to the EoS comportment with respect to n along each of the cylinder solutions (45a)-(45c) (from top todown).4 Antonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology? Finally, let us discuss the cosmological evolution of thefour-dimensional sections determined by each of solution(45a)-(45c) and the behaviour of the extra dimension. Fig-ure 3 allow to draw significant informations about this be-haviour describing the dependence of a ( t, y ) and Φ ( t, y )with respect to n in the different cases.In order to display an accelerating dynamics, a power lawscale factor requires that the power index has to be nega-tive or greater than one. In particular, universes expand-ing at an accelerating rate need α > aa = − πG ρ tot + 3 p tot ) . Therefore, barotropic fluids like ones outlined by our model,with ρ tot = P K ρ K and p tot = P K p K , imply that such arelation can be written as¨ aa = − πG X K (1 + 3 ω K ) ρ K , (48)and the condition for accelerating cosmologies becomes Υ = P K (1 + 3 ω K ) ρ K <
0. As one can see in Fig. 4 forour model the regions of the parameters space that ful-fill this requirement exactly determine accelerating expan-sions (the deceleration parameter q = − ¨ aa ˙ a is negative).Let us remember that the observations suggest q ≈ − . q ≈ − . q ≥ – Solution ( α , γ )-(45a);This solution is allowed in the interval < n < .Values of n < n = 1) implies a triv-ial behaviour with vanishing ρ K . For n ∼ . T Mixµν . In such acase ρ IMT , ρ
Curv > ρ Mix <
0, the whole cos-mic mass-energy budget is positively defined as alreadysaid above.Looking to the cosmological dynamics one can noticethat this solution implies a negative power law indexfor the scale factor within the interval < n < ≤ α < < n < . According with Fig.3-4this behaviour indicates accelerated contracting uni-verses in the first interval and standard matter-likedominated cosmologies in the second one. The extra-dimension is always decreasing since is 0 < γ <
1. Itis interesting to notice that the ordinary matter-likeregime is achieved: in the interval 1 < n < .
25 withtwo of the three fluids that behave as sources with anegative defined EoS (the other one is a stiffed fluidwith ω IMT >
1) and in the range 1 . < n < withtwo standard matter-like sources (0 < ω IMT < ω Curv >
1) together the cosmological constant con-tribute. The divergence of the barotropic factor ω Curv for n = 1 .
25 is regulated by the correspondent vanish-ing of ρ Curv . – Solution ( α , γ )-(45b);We have again < n < . For this solution, the scalefactor shows a standard matter rate since is always0 < α < , there is no acceleration ( q ( α , γ ) > ρ IMT is always a wellbehaved thermodynamical fluid with 0 ≤ ω IMT ≤ / − / ≤ ω Curv ≤ /
2. For values n & . n ∼ . ρ i , while ρ Mix is negative, ρ tot is always ≥
0. For n = 1,the two curvature induced sources collapse to a zero-valued energy density, while the induced matter tensorplays the role of a radiation fluid with ω IMT = 1 / < n < and anincreasing evolution for < n < , is static when γ = , . – Solution ( α , γ )-(45c);Admissible values for n ∈ R − { / } . This solutionexplicitly resembles the standard 4-D f ( R )-gravity so-lution given in [38]. The scale factor and the extra-dimension show the same dynamics since α = γ .There are two interesting regions around n ∼ . n ∼ .
6. In the first case we have two standard mattercomponents, dust-like and radiation-like, with ω Curv ≈ , ω IMT ≈ .
26 together a cosmological constant-likeone with ω Mix = −
1. The correspondent scale factor ishowever non accelerating since q is positive. The sec- ntonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology? 15 - - Ρ Ρ TOT Ρ Mix Ρ Curv Ρ IMT - - Ρ Ρ TOT Ρ Mix Ρ Curv Ρ IMT - - - - - Ρ Ρ TOT Ρ Mix Ρ Curv Ρ IMT
Fig. 2.
Energy densities for the different species of matter-energy obtained in a 5D f(R)-gravity model when reduced to thestandard 4-D spacetime. Curves labelled with ρ tot describe the behaviour of the whole matter-energy cosmological budget;natural units are adopted. The upper panel refers to solution (45a), the middle one to (45b), finally the last one to solution(45c).6 Antonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology? - - - - Α - - - - Α Α Α Α - - - - Γ - - - Γ Γ Γ Γ Fig. 3.
The scale factor and the extra coordinate power law indexes vs. n for the three cylinder solutions obtained along the5-D to 4-D reduction of higher order gravity power law models. Both plots contain a detail that enlightens the behaviour in theregion n ≈ ond region shows a dust-like source with ω Curv ≈ ω IMT ≈ − .
6. In such a case the 4-D section shows an accelerating expansion since q < q = − . n ≈ .
69) as well as an increasing extra-dimension. The zeros of ω IMT (dust-like behaviour ofthis source) are trivial since the correspondent valueof the energy density is vanishing.Actually, this solution allows a plethora of differentbehaviours. In particular, expanding solutions with noacceleration are obtained for − ≤ n < and 1 < n ≤ with two standard matter-like sources plus the effec-tive cosmological constant contribute given by T Mixµν ;on the other side accelerated expansions are given bymodels with n < − and < n < . In these in- tervals T Curvµν plays the role of a baryonic componentand the other two matter-energy sources act as darkenergy-like sources. The energy densities have, again,both negative and positive values with ρ tot ≥
0. For n > . ρ K with ω K ∼ −
1, insuch a case the three induced components behave allas cosmological constant-like sources.As a matter of fact, fourth order gravity provide, in thecylindricity case, a completely different framework withrespect to the standard Induced-Matter Theory. The stan-dard scheme implies, in the 4-D limit, that induced matterbehaves as a radiation-like fluid ( p = ρ/
3) [19] accord-ing with the result obtained with an initial condition ofcylindricity by Kaluza [32]. In the case of higher orderKaluza-Klein models the energy-momentum tensors en- ntonio Troisi: Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology? 17 - - - - - - q H Α , Γ L q H Α , Γ L q H Α , Γ L - - - - - U U H Α , Γ L U H Α , Γ L U H Α , Γ L Fig. 4.
The deceleration parameter and the Υ term, which enters the scale factor acceleration equation, vs. n. Again the threeplot are related to the cylinder solutions (45a)-(45c). tering the 4-D induced field equations can describe moregeneral matter sources. Therefore one can obtain dynam-ically significant cosmologies also in presence of a Killingsymmetry for the extra-coordinate. Of course, the signifi-cance of such solutions have to be corroborated with cos-mological and astrophysical data. We discussed the possibility to develop a higher orderand higher dimensional gravity model. Within such anapproach, after the 5-D to 4-D reduction procedure, ithas been possible to suggest a new interpretive schemefor the 4-D cosmological phenomenology. Specifically, wehave considered a vacuum 5-D f ( R )-gravity model. The 4-D effective reduction provides, in this case, a GR-like cos- mology characterized by means of three induced matter-energy tensors of geometrical origin. Within this mattergeometrization paradigm the cosmological fluid sourcescan be achieved considering: the standard Induced MatterTensor of non-compactified KK theory T IMTµν , the Curva-ture Quintessence tensor T Curvµν coming from higher or-der derivative terms of f ( R )-gravity and T Mixµν a mixedquantity that arises from both higher dimensional andhigher order curvature counter-terms. This last quantityembodies the 5-D to 4-D reduction of the two combinedapproaches. We have obtained the complete field equa-tions formalism and in order to check model predictionwe have worked out a routine to search for power law so-lutions in the case f ( R ) = f R n . The whole scheme pro-vides the right KK-GR limit once f ( R ) → R . In such acase, on shell, radiation dominated cosmologies ( p = 1 / ρ ) are obtained if no dependence on the extra-coordinateis taken into account. More generally, the KK-GR limitfurnishes solutions that agree with similar results alreadypresent in literature. In particular, the curvature inducedmatter-energy tensors vanish while the Induced MatterTensor, that collects all metric terms depending on extra-coordinate, turns out to be the only one source of 4-D fieldequation.Relaxing the hypotheses on n , i.e. by assuming a genericpower law fourth order gravity model, it is possible tosearch for more general solutions. As a first preliminarystep, cylindricity has been taken into account. A more gen-eral study has been left to a dedicated forthcoming publi-cation. Differently from the standard KK-GR theory, thecombination of fourth order gravity with higher dimen-sions implies, in such a case, non trivial solutions that cangive rise to interesting cosmologies. We have obtained aset of solutions, parameterized by the f ( R ) power index n , that determine a variety of possible barotropic fluidsin the 4-D mass-energy sector. In relation to the valueof n one can have both standard matter components anddark energy-like ones. The cylinder condition implies that T Mixµν , according to its definition, behaves as an effectivecosmological constant-like term with ω Mix = −
1. Actu-ally, in such a case, the energy density ρ Mix depends ontime with an inverse square law. This fact suggests theintriguing possibility to have a time varying cosmologicalconstant. The cosmological behaviour of the 4-D sectionsshows that it is possible to have both accelerating and nonaccelerating expansions. In particular, non accelerated ex-pansion can be obtained also in presence of subdominantdark-energy like components. On the other side, it canbe easily framed a dark energy dominated speeding upuniverse generalizing results obtained in the standard 4-D f ( R )-gravity. The interesting aspect of this last solutionsis that geometry can provide both standard matter-likesources and dark-energy-like ones, one of this can mimica phantom fluid. The extra-dimension follows an indepen-dent evolution, that, in the third solution, is equivalentto one of the scale factor. Therefore, in such a case, ex-panding accelerated evolutions are characterized also by agrowing fifth coordinate.The price to pay for geometrizing matter-energy sourcesis that cosmological energy densities can assume negativevalues violating the WEC. We have shown that althoughthis is true for some values of n , also in interesting regionsof the parameters space, the whole mass-energy budget ρ tot is always positive definite and, therefore, total top-down geometrized 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 cosmologi-cal observations. In particular, this achievement seems todeserve some more studies in the direction of dark en-ergy and dark matter interpretation. Of course, more in-sights are necessary in order to evaluate such a theoreticalscheme. In particular, the matter sector (i.e. matter-likecomponents), which has been satisfactory checked within standard IMT [69,75], require a careful investigation sincematter properties should be completely induced by grav-ity. In addition, model predictions have also to be checkedalso against cosmological and astrophysical data.Furthermore, if f ( R )-Kaluza-Klein gravity determine,in the 5-D to 4-D reduction of cylinder solutions, cos-mological models with a suitable top-down matter ge-ometrization, in principle, more general solutions couldallow to develop cosmological models with a wider phe-nomenology. On the other side, the fifth dimension role re-quires a careful analysis when the extra-coordinate Killingsymmetry is discarded. In fact, if the cylinder condition isrelaxed one naturally gets into the game of a length scalerelated with the extra-coordinate. In this sense, it has beenshown that 5-D f ( R ) gravity suggests that, eventually,there is a strict interconnection between the fifth dimen-sion dynamics and the higher derivative counter-terms. Acknowledgments : The author is thankful with Prof.S. De Pasquale and Prof. S. Pace for their help and sup-port.
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