Resembling dark energy and modified gravity with Finsler-Randers cosmology
Spyros Basilakos, Alexandros P. Kouretsis, Emmanuel N. Saridakis, Panayiotis Stavrinos
aa r X i v : . [ g r- q c ] D ec Resembling dark energy and modified gravity with Finsler-Randers cosmology
S. Basilakos, ∗ A.P.Kouretsis, † Emmanuel N. Saridakis,
3, 4, ‡ and P.C. Stavrinos § Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efesiou 4, 11527, Athens, Greece Section of Astrophysics, Astronomy and Mechanics,Department of Physics Aristotle University of Thessaloniki, Thessaloniki 54124, Greece Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece Instituto de F´ısica, Pontificia Universidad de Cat´olica de Valpara´ıso, Casilla 4950, Valpara´ıso, Chile Department of Mathematics, University of Athens, Athens 15784, Greece
In this article we present the cosmological equivalence between the relativistic Finsler-Randerscosmology, with dark energy and modified gravity constructions, at the background level. Start-ing from a small deviation from the quadraticity of the Riemannian geometry, through which thelocal structure of General Relativity is modified and the curvature theory is extended, we extractthe modified Friedmann equation. The corresponding extended Finsler-Randers cosmology is veryinteresting, and it can mimic dark-energy and modified gravity, describing a large class of scale-factor evolutions, from inflation to late-time acceleration, including the phantom regime. In thisrespect, the non-trivial universe evolution is not attributed to a new scalar field, or to gravitationalmodification, but it arises from the modification of the geometry itself.
PACS numbers: 98.80.-k, 95.36.+x, 04.50.Kd
I. INTRODUCTION
Since the discovery of the accelerated expansion of theUniverse (see [1] and references therein) a lot of efforthas been made in order to understand the physical mech-anism which is responsible for such a cosmological phe-nomenon. There are two basic directions one can followin order to obtain its explanation. The first is to intro-duce the concept of dark energy (hereafter DE) withinthe framework of General Relativity (for reviews see forinstance [2]), while the second is to modify the gravita-tional sector itself (see [3] and references therein).From the DE viewpoint the simplest way to fit thecurrent cosmological data is to include in the Friedmannequations the cosmological constant [1]. However, thedisadvantage of the so-called concordance Λ-cosmologyis the fact that it suffers from the cosmological constantproblem itself [4]. This intrinsic problem appears as adifficult issue which includes many aspects: not only theproblem of understanding the tiny current value of thevacuum energy density ( ρ Λ = c Λ / πG ≃ − GeV )[4] in the context of quantum field theory or string theory,but also the cosmic coincidence problem, namely why thedensity of matter is now so close to the vacuum density[5]. Unfortunately, the alternative and more complex DEscenarios, for instance quintessence [6–8], phantom [9],quintom [10] etc, are not free from similar fine-tuning andother no less severe problems (including the presence ofextremely tiny masses and peculiar forms of the scalarfield kinetic energy). ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: Emmanuel [email protected] § Electronic address: [email protected]
The above problems have inspired many authors toproceed to the alternative direction of modified gravity,such as the braneworld Dvali, Gabadadze and Porrati [11]model, f ( R ) gravity [12], f ( T ) gravity [13, 14], scalar-tensor theories [15], Gauss-Bonnet gravity [16], Hoˇrava-Lifshitz gravity [17], nonlinear massive gravity [18] etc.The underlying idea is that the accelerated expansion,either during inflation or at late times, can be driven bya modification of the Einstein-Hilbert action, while thematter content of the universe remains the same (rela-tivistic and cold dark matter). However, the majorityof modified gravity models are plagued with no physicalbasis and/or many parameters.On the other hand, the last decade the Finslerian rel-ativistic extensions have gained a lot of attention, sinceFinsler geometry naturally extends the traditional Rie-mannian geometry [19]. In this formulation, in generalone starts with the Lorentz symmetry breaking, which isa common feature within quantum gravity phenomenol-ogy. Such a departure from relativistic symmetries ofspace-time, leads to the possibility for the underlyingphysical manifold to have a broader geometric struc-ture than the simple pseudo-Riemann geometry. In theselines, Finsler geometry is the simplest class of extensions,since it generalizes Riemann geometry. Note that theRiemannian geometry itself is a special type of the Fins-lerian one.One of the most characteristic features of Finsler ge-ometry is the dependence of the metric tensor to theposition coordinates of the base-manifold and to the tan-gent vector of a geodesic congruence, and this velocity-dependence reflects the Lorentz-violating character of thekinematics. Additionally, Finsler geometry is stronglyconnected to the effective geometry within anisotropicmedia [20] and naturally enters the analogue gravity pro-gram [21]. These features suggest that Finsler geome-try may play an important role within quantum gravityphysics.From the cosmological viewpoint, in a series of works[22, 23] it was reported that in the osculating Riemannianlimit the cosmic expansion of the flat Finsler-Randers(hereafter FR) gravity is identical to that of flat DGP,despite the fact that the geometrical origin of the twocosmological models is completely different. The lattermeans that the flat FR model inherits all the advan-tages and disadvantages of the flat DGP gravitationalconstruction. However, the fact that DGP gravity is un-der observational pressure [24] implies that the flat FRmodel faces the same problems [23].Therefore, in the present work we are interested in ex-tending the results of [22] and [23] in order to derivean extended version of the FR model (hereafter EFR),free from the observational inconsistencies. To achievethat, instead of the osculating Riemannian limiting pro-cesses [22], which is a metric-based approach, we use thecovariant 1+3 formalism [25], that under certain condi-tions can be naturally extended in the Finslerian frame-work [26]. In this less restrictive case, we can mimic allnon-interacting DE models and the majority of modifiedgravitational constructions, and we are able to describea large class of cosmological evolutions.The plan of the work is as follows. In Sec. II wepresent the metrical extension of Riemannian geometry,and we discuss the evolution of the kinematical variablesand the Finsler-Randers geometrical structure. In Sec.III we focus on the isotropic expansion and we developthe cosmological model. We prove the equivalence be-tween the EFR and DE, as well as with some classes ofmodified gravity, at the expansion level, and we discusssome particular examples. Finally, in Sec. VI we drawour conclusions. II. RELATIVISTIC FINSLER GEOMETRY
Recently, there is an increasing interest in Finsler ge-ometry since it has been reported within different as-pects of quantum gravity. The effective metric dependseither on velocity-like variables or on the tangent vec-tor field of the observers’ cosmic lines. A representa-tive example of the first case is the stochastic space-timeD-foam where the effective metric depends on the ve-locity of D-particles that recoil on the world-sheet [27].Another scenario where Finsler geometry emerges, andthe metric depends on fiber coordinates, is the covariantGalilean transformations in curved space-times [28]. Onthe other hand, dependence of the metric on the parti-cle’s 4-velocity arises in other Lorentz-violating theories,such as the Hoˇrava-Lifshitz gravity [29]. Additionally,a Finslerian line-element has been encountered in defor-mations of Cohen and Glashow’s very special relativity[30], as well as in holographic fluids [31]. Moreover, bi-metric constructions can be naturally incorporated in theFinsler framework [32]. Finally, we mention that Finslergeometry can be closely related to the standard-model extension [33]. Before proceeding to the cosmological ap-plication of Finsler geometry, in the following subsectionswe briefly present its basic features.
A. Finsler congruences
The main object in Finsler geometry is the fundamen-tal function F ( x, dx ) that generalizes the Riemannian no-tion of distance (see for example [34–36]). In Riemanngeometry the latter is a quadratic function with respectto the infinitesimal increments dx a between two neigh-boring points. Keeping all the postulates of Riemann ge-ometry but accepting a non-quadratic distance measure,a metric tensor can be introduced as g ab ( x, y ) = 12 ∂ F ∂y a ∂y b , y a = 0 , (1)for a given connecting curve with tangent y a = dx a dτ . Notethat when the generating function F ( x, y ) is quadratic,the above definition is still valid and leads to the met-ric tensor of Riemann geometry. The dependence of themetric tensor to the position coordinates x a and to thefiber coordinates y a suggests that the geometry of Finslerspaces is a geometry on the tangent bundle T M . In otherwords, the Finsler manifold is a fiber space where tensorfields depend on the position and on the infinitesimal co-ordinate increments y a . Therefore, the position depen-dence of Riemann geometry is replaced by the so called element of support , which is the pair ( x a , y a ).In relativistic applications of Finsler geometry the roleof the supporting direction y a must be explicitly given.For example, it may stand as an internal variable, as anexplicit or implicit violation of Lorentz symmetry, as anaether-like direction or simply as the velocity of the fun-damental observer. In this article we restrain our analysisto the latter case, where the supporting direction y a isthe tangent to the cosmic flow lines. Using only varia-tional arguments we can arrive to the deviation equationfor the supporting congruence y a . The deviation equa-tion directly provides all the information for the internaldeformation of the time-like geodesic flow y a . Followingthe same procedure with GR, we can extract the propa-gation formulas for the expansion, shear and vorticity ofan infinitesimal cross-section of the cosmological flow.The infinitesimal distance between two neighboringpoints on the base manifold (position space) is given bythe small displacement along the connecting curve γ ( τ ),that depends on the position x a and on the coordinateincrements dx a : dτ = F ( x, dx ) , (2)where in Finsler geometry F ( x, dx ) does not necessarilydepend quadratically on the dx a increments. The actualdistance traveled on the base manifold along a given di-rection is I = Z F ( x, dx/dτ ) dτ, (3)where the metric function F ( x, y ) is homogeneous offirst order with respect to the displacement arguments y a = dx a /dτ . Applying the least action principle on theprevious integral we arrive to the geodesic equation forthe supporting direction˙ y a ≡ y a ∇ a y b = dy a dτ + 2 G a ( x, y ) = 0 , (4)where G a are the spray coefficients with respect to the F ( x, y ) fundamental function and they are given by G a = 14 g ab (cid:18) ∂ F ∂x c ∂y b y c − ∂F ∂x b (cid:19) . (5)When y a stands for the velocity of the fundamental ob-server then the dot operator in relation (4) is a directgeneralization of the time-propagation of GR relativis-tic kinematics. In other words, if the supporting di-rection y a = dx a dτ is the observers’ 4-velocity then theaffine parameter τ is the proper time. Note, that we canrecast relation (4) in the familiar form with respect tothe Christofell symbols, with the only difference that themetric will depend on the supporting element.The relative acceleration between two neighboring ob-servers is given by the second variation of the distancemodule. Focusing the analysis along the y a direction, thesecond variation leads to the Jacobi equation. The rela-tive acceleration between nearby geodesics is monitoredby an infinitesimal connecting vector (the deviation vec-tor) defined as ˜ x a = x a + ξ a , (6)where the tilde stands for the neighboring referenceframe. Then, substituting the previous expression to theEuler-Lagrange equations (4), and keeping up to first-order terms with respect to the deviation vector ξ a , leadsto the following formula¨ ξ a + H ab ( x, y ) ξ b = 0 , (7)where H ab is a tensor field that incorporates the relativedisplacement of nearby geodesics in a Finslerian frame-work, given by H ab = 2 ∂G a ∂x b − y c ∂ G a ∂y b ∂x c + 2 G c ∂ G a ∂y b ∂y c − ∂G a ∂y c ∂G c ∂y b . (8)The first order homogeneity of the metric function leadsto the constraint ∂g ab ∂y c y c = 0. The latter guarantees thatfor most connection structures (for example Chern, Car-tan or Berwald) the Jacobi field (7) remains the same(see for example [34–37]).The coefficients of the tensor field H ab are directly de-termined by the metric function F ( x, y ) through the leastaction principle that gives back the spray coefficients (5).As in Riemann geometry, expression (8) is second orderhomogeneous with respect to y a , but the dependence is non-quadratic. It’s eigenvalues correspond to the sec-tional curvatures in the principal directions and desig-nate the relative motion between neighboring integralcurves. Relation (8) encloses all the relevant informationfor Finslerian tidal effects on the y a congruence. The ten-sor field H ab is responsible for the relative accelerationbetween nearby observers and will generate expansionand shear on the time-like y a -congruence. Apparently,the y a -deformable kinematics will be modified due to thenon-quadratic dependence of H ab on the velocity of thefundamental observer. B. Deformable kinematics
The observers’ time-like congruence introduces a uni-direction in the physical manifold. This asymmetry isencoded in the metric function F ( x, y ) and induces the1+3 “threading” of space-time [25]. In the covariant 1+3formalism the metric is not the central object, since we donot use a particular coordinate system. Instead, we usethe kinematic quantities, the irreducible components ofcurvature and conservation arguments, while Einstein’sfield equations enter as simple algebraic relations betweencurvature and matter [25]. The deviation of geodesicsis of central importance since it monitors the internaldeformation of the cosmic medium in a covariant way.From the Finslerian perspective the space and timedecomposition is directly related to the first-order homo-geneity of F ( x, y ). In particular, the fundamental ob-server’s velocity y a defines a family of integral curves onthe space-time manifold. With respect to this 4-velocitywe can decompose tensor fields along y a and on the per-pendicular spatial hyper-surface. In fact, we can recastthe metric tensor (1) in the following form g ab = F ∂ F∂y a ∂y b + ∂F∂y a ∂F∂y b , (9)where we have split the space-time metric in two partsby using the quantities l a = ∂F∂y a , h ab = F ∂ F∂y a ∂y b . (10)Using the first order homogeneity of the metric function F ( x, y ) we can prove that l a is the normalized velocityof the observers’ flow-lines, l a = y a /F . In addition, thefirst order homogeneity of the fundamental function im-plies that h ab l b = 0 and also that the rank is ( h ab ) = 3.Therefore, the tensor h ab stands for the projection tensorof relativistic kinematics.With the space-time split (9) in hand we can decom-pose tensor fields to their irreducible parts, in direct anal-ogy to the standard gravitational physics, for example X a = g ba X b = ( h ba + l b l a ) X b = Xl a + X a , (11)where X = l a X a is the time-like part and X a = h ba X b is the space-like part. Using the 1 + 3 covariant formal-ism we will track the internal motion of the normalizedsupporting direction l a (the congruence l a is consideredto be time-like, l a l a = 1 [38]). Restraining the analysisalong the l -time-like flow, the propagation equation ofthe deviation vector at first order is given by˙ ξ a = B ab ξ b , (12)where it is straightforward to prove that B ab = ∇ b l a ,that is the tensor field B ab is the distortion tensor ofthe time-like congruence. Following relation (10), we candecompose the distortion tensor to it’s irreducible parts ∇ b l a = 13 Θ h ab + σ ab + ǫ abc ω c , (13)where for the 3D spatial derivative D a = h ba ∇ b theirreducible components are: the expansion Θ = D a l a that tracks volume changes, the shear σ ab = D h b l a i that incorporates shape distortions, and the vorticity ω a = ǫ abc D b l c / l a .Taking the time derivative of relation (12) and substi-tuting in relation (7) we arrive to the evolution equationfor the internal deformations of the time-like flow:˙ B ab + B ac B cb = −H ab . (14)This propagation law reflects the effect of the Finsleriancurvature tensor H ab on the deformable kinematics ofa time-like flow. The irreducible parts of relation (14)provide the evolution equation for the expansion (Ray-chaudhuri’s equation)˙Θ + 13 Θ = −K − σ − ω ) , (15)the propagation of shear (which describes kinematicanisotropies)˙ σ h ab i = −
23 Θ σ ab − σ c h a σ cb i − ω h a ω b i − H h ab i , (16)and finally the propagation of vorticity˙ ω a = −
23 Θ ω a + σ ab ω b , (17)where K = H ab h ab is the scalar flag curvature of theFinslerian manifold when H h ab i = 0 [26, 34].The above system of propagation equations is very sim-ilar to the analogous expressions in General Relativity.In particular, they are the same except that in the Rie-mannian limit the tensor (8) depends quadratically onthe observers’ 4-velocity. Thus, the key difference is thenon-trivial dependence of the curvature tensor H ab to thevelocity of the fundamental observer, which modifies theway that curvature generates deformations on a time-likemedium. Angle brackets stand for the projective, symmetric and trace-freepart of a second rank tensor X h ab i = h c ( a h db ) X cd − X cd h cd h ab . C. Finsler-Randers metric function
In Finsler geometry the form of the metric function F ( x, y ) is of central importance since it generates all theother geometric quantities. One of the most simple casesafter the Riemann limit is the Randers norm [39], whichis given by F = α + β, (18)where α = p α ab y a y b is a Riemann metric function and β = b a y a stands for an arbitrary 1-form. The funda-mental function (18) interfaces a Riemann space-timewith a Finslerian one in a simple way, since the Ran-ders metric is the limiting case of a large number ofFinsler space-times, when we consider small departuresfrom GR. For example, in a large class of ( α, β )-metricswhere F = αφ ( β/α ), the almost Riemannian limit φ ∼ β/α ,when φ ∼ β/α .Relation (18) has the important consequence that onecan separate geometric quantities to the purely Riemannpart with respect to the α metric function, and to theFinsler contribution. In this case, for specific examples,we can directly inspect the effect of non-quadraticity onthe space-time medium.In particular, the geometric entity that accurately in-corporates the non-quadraticity of the metric function isthe indicatrix F ( x, y ) = 1, which represents an arbitrarylocus on the tangent bundle [36]. This locus in the Rie-mann case defines a quadratic hyper-surface. In case ofa Randers type geometry (18) the hyper-surface is stillquadratic but becomes eccentric [39]. In other words, theRanders metric function (18) assigns at each space-timepoint a vector b a that describes the displacement of thecenter of the indicatrix. This property translates to adisformal correlation between the Finslerian metric ten-sor (1) and the Riemannian a ab given in relation (18). Infact, substitution of relation (18) into (1) yields g ab = Fα ( α ab − ¯ l a ¯ l b ) + l a l b , (19)where ¯ l a = ∂α∂y a is the normalized velocity on the Riemannsector. The disformal relation (19) introduces an explicitdependence of the space-time metric on the velocity ofthe cosmic flow lines. Note, that similar behavior of theeffective geometry is commonly reported in investigationsof anisotropic media [20].Concerning the signature of the Finsler-Randers space-time it is useful to introduce the non-holonomic frame on T M , namely Y ba = r αF (cid:18) δ ba + r αF l a l b − ¯ l a ¯ l b (cid:19) (20)and it is straightforward to prove the identity Y ca Y bc = δ ba . Then, using relation (20) we can recast the Finsle-rian metric tensor to the following form [40] g ab = Y ca Y db α cd . (21)Thus, from the definition (20) and the above relationwe conclude that the Finsler-Randers metric tensor andthe Riemann metric α ab have the same signature. Tak-ing into account that the metric tensor (19) must bereal, we deduce that the time-like y a -congruence is posi-tive definite or equivalently the signature is (+ , − , − , − ).This restrains the y a -bundle of geodesics to be time-like [32], however we can define first-order Finsleriantensor fields as space-like U ( x, y ) a U ( x, y ) a <
0, null U ( x, y ) a U ( x, y ) a = 0, and time-like U ( x, y ) a U ( x, y ) a > α, β )-metric and we as-sume that the velocity of the fundamental observer isgiven by the normalized vector l a = y a /F . Then, if b a is a closed form with respect to the Riemann covariantderivative of the α -metric, b [ a ; b ] = 0, the spray coefficients(5) for the normalized velocity l a take the simplified form[34] G a = ¯ G a + 12 Φ l a , (22)where bars denote the Riemann parts with respect to the α -metric, and we define Φ = b a ; b l a l b . Substituting rela-tion (22) into the covariant expression (8), the curvaturetensor takes the simplified form H ab = ¯ H ab + 14 (cid:0) − (cid:1) h ab , (23)where the last two scalars are given with respect to theRiemann covariant derivative of the α -metric:Ψ = b a ; b ; c l a l b l c , (24)and we have defined ¯ H ab = F − ¯ R acbd y c y d for the part ofthe curvature coming from the Riemann metric function α of relation (18). The second rank tensor (23) clearlyincorporates the relation between a part of the Riemanncurvature and a part the actual curvature of the fore-ground manifold. As we have already mentioned, thelatter curvature generates deformations in the assumed“physical” space-time which is of Finsler type, while theRiemann curvature represents the gravitational sector.Note that Ψ, that is the nature of the Finsler-Randerscontribution to the curvature, is defined on a geometri-cal basis, since it directly originates from the curvatureof the Finsler-Randers geometry (23). Therefore, fromrelation (23) and the deformable kinematics given in re-lations (15)-(17), we conclude that the Riemann curva-ture of the gravitational sector generates deformations inthe foreground space-time in a modified way. III. FINSLER-RANDERS COSMOLOGY
In this section we investigate the conditions underwhich the Finsler-Randers cosmology can provide a cos-mic acceleration equivalent to the traditional scalar fieldDE or classes of modified gravity. We assume that the“physical” geometry is represented by the non-quadraticmetric function (18), while the gravitational geometryis given by its’ Riemannian part. Thus, in a FRW-likescenario the Riemann curvature ¯ R abcd is related to theenergy-momentum tensor of a perfect fluid through theEinstein’s field equations¯ R ab −
12 ¯ Rα ab = T ab = ρ ¯ l a ¯ l b + p ¯ h ab , (25)where we define the projection tensor of the Riemanniansector as ¯ h ab ≡ α ab − ¯ l a ¯ l b , the overall energy densityas measured in the ¯ l a frame as ρ = T ab ¯ l a ¯ l b , the totalisotropic pressure as p = T ab ¯ h ab / πG ≡
1. Furthermore,for our setup it is natural to assume a homogeneous andisotropic Riemannian sector for the gravitational geom-etry. Hence, we can neglect the non-local gravitationaldegrees of freedom and the Weyl curvature becomes neg-ligible. In this case the Riemann curvature depends onlyon its’ local parts¯ R abcd = 12 ( α ac ¯ R bd + α bd ¯ R ac − α bc ¯ R ad − α ad ¯ R bc ) −
16 ¯ R ( α ac α bd − α ad α bc ) . (26)We consider shear and vorticity free evolution forthe cosmic fluid, which is in agreement with the tightconstrains of the cosmic microwave background (CMB)anisotropies (see for instance [43]). In fact, using rela-tions (23),(25) and (26) we obtain that H h ab i ∼ b h a b b i ,and the source term in the propagation of shear (16) isnegligible if b a tends to be purely time-like. Thus, ourkinematical setup is consistent with a shear and vorticityfree bulk flow, since there are no source terms in relations(16) and (17). Then, keeping up to first-order terms withrespect to b a in (23), and using the field equations (25)and the decomposition (26), Raychaudhuri’s formula (15)acquires the simplified form˙Θ + 13 Θ = −
12 (1 − β )( ρ + 3 p ) −
32 Ψ , (27)where we have used the auxiliary relation ¯ H ab h ab = (1 − β )( ρ + 3 p ). The Raychaudhuri’s equation (27) isthe fundamental equation that describes the cosmologi-cal evolution. The crucial point is the sign of its righthand side. In particular, negative terms align with thegravitational pull, while positive terms accelerate the ex-pansion. The Finsler contribution in the first term ofthe rhs of (27) acts as an effective coupling constant. Asan example, if we neglect this term then the significantterm that incorporates the effects of non-quadraticity isthe last one, and when Ψ < > SO (4) symme-try is broken, and hence it implies new kinematic effectsfor the bulk flow of matter, by modifying the curvaturetheory. Apparently, by discarding the local flatness ofGeneral Relativity we acquire long-range modificationsin the Finslerian geometrodynamics.Let us discuss here the energy conservation in the sce-nario at hand. The energy density and the isotropic pres-sure as measured in the Riemannian frame ¯ l a are relatedto the “physical” frame l a by the following relations ρ = F α ρ ( f ) , p = Fα p ( f ) , (28)where we have defined ρ ( f ) = T ab l a l b and p ( f ) = T ab h ab / F/α = 1 + β/α ∼
1, we obtain that at first or-der in the two frames the energy density and pressure arethe same, namely ρ ∼ ρ ( f ) and p ∼ p ( f ) 2 . At early timesour first-order approximation is no-longer valid since thetwo frames will start to diverge, having a direct impact onthe effective equation of state. Additionally, in the pres-ence of pressure the spatial part of the energy-momentumconservation h ca ∇ b T cb = 0 yields( ρ + p ) l b ∇ b l a = − D a p. (29)However, by construction the l a -congruence is geodesic(4) and the previous relation implies that D a p = 0. Thelatter condition is valid in an isotropic and homogeneousbackground, but considering cosmological perturbationsnon-geodesic congruences will be involved in the calcu-lations (for an 1+3 treatment of non-geodesic flows see[26]). Hence, our model is consistent at late times of thecosmological history (for example at dust and radiationdominated eras) and for vanishing gradients of pressure.On the other hand, taking the time-like part of theenergy momentum conservation, l a ∇ b T ab = 0, and de-composing it to the irreducible parts with respect to the l a -congruence, we obtain˙ ρ = − Θ( ρ + p ) (30)for the total energy density. Here we mention that theabove relation is valid for the first-order approximation, In relativistic cosmology a similar limiting process between rela-tive frames is used to study “peculiar” frames and the Zeldovichapproximation (see for example [44]). where the energy density is almost the same in the Re-mannian frame ¯ l a and in the Finslerian one l a that rep-resents the bulk flow of matter. Introducing the charac-teristic length scale a ( scale factor ) of the spatial volumeby dV ∝ a , we extract that for the expansion we haveΘ = ( dV )˙ /dV = 3 ˙ a/a . Using this expression we can re-cast Raychaudhuri’s formula (27) in terms of the scalefactor for late times of the cosmological evolution as3 ¨ aa = −
12 ( ρ + 3 p ) −
32 Ψ , (31)where the total matter fluid itself is in general a mix-ture of relativistic matter (i.e. radiation, ρ r with p r = ρ r / ρ m with p m = 0) components, implying ρ = ρ m + ρ r and p = p m + p r = ρ r /
3. Now using the continuity equation(30) together with the Raychaudhuri’s formula (31), weretrieve the modified Friedmann equation: H ( a ) = 13 ρ − a − Z a Ψ( a ) da − C a . (32)In the above expression C is an integration constant,which in the FRW limit coincides with the spatial curva-ture, and thus without loss of generality in the followingwe set it to zero. For the rest of our analysis we focuson the matter dominated era (well after radiation-matterequality) in which the radiation component is considerednegligible and thus we use ρ ≡ ρ m .Equation (32) incorporates the effects of Finsler-Randers geometry in the expansion of the universe. Weremind that Ψ( a ), which is the nature of the Finsler-Randers contribution to the curvature, is defined on a ge-ometrical basis, since it directly originates from the cur-vature of the Finsler-Randers geometry (23). Since fromfirst principles the evolution of Ψ remains unconstrained(this could be achieved by relating the Finsler structureto a particular Quantum Gravity scenario, which lies be-yond the scope of the present work) any Ψ( a ) profile ispossible. Thus, from (32) on can deduce that a largeclass of scale-factor evolution can be realized within thecontext of Finsler-Randers geometry.Let us examine the condition of a local small departurefrom quadraticity, in relation to the accelerated cosmo-logical expansion. In a first approach we may writeΨ = b a ; b ; c l a l b l c ∼ βλ F , (33)where λ F is a characteristic length scale related to thevariation of β . A small value of β corresponds to a sortlength scale of the modification. Using the approxima-tion R a u Ψ( u ) du ∼ Ψ a , together with (33), the Fried-mann equation (32) for an accelerated phase leads to theapproximate relation | β | ∼ (cid:18) λ F λ H (cid:19) , (34)where λ H = H − is the Hubble horizon. The aboverelation is a rough estimation of the β -parameter, withrespect to the Hubble horizon, in order to obtain an ac-celerated expansion. Thus, the length λ F represents acharacteristic scale above which the gravitational physicsis affected. The condition | β | ≪ λ F is some orders of magnitude bellow the Hubble hori-zon. For example, if we assume that the gravitationalsector is modified above galactic scales (kpc) and tak-ing into account that λ H ∼ pc, relation (34) leadsroughly to | β | ∼ − . Hence, interestingly enough,even small departures from Lorentz invariance can leadthe cosmic flow to the accelerated phase.Let us make a comment here on the Lorentz invari-ance violation. The effective geometric formulation ofthe present work stands for the geometry of space-timeas measured by the comoving observers of the self gravi-tating cosmic medium. Thus, the β variable parameter-izes possible departures from Lorentz invariance in thegravitational sector. The most stringent constraints ofLorentz violation in the gravitational sector arise fromparametrized post-Newtonian (PPN) analysis using solarsystem data [45], and the most recent results from Grav-ity Probe B put an upper bound at 10 − [46]. There-fore, the above representative example lies far inside thiswindow. Note that during the last years, the PPN anal-ysis in Finsler geometry has been developed in Ref.[47],however to the best of our knowledge the metric func-tions that have been used are not of Randers type. Fur-thermore, the study of Lorentz violation in the gravitysector involves possible future detection of gravitationalwaves and possible Lorentz violation corrections (see forexample [48]), constraints on the inverse square law andgravitomagnetic effects, CMB anisotropies and black holephysics [49]. This detailed analysis eventually will alsoconstrain the ’Finslerity’ of the gravitational sector butlies beyond the scope of this work. A. Analogue to dark energy and modified gravity
In this subsection we show that the above Finsler-Randers-modified Friedmann equation (32), can mimicany dark energy scenario, through a specific reconstruc-tion of the Ψ( a ), that is of the Finsler-Randers contribu-tion to the curvature. For this shake we writeΨ( a ) = Ψ X ( a ) / , Ψ < H ( a ) = H E EF R ( a ) , (36)and using also that ρ m = ρ m a − the Friedmann equa-tion (32) writes as E EF R ( a ) = Ω m a − + Ω Ψ a − Q ( a ) . (37)In this expression we have defined Q ( a ) = Z a uX ( u ) du, (38) while the density parameters read as Ω m = ρ m / H and Ω Ψ = − Ψ / H , with Ω m +Ω Ψ = 1. We mentionthat for mathematical convenience Q ( a ) is normalized tounity at the present time.Now we can return to the aforementioned basic ques-tion: Under which circumstances equation (37) can re-semble that of dark energy ? In order to address this cru-cial question we need to calculate the effective equation-of-state parameter (hereafter EoS) w ( a ) for the EFR cos-mology introduced above. We proceed as though wewould not know that the original Hubble function is theone given by equation (37) and we assume that it behavesaccording to the typical expansion rate of the universewhere the DE is caused by a scalar field with negativepressure, namely P D = w ( a ) ρ D ( a ). Therefore, for homo-geneous and isotropic cosmologies, driven by non rela-tivistic matter and a scalar field DE, the first Friedmannequation is given by E DE ( a ) = (cid:2) Ω m a − + Ω DE f ( a ) (cid:3) (39)with f ( a ) = exp (cid:26) − Z a (cid:20) w ( u ) u (cid:21) d u (cid:27) , (40)where Ω DE = ρ DE / H is the DE density parameterat present time, which obeys Ω m + Ω DE = 1.The next step is to require the equality of the expan-sion rates of the original EFR picture (37) and that ofthe DE picture (39), namely E RF ( a ) = E DE ( a ) for everyscale factor, and doing so we extract the integral equation Q ( a ) = a f ( a ) . (41)Differentiating the above equation, and using (38) and(40), we obtain the function X ( a ) [and thus Ψ( a )] interms of the EoS parameter w ( a ), as X ( a ) = − [1 + 3 w ( a )] f ( a ) . (42)In this viewpoint, if we know apriori the effective EoSparameter then we can obtain via Eq.(42) the Finser-Randers function X ( a ) and vice-versa. Finally, inverting(42) and utilizing again (38),(40), we find after some sim-ple algebra that w ( a ) = − − a (cid:20) − a + d ln Qda (cid:21) . (43) Practically, defining Ω m and Ω r as the standard nonrelativisticand radiation density parameters at the present time, we canhave that the complete Hubble function reads as E EF R ( a ) =Ω m a − + Ω r a − + Ω Ψ a − Q ( a ) in the limit of F/α = 1 + β/α ∼ z CMB ) the Hubble horizon is λ H ∼ . × pc which impliesthat | β | ≪ m + Ω r + Ω Ψ = 1. Relation (43) is one of the basic results of our work. Itprovides the relation of any EoS evolution with the nec-essary form of the Finsler-Randers geometry. In particu-lar, for a given desired form of w ( a ) we use (43) in orderto find the corresponding Q ( a ). Then through (41),(42)we calculate X ( a ), and using (35) we obtain Ψ( a ). Fi-nally, with the profile of Ψ( a ) in hand we determine theFriedmann equation of motion (32) that together withthe continuity equation (30) fully determines the cosmo-logical evolution.We stress here that there is not any restriction atall, namely the above procedure can be applied for any w ( a ), as long as the corresponding Hubble function isgiven by (39), for instance including the quintessenceand phantom regimes, the phantom-divide crossing fromboth sides, etc. In the following subsection, without lossof generality, we reconstruct X ( a ) of the Finsler-Randersmetric function, for the most familiar cosmological sce-narios. B. Specific examples
In order to proceed to specific examples, the precisefunctional form of X ( a ) has to be determined. How-ever, note that this is also the case for any dark-energymodel, as far as the equation of state (EoS) parame-ter in concerned. Potentially, in the current work wecould phenomenologically treated X ( a ) [and thus Q ( a )and Ψ( a )] either as a Taylor expansion around a = 1[ X ( a ) = X + X (1 − a )] or as a power law X ( a ) ∝ a ν .Instead of doing that we have decided to mathemati-cally investigate the conditions under which the Finsler-Randers cosmological model can produce some of thewell known DE models. Bellow we provide some spe-cific examples along the above lines. In particular, wefirst consider some literature scalar-field DE models thatemerge from FRW cosmology with General Relativity,and for these models we reconstruct the functional formsof Ψ( a ) = Ψ X ( a ) / • Cosmological ConstantInserting w Λ = − const. into (43) we obtainthat Q ( a ) = a , which leads to f ( a ) = 1 and thusto X ( a ) = 2. • Quintessence and Phantom models with constant w In these constant- w scenarios [2, 7, 9] DE is at-tributed to a homogeneous scalar field, with a suit-able potential in order to keep the EoS constant,which requires a form of fine tuning. Specifically,the DE models with a canonical kinetic term of thescalar field lead to − ≤ w , while models of phan-tom DE ( w < −
1) require an exotic nature, namelya scalar field with negative kinetic energy, which could lead to unstable quantum behavior [50]. Sub-stituting w ( a ) = w = const. into (43) we find Q ( a ) = a − (1+3 w ) , (44)and thus X ( a ) = − (1 + 3 w ) a − w ) . (45)In other words, if we desire to construct aQuintessence or Phantom look-alike Hubble expan-sion (frequently used in cosmological studies), weneed to write X ( a ) as in (45). • Chevalier-Polarski-Linder DEWe consider the Chevalier-Polarski-Linderparametrization [51], in which the dark en-ergy EoS parameter is defined as a first-orderTaylor-expansion around the present epoch: w ( a ) = w + w (1 − a ) . (46)In this case we straightforwardly obtain Q ( a ) = a − (1+3 w +3 w ) exp [ − w (1 − a )] (47)and therefore X ( a ) = 3 w ( a − − (1 + 3 w ) a Q ( a ) . (48)Similarly to the previous example, if we want tobuild a CPL look-alike Hubble expansion in thecontext of Finsler-Randers geometry then the cor-responding functional form of X ( a ) needs to obey(48). • Pseudo-Nambu Goldstone boson scenarioIn the Pseudo-Nambu Goldstone boson model [52]the dark energy EoS parameter is found with theaid of the potential V ( φ ) ∝ [1 + cos( φ/p )] and itreads w ( a ) = − w ) a p , (49)where p is a free parameter of the model. Based onthis parametrization the basic ERF functions aregiven by Q ( a ) = a exp (cid:20) − w p ( a p − (cid:21) (50)and X ( a ) = 2 − w ) a p a Q ( a ) . (51) • f ( T ) gravityLet us now give an example of how we can recon-struct the functional forms of X ( a ) and Q ( a ) ofthe equivalent Finsler-Randers cosmology, in thecase of a modified gravitational model. As a spe-cific case we choose the f ( T ) construction, whichis based on the teleparallel equivalence of GeneralRelativity. In this formulation the gravitationalinformation is included in the torsion tensor andthe corresponding torsion scalar T , and one ex-tends the Lagrangian considering arbitrary func-tions f ( T ) [13, 14]. Within such a framework theHubble function is written as H = 8 πG ρ m − f ( T )6 − f T H , (52)where T = − H is the torsion scalar and f T = ∂f ( T ) /∂T . Based on the matter epoch, defining E F T ( a ) = H ( a ) /H and using ρ m = ρ m a − , theabove equation always becomes E F T ( a ) = Ω m a − + Ω F y ( a ) (53)where Ω F = 1 − Ω m . The function y ( a ) is scaledto unity at present time and is given by y ( a ) = − F (cid:20) f ( T )6 H + 2 f T E ( a ) (cid:21) . (54)Comparing relations (53), (54) with equation (37),we find that Q ( a ) = y ( a ) a (55)and X ( a ) = dyd ln a − y ( a ) a . (56)As an example we use the power-law model of Ben-gochea & Ferraro [13] with f ( T ) = α ( − T ) b , α = (6 H ) − b Ω F b − , (57)where b is the free parameter of the model whichhas to be less than unity in order to ensure a cosmicacceleration. Inserting (57) into (54) we arrive at y ( a ) = E b ( a ) = a Q ( a ). Obviously, for b = 0the power-law f ( T ) model reduces to the ΛCDMmodel, while for b = 1 / f ( T ) power-law gravity models. Note that, aswe said in the Introduction, the equivalence of DGPwith Finsler-Randers cosmology was already foundby some of us in [23].In summary, from the above analysis and the specificexamples, it becomes clear that the DE scenarios (includ-ing some modified gravity models) that satisfy (39), canbe seen as equivalent to the geometrical EFR cosmolog-ical model.Finally, in a forthcoming publication we attempt tophysically derive the precise functional form of X ( a ), aswell as to provide a full perturbation analysis, which canbe used in order to distinguish the Finser-Randers sce-nario from other DE and modified gravity models [53]. IV. CONCLUSIONS
In the present work we investigated an extended formof Finsler-Randers cosmology, and we showed that it canmimic any non-interacting dark-energy scenario, as wellas modified gravity models, at the background level. Inparticular, we started from a small deviation from thequadraticity of the Riemannian geometry, and we ex-tracted the modified Friedmann equation that determinesthe universe evolution.The effect of the Finsler-Randers modification is toproduce correction terms to the Friedmann equation,that can lead to a large class of scale-factor evolution,including the quintessence and phantom regimes, thephantom-divide crossing from both sides, etc. As weshowed, for a given dark-energy equation-of-state param-eter we can reconstruct the corresponding functions ofthe Finsler-Randers space that indeed give rise to such abehavior, and vice versa. Therefore, the present work isa completion of the previous works of some of us [22, 23],where we had showed the equivalence of Finsler-Randerscosmology with particular modified gravitational modelsas the DGP one, since we now show that the extendedFinsler-Randers cosmology can resemble a large class ofcosmological scenarios.In this respect, the non-trivial universe evolution, andespecially its accelerated phase either during inflation orat late times, is not attributed to a new scalar field,or to gravitational modification, but it arises from themodification of the geometry itself. In particular, even avery small non-quadraticity of the Finsler-Randers geom-etry, in which the local structure of General Relativity ismodified and the curvature theory is extended, can leadto significant implications to the cosmological evolution.One should still provide an explanation for the origin ofthe Finsler-Randers geometry itself, and the small depar-ture from the Riemann one. Although there are indica-tions that this must be related to quantum gravity effects[21, 27–33], this issue lies beyond the scope of the presentwork and it is left for a future investigation.We close this work by making two comments. Thefirst is that, as we discussed in the text, our analysis isvalid at intermediate and late times, including the radia-tion era, where all the energy conservations hold as usual.The second is that the above equivalence between Finsler-Randers geometry and dark energy and modified gravitymodels, has been obtained at the background level, thatis demanding the same scale-factor evolution. However, anecessary step is to proceed to a detailed analysis of thecosmological perturbations, and see whether the afore-mentioned equivalence breaks, which would allow to dis-tinguish between the various scenarios (this was indeedthe case in the equivalence of the simple Finsler-Randersgeometry with the DGP model [23]), or whether it ismaintained, in which case the degeneracy of the aboveconstructions would be complete. This complicated anddetailed investigation is in progress [53].0
Acknowledgments
The authors would like to thank an unknown refereefor his valuable comments and suggestions. SB acknowl-edges support by the Research Center for Astronomyof the Academy of Athens in the context of the pro-gram “
Tracing the Cosmic Acceleration ”. The researchof ENS is implemented within the framework of the Ac-tion “Supporting Postdoctoral Researchers” of the Oper-ational Program “Education and Lifelong Learning” (Ac-tions Beneficiary: General Secretariat for Research andTechnology), and is co-financed by the European SocialFund (ESF) and the Greek State. PS receives partial sup-port from the “Special Accounts for Research Grants” ofthe University of Athens.
Appendix: The 1+3 covariant formalism
We briefly summarize the 1+3 covariant formalism asdeveloped by J.Ehlers and G.F.R.Ellis [25]. The nota-tion in this Appendix is for the Riemannian limit thatwill serve as guidance through our calculations to theFinslerian case. The covariant approach employs a time-like vector field u a with u a u a = 1. With respect to thisnormalized vector field we split spacetime to time andspace. The 1+3 split is a particular case of the tetradformalism where the u a congruence represents the frameof comoving observers. With respect to the 4-velocity u a we can decompose all tensors to their irreducibleparts. In particular, using the projection 2nd-rank ten-sor h ab = g ab − u a u b we can covariantly define the timederivative and the spatial gradient of an arbitrary tensorfield, namely˙ S cd..ab.. = u e ∇ e S cd..ab.. (A.1)D e S cd..ab.. = h se h fa h pb h cq h dr ... ∇ s S qr..fp.. . (A.2)Instead of writing the metric to a particular coordinatesystem the geometry as measured by the u a family ofobservers is described by the irreducible parts of the fol-lowing tensor fieldD b u a = 13 Θ h ab + σ ab + ω ab , (A.3) where we define the kinematic quantities: the expansionΘ = D a u a , the shear σ ab = D h b u a i and the vorticity ω ab = D [ b u a ] . The projective symmetric and trace freepart is defined as X h ab i = h c ( a h db ) X cd − X cd h cd h ab , (A.4)where indices in squared brackets is the symmetrisedpart.In case of a shear and vorticity free expanding congru-ence of geodesics the evolution of the deviation vectorthat connects nearby observers is ˙ ξ a = Θ ξ a [25]. Takingthe time derivative of the previous expression, and substi-tuting to the deviation of geodesics ¨ ξ a + R acbd u c u d ξ b = 0,gives back the Raychaudhuri’s equation˙Θ + 13 Θ = − R ab u a u b . (A.5)The energy momentum tensor of pressureless matter is T ab = ρu a u b , and the contracted Einstein’s field equa-tions along the observers 4-velocity give back the aux-iliary expression, R ab u a u b = ρ . Moreover, an impor-tant geometric entity is the characteristic length scaleof the expanding 3D cross-section, namely the scale fac-tor a given by dV ∝ a . The reader should notice thatthe scale factor is covariantly defined, in contrast to themetric based approach where it is introduced through aparticular coordinate system. Thus, for the expansionwe obtain Θ = ( dV )˙ /dV = 3 ˙ a/a , and therefore we canrewrite relation (A.5) in the form3 ¨ aa = − ρ. (A.6)The physical requirement of pressureless matter impliesthat for a conservative system the matter energy densityscales with the volume element, that is ρdV = const .Furthermore, using the definition for the scale factor weacquire ( ρa )˙ = 0 (alternatively one may decompose theenergy momentum conservation law ∇ b T ab = 0 to its’irreducible parts [25]). The latter together with rela-tion (A.6) fully determines the evolution of the dust-likemedium (for further details see for example [43]). [1] M. Tegmark et al. , Astrophys. J. , 702 (2004);D. N. Spergel et al. , Astrophys. J. Suplem. , 377(2007); T. M. Davis et al. , Astrophys. J. , 716 (2007);M. Kowalski et al. , Astrophys. J. , 749(2008); G. Hin-shaw et al. , Astrophys. J. Suplem. , 225 (2009);J. A. S. Lima and J. S. Alcaniz, Mon. Not. Roy. As-tron. Soc. , 893 (2000); J. F. Jesus and J. V. Cunha,Astrophys. J. Lett. , L85 (2009); S. Basilakos andM. Plionis, Astrophys. J. Lett. , 185 (2010).[2] E. J. Copeland, M. Sami and S. Tsujikawa, Intern. Jour-nal of Modern Physics D, , 1753,(2006); L. Amendola and S. Tsujikawa, Dark Energy Theory and Observations ,Cambridge University Press, Cambridge UK, (2010).[3] S. Capozziello and M. De Laurentis, Phys. Rept. ,167 (2011).[4] S. Weinberg, Rev. Mod. Phys. , 1 (1989).[5] P. J. Steinhardt, Critical Problems in Physics, ed. byV. L. Fitch et al. D. R. Marlow & M. A. E. Dementi(Princeton Univ. 1997); P. J. Steinhardt, Phil. Trans.Roy. Soc. Lond. A , 2497 (2003).[6] R. R. Caldwell, R. Dave, and P.J. Steinhardt, Phys. Rev.Lett., , 1582 (1998). [7] P. J. Peebles, B. Ratra, Rev. Mod. Phys., , 559, (2003).[8] T. Padmanabhan, Phys. Rept., , 235, (2003);J. Si-mon, L. Verde, R. Jim´enez, Phys. Rev. D71 , 123001,(2005).[9] R. R. Caldwell, Phys. Lett. B , 23 (2002); S. Nojiriand S. D. Odintsov, Phys. Lett. B , 147 (2003); H.K. Jassal, J.S. Bagla, T. Padmanabhan, Phys. Rev.
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