Cosmological solutions and finite time singularities in Finslerian geometry
aa r X i v : . [ g r- q c ] F e b Cosmological solutions and finite time singularities in Finslerian geometry
Nupur Paul, ∗ S. S. De, † and Farook Rahaman ‡ Department of Mathematics, Jadavpur University, Kolkata − Department of Applied Mathematics, University of Calcutta, Kolkata − We consider a very general scenario of our universe where its geometry is characterized by theFinslerian structure on the underlying spacetime manifold, a generalization of the Riemanniangeometry. Now considering a general energy-momentum tensor for matter sector, we derive thegravitational field equations in such spacetime. Further, to depict the cosmological dynamics insuch spacetime proposing an interesting equation of state identified by a sole parameter γ which forisotropic limit is simply the barotropic equation of state p = ( γ − ρ ( γ ∈ R being the barotropicindex), we solve the background dynamics. The dynamics offers several possibilities dependingon this sole parameter as follows − (i) only an exponential expansion, or (ii) a finite time pastsingularity (big bang) with late accelerating phase, or (iii) a nonsingular universe exhibiting anaccelerating scenario at late time which finally predicts a big rip type singularity. We also discussseveral energy conditions and the possibility of cosmic bounce. Finally, we establish the first law ofthermodynamics in such spacetime. PACS numbers: 98.80.+k; 95.36.+xKeywords: Finsler geometry; Singularities; Current acceleration
1. INTRODUCTION AND PRELIMINARIES
To understand the current accelerating expansion his-tory of the universe, mainly two different approaches areconsidered. The first one is to introduce some dark en-ergy fluid in the framework of general relativity, while thelater introduces some modifications into the gravitationalsector. Now, considering the dark energy concept, obser-vations favor the existence of ΛCDM cosmology whereΛ is the cosmological constant and it is considered tobe responsible for this accelerated expansion. However,this cosmology has one serious issue known as cosmo-logical constant problem which inspired several authorsto consider some time varying dark energy fluid in thename of quintessence, phantom, Chaplygin gas and soon, although it should me mentioned that they intro-duced some other issues, for instance the cosmic coinci-dence problem and so on. Probably, both the cosmologi-cal constant as well as the time varying dark energy mod-els inspired cosmologists to introduce the modificationsinto the gravitational sector which finally appeared witha large number of gravitational modifications, such as f ( R ), f ( T ), etc. However, the basic problem in all suchmodified gravity theories is to first fix a functional formof f ( R ) and f ( T ) and then we study their viability withthe theoretical bounds and finally with the observationaldata . That means essentially in these two formalisms,we start with some phenomenological grounds. Thus, noone can exclude the possibility of a new dark energy fluidor some new modified gravity theory, or even some new ∗ Electronic address: [email protected] † Electronic address: ssddadai08@rediffmail.com ‡ Electronic address: [email protected] Exactly what we do while dealing with several dark energy fluids. direction of research which may positively account of thecurrent astrophysical and cosmological issues.In this article we have discussed the evolution of theuniverse in the context of Finslerian geometry which isa generalization of the traditional Riemannian geometry[1, 2] . The basic concept in such theories is to considerthe violation of Lorentz symmetry which is one of thecommon features in the quantum gravitational regimeand it may introduce some new directions in the moderncosmological research. The metric in this space is definedby a norm F ( x, y ) (where y ∈ T x M is the tangent vec-tor on a spacetime point x ) on a tangent bundle of thespacetime instead of an usual inner product structure onthe corresponding spacetime, for a detailed descriptionwe refer [3, 4]. It has been found that during last cou-ple of years a considerable attention has been paid onthis extended geometry to address some issues related toastrophysics and cosmology [5–10, 12–15], specifically, ithas been argued that the dark matter and the dark en-ergy problem can be addressed in such a context [6, 7, 10].So, essentially, the cosmology in this space-time will beworth exploring for new physical results.A Finslerian structure on a smooth 4 − dimensionalmanifold M is defined on the tangent bundle T M of M by Finsler metric F = ˜ T M = T M − { } and F is smoothon ˜ T M . In addition F is positively homogeneous of de-gree one with respect to ( y a ), i.e. F ( x, ky ) = kF ( x, y ), k >
0. The Finslerian metric tensor g ij is given by g ij = 12 ∂ F ( x, y ) ∂y i ∂y j (1)defined on ˜ T M . The causality in Finsler spacetime can We remark that this geometry was in fact suggested by Riemannhimself in his book [20] be given by the metric function F . A positive, zero ornegative value of F corresponds to timelike, null or space-like curves. In many cases it is useful that a Lorentzsignature to be taken under consideration. A non-linearconnection N on T M is a distribution on
T M , supple-mentary to the vertical distribution V on T M as T ( x,y ) ( T M ) = N ( x,y ) ⊕ V ( x,y ) . (2)A non-linear connection can be defined as N aj = ∂G a ∂y j , (3)where G a is given by G a = 14 g aj (cid:18) ∂ F ∂y j ∂y k y k − ∂ j F (cid:19) . (4)The geodesic equation for the Finsler space follows fromthe Euler-Lagrange equations dds (cid:18) ∂F∂y a (cid:19) − ∂F∂x a = 0 , (5)and it has the form dy a ds + 2 G a ( x, y ) = 0 . (6)A complete description of Finslerian structure of space-time manifold M is usually given with a metric func-tion F , a nonlinear connection N and compatible (met-rical) connections in the framework of a tangent bundle T M of spacetime [4]. However in some cases we canrestrict our consideration in order to describe the localanisotropic ansantz of gravitational field equations on abase 4 − dimensional manifold [5]. In such a case we areable to compare different scenarios of Riemannian-typecosmological models as FLRW with Finslerian ones, e.g[11].Thus, being motivated by the new generalized space-time governed by a Finslerian metric, in the present workwe have tried to address the current evolution of the uni-verse. Considering the background matter distributionto be anisotropic we wrote the gravitational equations insuch spacetime. Then we introduce an interesting equa-tion of state which for the isotropic limit simply assumesthe barotropic equation of state. We show that undersuch equation of state, the field equations can be analyt-ically solved which depending on the barotropic index ofthe fluid exhibits several cosmological issues. We foundthat depending on the barotropic index of the fluid thecosmological solution can predict a finite time past sin-gularity and currently it can exhibit an accelerating sce-nario of our universe. On the other hand, the model canoffer a singularity free universe in the past while it can notescape from a finite time future singularity. Moreover, wealso show that the violation of the energy conditions canlead to the cosmic bounce in such geometric structure. Finally, we establish that the first law of thermodynam-ics can hold under a simple energy conservation relation.Overall, the current picture offers some interesting pos-sibilities.We have organized the paper as follows: In section 2,we have presented the modified field equations in Finslerspace-time. Section 3 deals with a toy model of an equa-tion of state connecting the anisotropic matter distribu-tion and evaluates the cosmological solutions. In 4, webriefly describe the energy conservation relation and thefirst law of thermodynamics. Finally, we close our workin section 5 with a brief summary.
2. GRAVITATIONAL EQUATIONS IN FINSLERSPACE-TIME
To explain the dynamics of the universe, one desires tointroduce a metric specifying the geometry of the space-time which connects with its matter distribution by theEinstein’s gravitational equation. In order to realize thedynamics, we consider that our universe is described bythe Finsler metric which is of the form [12] F = y t y t − R ( t ) y r y r − r R ( t ) ¯ F ( θ, φ, y θ , y φ ) , (7)which has been taken inspired by the known fact thatat large scales our universe is well described by the flatFriedmann-Lemaˆıtre-Robertson-Walker (FLRW) line el-ement. We construct a cosmological model by inserting¯ F as a quadric in y θ and y φ . Note that the presentFinsler space ( for the case ¯ F as quadric in y θ and y φ )can be obtained from a Riemannian manifold ( M, g µν ( x ))as we have F ( x, y ) = q g µν ( x ) y µ y ν . One can notice that, this is a semi-definite Finslerspace-time and consequently, one can use the covariantderivative of the Riemannian space. It is to be noted thatthe Bianchi identities overlap with those of the Rieman-nian space (being the covariant conservation of Einsteintensor). Since the present Finsler space reduces to theRiemannian space, therefore, the gravitational field equa-tions can be obtained readily. The base manifold of theFinsler space regulates the gravitational field equation inFinsler space and the fiber coordinates y i play the roleof the velocities of the cosmic components i.e. velocitiesin the energy momentum tensor. Hence, one can derivethe gravitational field equations in Finsler space from theEinstein gravitational field equation in the Riemannianspace-time with the metric (7) in which the metric ¯ g ij isgiven by (see Appendix A)¯ g ij = diag(1 , sin √ λ θ ) , i.e. g µν = diag(1 , − R ( t ) , − r R ( t ) , − r R ( t ) sin √ λ θ ) , where λ > F is specified as a constant flag curvature space,that is it is assume that ¯ R ic = λ . This Ricci scalar “ ¯ R ic”is that of the Finsler structure ¯ F and two dimensionalFinslerian structure is specified by the constancy of theflag curvature. This flag curvature of Finsler space isin fact, the generalization of the sectional curvature ofRiemannian space. It will be apparent latter that thesolution of vacuum field equation must lead to the con-stancy of the flag curvature with its value λ = 1. But formore general case of Finsler structure ¯ F , it can be speci-fied by the constant flag curvature having any real valuefor λ . In fact, for λ = 1, we can get the usual Friedmannequations of the FLRW universe.Let us now assume the general energy-momentum ten-sor for the matter sector as T µν = ( ρ + p t ) u µ u ν − p t g µν + ( p r − p t ) η µ η ν , (8)where u µ u µ = − η µ η µ = 1, p r , p t are respectively de-note the pressures of the anisotropic fluid in the radialand transversal directions. The modified gravitationalfield equations in Finsler space-time are obtained as (seeAppendix B)8 π F Gρ = 3 ˙ R R + λ − r R , (9)8 π F Gp r = − RR − ˙ R R − λ − r R , (10)8 π F Gp t = − RR − ˙ R R . (11)Note that, λ = 1 implies p r = p t , that means, it helps torecover the gravitational field equations for flat FLRWuniverse. On the other hand, if we put p r = p t in theabove field equations, we readily find λ = 1. Hence, wefind that, λ = 1 ⇐⇒ flat FLRW universe. Moreover,one can easily see that when t −→ ∞ , we again findthat both the anisotropic pressure components becomeequal and the usual Friedmann equations in the spatiallyflat FLRW universe in presence of a perfect fluid withenergy-momentum tensor T µν = ( p + ρ ) u µ u ν + pg µν arerecovered.Now, introducing the Hubble parameter H = ˙ R/R ,the field equation (9) can be written as (for simplicity,from now we work in the units where 8 π F G = 1) ρ = 3 (cid:18) H + λ − r R (cid:19) ⇐⇒ Ω + Ω k = 1 , (12)where Ω = ρ/ H is the density parameter representingthe matter sector, and in compared to the Friedmann universe, the quantity Ω k = − ( λ − r R H ) can be lookedas the density parameter for the scalar curvature in theFinslerian geometry. Also, for both the directions, theRaychaudhuri equation can be written as˙ H = −
12 ( ρ + p r ) , (13)˙ H = −
12 ( ρ + p t ) + λ − r R , (14)where again we note that, for λ = 1, it reduces to onlyone equation.
3. EVOLUTION AND DYNAMICS: A TOYMODEL
In general, the equation of state of this anisotropic fluidtakes a general form f ( p r , p t , ρ ) = 0. The exact formof the equation of state is not known and hence, it is achallenge for modern cosmology to derive the cosmologi-cal evolution correctly. Still, we adopt mainly two possi-ble ways. One is to assume a very simple formulation ofthe equation of state in order to derive the evolutionaryparameters so that we can match them with the obser-vational data, and on the other hand, the reconstructionof any quantity from observed data is of worth exploring.However, in the present work, we adopt the first possi-bility, and thus, we start with the following equation ofstate γ ( p t − p r ) + p r = ( γ − ρ (15)where γ ∈ R is simply a constant. The essence of thisequation of state is that for p t = p r = p (say), the equa-tion of state in (15) is simply reduced to p = ( γ − ρ ,representing the barotropic equation of state. Further,we notice that, for γ = 0, equation (15) implies p r = − ρ ,and hence p t = − R /R . We are interested in the cos-mological solutions for the above choice of the equationof state. Now, using the field equations (9), (10), (11),we can exactly solve the scale factor R as R = R (cid:20) γ H ( t − t ) (cid:21) γ , ( γ = 0) (16) R = R exp ( H ( t − t )) , ( γ = 0) (17)where t , H are respectively the present cosmic timeand present day value of the Hubble parameter, and itis worth noting that the solutions obtained in Eqns (16),(17) exactly match with the solutions obtained for theisotropic matter distribution in the FLRW geometry withthe equation of state p = ( γ − ρ . Consequently, theHubble parameter can be derived as H = H γ H ( t − t ) , ( γ = 0) (18) H = H = Constant , ( γ = 0) (19)Therefore, it is clear that for γ = 0 we realize an expo-nential expansion of the universe. On the other hand, wecan divide γ = 0 into the following two conditions when γ > γ < γ > In this case, the solutions for the scale factor and theHubble parameter take the forms as in equations (16)and (18). The solutions offer the following scenario ofour universe. At t f = t − H γ , lim t → t f R ( t ) = 0 , and lim t → t f H ( t ) = ∞ , (20)which clearly shows that the universe attains a big bangsingularity in the past ( t f < t ). On the other hand, atlate time we find thatlim t →∞ R ( t ) = ∞ , and lim t →∞ H ( t ) = 0 . (21)Now, introducing the deceleration parameter q = − − ˙ H/H , we find that, for this cosmological solution onehas q = − γ/
2, which represents an accelerating uni-verse (i.e. q <
0) for γ < /
3. Thus, the model withthe equation of state in (15) presents a model of ouruniverse which predicts the big-bang singularity (a finitetime singularity, but independent of γ ), and describes anaccelerating universe for γ < / γ < Now, we consider the cosmological solutions for γ < γ = − α (where α > R = R h − α H ( t − t ) i α , (22) H = H − α H ( t − t ) . (23)We find that the scale factor can not be zero at anyfinite cosmic time in the past evolution of the universe, inother words it gives a solution to the nonsingular universethat has been proposed in several contexts in moderncosmology with great interests, see Refs. [20–27], but onthe other hand, from the solution of the scale factor ineqn. (22), it is seen that it diverges at some finite cosmictime in future, that means for t s = t + αH , R ( t ) −→∞ . Thus, we find that this cosmological solution givesa realization of a nonsingular universe in the past whichconsequently predicts a “big rip” singularity. Now, we find that we arrive at two different cosmo-logical scenarios, one which starts with big bang but atlate time it does not have any singularity, the other hasa nonsingular nature in past but at future it has a big ripsingularity. Therefore, in the next section we constrainthe model parameters for the first cosmological solution(i.e. for γ > p r = ω r ρ , p t = ω t ρ , and where we assume that ω r and ω t are the constants. Now, it is easy to see thatunder such conditions, one can exactly solve the energydensity using the above two equations, where the explicitform for ρ is ρ = ρ (cid:18) r ω t − ω r ) R (3+ ω r +2 ω t ) (cid:19) (24)where ρ > ω r = ω t . Additionally, the energy conditions inthis spacetime can be written in the following way: WEC: ρ ≥
0, and ρ + p r ≥ ρ + p t ≥ ˙ R R ! + λ − r R ≥ , ¨ RR ≤ ˙ R R , and ¨ RR ≤ ˙ R R + λ − r R SEC: ρ + p r ≥ ρ + p r + 2 p t ≥ RR ≤ ˙ R R , and ¨ RR ≤ NEC: ρ + p r ≥
0, and ρ + p t ≥ RR ≤ ˙ R R , and ¨ RR ≤ ˙ R R + λ − r R DEC: ρ ≥
0, and − ρ ≤ p r ≤ ρ , − ρ ≤ p t ≤ ρ In a similar way, the inequalities respectively reducedto 3 ˙ R R ! + λ − r R ≥ , − ˙ R R ! − λ − r R ≤ ¨ RR ≤ ˙ R R , and − ˙ R R ! − λ − r R ≤ ¨ RR ≤ ˙ R R + λ − r R Clearly the term (cid:0) λ − r R (cid:1) makes a significant contributionto the energy conditions, where we note that for λ = 1,the above conditions are simply reduced to the energyconditions as we find for a flat FLRW universe with aperfect fluid distribution. In this section we will seek for bouncing conditions inthis spacetime. Since the expansion scalar in this space-time is defined to be H = ˙ R/R , therefore one may recallthe bounce condition [19] H ( t b ) = 0 , and ˙ H ( t b ) > t b is the time where universe has bounced. Actu-ally, the above conditions can also be written as ˙ R ( t b ) = 0and ¨ R ( t b ) >
0. From the above energy conditions, fol-lowing the bounce conditions, an immediate solution forthe bouncing universe is that λ > • WEC : ρ + p ( r ) < , ρ + p ( t ) < , λ >
1, so WECis violated. • SEC : ρ + p ( r ) < , ρ + p ( r ) + 2 p ( t ) <
0, so SEC isviolated. • NEC and DEC are also violated.These cases are necessary conditions in order to have acosmic bounce.
4. ENERGY CONSERVATION RELATIONS
In this section we shall devote our discussions on theenergy conservation relation. Let us propose the energyconservation equation as follows d ( ρV ) = − P dV − V F a dr. (26)The additional term in R.H.S. is due to the anisotropicforce F a given by F a = 2( p t − p r ) r = 2( λ − π F G r R , (27) and the pressure P is the average pressure which is givenby P = p r + p t + p t p r p t . (28)The proposed conservation equation (26) can be writtenas V dρ + ρdV + P dV + V F a dr = 0 , or, alternatively as dρ + ρ dVV + P dVV + F a dr = 0 , which then turns into ∂ρ∂t dt + ∂ρ∂r dr + ( ρ + P ) dVV + F a dr = 0 , which again can be recast as dt ∂ρ∂t + ( ρ + P ) ˙ VV ! + dr (cid:18) ∂ρ∂r + F a (cid:19) = 0 , which clearly reports the following two separate equa-tions: ∂ρ∂t + 3 H ( ρ + P ) = 0 , (29) ∂ρ∂r = − F a . (30)Note that, the first equation is the usual energy conserva-tion equation for the homogeneous and isotropic universewith the effective pressure P . The second equation canbe derived from the above field equations (9), (10), (11).That means, the proposed conservation relation (26) isconsistent and well motivated with the energy conserva-tion relation in the context of general relativity.Also, if we define a pressure P as the weighted averageof p t and p r , i.e. if P = γp t − ( γ − p r (note that thesum of the weight is γ − ( γ −
1) = 1) then we see thatthe present equation of state is the barotropic equation ofstate P = ( γ − ρ . In equation (28) we have taken P asthe simple arithmetic average of the radial and transversepressure, i.e. the pressure in three orthogonal directions.This case corresponds γ = . With this, the conservationequation becomes d ( ρV ) + P dV + (cid:18) γ − (cid:19) ( p r − p t ) dV + ˜ F dr = 0 (31)where, ˜ F = V F a is an anisotropic force. In fact, theequation (31) can be written as dU + dW = dQ = 0 , (32)where U = ρV is the internal energy, W is the work doneand Q is the heat transfer and using these terminology,eqn. (31) is nothing but the following equation dW = P dV + (cid:18) γ − (cid:19) ( p r − p t ) dV + ˜ F dr. (33)The first two terms in the right hand side being thechange of work from pressures and the third being thatdue to the anisotropic force. This equation representsthe first law of thermodynamics for the case of adiabaticheat transfer.
5. CONCLUDING REMARKS
In this work we considered the spacetime of our uni-verse described by the Finsler geometry, a generalizationof the Riemannian geometry [1, 2]. A number of stud-ies [5–10, 12–15] in this framework have been performedin order to offer some explanations on some recent as-trophysical and cosmological evidences. Now, consider-ing a general matter distribution which by constructionstands for an anisotropic matter distribution, we rewrotethe modified gravitational field equations. We solved thedynamics of the universe for a simple equation of state ofthe matter sector (see eq. (15)) characterized by a soleparameter γ and which is motivated from the fact that incase of a perfect fluid matter distribution the barotropicequation of state, is recovered. Since γ has been keptfree for our analysis, so depending on it, we found threedistinct cosmological scenarios with current interests: • For γ = 0, we realize an exponential expansion ofthe universe. But, we do not have any such otherinformation for such specific value of γ . • γ > γ < /
3. So, this model can trace back the earlyphase of the universe as well as stands for the latetime accelerated phase. • The case γ <
Acknowledgments
FR would like to thank the authorities of the Inter-University Centre for Astronomy and Astrophysics,Pune, India for providing research facilities. FR is alsograteful to DST-SERB and DST-PURSE, Govt. of Indiafor financial support. We wish to thank Panayiotis C.Stavrinos and Supriya Pan for helpful discussion.
Appendix A: Basics of Finslerian geometry
We construct a cosmological model by inserting ¯ F inthe following form¯ F = y θ y θ + f ( θ, φ ) y φ y φ . (A1)Here,¯ g ij = diag(1 , f ( θ, φ )) , and ¯ g ij = diag(1 , f ( θ, φ ) ); [ i, j = θ, φ ]The geodesic equations in Finsler space-time are d x µ dτ + 2 G µ = 0 , where G µ = g µν (cid:16) ∂ F ∂x λ ∂y ν y λ − ∂F ∂x ν (cid:17) are the geodesicspray coefficients which calculated from ¯ F as G θ = − ∂f∂θ y φ y φ G φ = 14 f (cid:18) ∂f∂θ y φ y θ + ∂f∂φ y φ y φ (cid:19) Hence, one can find¯ F ¯ R ic = y φ y φ h − ∂ f∂θ + 12 f ∂ f∂φ − ∂∂φ (cid:18) f ∂f∂φ (cid:19) − f (cid:18) ∂f∂θ (cid:19) i − y φ y φ h f (cid:18) ∂f∂φ (cid:19) + 14 f ∂f∂φ f ∂f∂φ + ∂f∂θ f ∂f∂θ − f (cid:18) ∂f∂φ (cid:19) i + y θ y θ " − ∂∂θ (cid:18) f ∂f∂θ (cid:19) − f (cid:18) ∂f∂θ (cid:19) + y φ y θ (cid:20) f ∂ f∂θ∂φ − f (cid:18) ∂f∂θ (cid:19) (cid:18) ∂f∂φ (cid:19)(cid:21) − y φ y θ (cid:20) ∂∂θ (cid:18) f ∂f∂φ (cid:19) + 12 ∂∂φ (cid:18) f ∂f∂θ (cid:19)(cid:21) (A2)The coefficient of y φ y θ is zero iff, f is independent of φ i.e. f ( θ, φ ) = f ( θ ) (A3)Note that other coefficients of y θ y θ & y φ y φ are non zero.For the above consideration, Eq.(A2) yields¯ F ¯ R ic = " − f ∂ f∂θ + 14 f (cid:18) ∂f∂θ (cid:19) ( y θ y θ + f y φ y φ )This yields ¯ R ic as¯ R ic = − f ∂ f∂θ + 14 f (cid:18) ∂f∂θ (cid:19) . (A4)It is obvious that ¯ R ic may be a constant, say λ , or afunction of θ .For ¯ R ic = λ , one can find Finsler structure ¯ F as¯ F = y θ y θ + A sin ( √ λθ ) y φ y φ , if λ > y θ y θ + Aθ y φ y φ , if λ = 0 ,y θ y θ + A sinh ( √− λθ ) y φ y φ , if λ < . We note that one can take A as unity without any loss ofgenerality. Thus, we get the following form of the Finslerstructure (7) as F = y t y t − R ( t ) y r y r − r R ( t ) y θ y θ − r R ( t ) sin θy φ y φ + r R ( t ) sin θy φ y φ − r R ( t ) sin ( √ λθ ) y φ y φ , = α + r R ( t )(sin θ − sin ( √ λθ )) y φ y φ , which implies F = α + r R ( t ) χ ( θ ) y φ y φ (A5) where χ ( θ ) = sin θ − sin ( √ λθ ), and α is a Riemannianmetric. Therefore, choosing b φ = rR ( t ) p χ ( θ ), we get F = αφ ( s ) , φ ( s ) = p s (A6)where s = ( b φ y φ ) α = βα , and b µ = (0 , , , b φ ) , b φ y φ = b µ y µ = β , ( β is one form). This readily shows that F is the metric of ( α, β )-Finsler space. One can write thekilling equation K V ( F ) = 0 in Finsler space by usingisometric transformations of Finsler structure as [12] (cid:18) φ ( s ) − s ∂φ ( s ) ∂s (cid:19) K V ( α ) + ∂φ ( s ) ∂s K V β ) = 0 , (A7)where K V ( α ) = 12 α (cid:0) V µ | ν + V ν | µ (cid:1) y µ y ν ; K V ( β ) = (cid:18) V µ ∂b ν ∂x µ + b µ ∂V µ ∂x ν (cid:19) y ν . The symbol “ | ” means the covariant derivative with re-spect to the Riemannian metric α . Now, we have K V ( α ) + sK V ( β ) = 0 , or αK V ( α ) + βK V ( β ) = 0 , which gives K V ( α ) = 0 , and K V ( β ) = 0 (A8)or V µ | ν + V ν | µ = 0 (A9)and V µ ∂b ν ∂x µ + b µ ∂V µ ∂x ν = 0 . (A10)Here, the second Killing equation constrains the firstone (Killing equation of the Riemannian space). Thistakes responsibility for breaking the symmetry (isomet-ric) of the Riemannian space. Appendix B: Gravitational field equations
The geodesic spray coefficients for the Finsler structure(7) are given by G t = 12 R ˙ R h r ¯ F + y r y r i (B1) G r = ˙ RR ! y r y t − r F (B2) G θ = ¯ G θ + y θ " y r r + ˙ RR y t (B3) G φ = ¯ G φ + y φ " y r r + ˙ RR y t (B4)The Ricci scalarRic ≡ R µµ = 1 F " ∂G µ ∂x µ − y λ ∂ G µ ∂x λ y µ +2 G λ ∂ G µ ∂y λ y µ − ∂G µ ∂y λ ∂G λ ∂y µ (B5)can be computed as F R ic = ¯ F " ¯ R ic − r (cid:16) R + R ¨ R (cid:17) + y r y r (cid:16) R + R ¨ R (cid:17) − y t y t ˙ R R + ¨ RR − ˙ R ! (B6)For Ricci tensor R ic µν = ∂ ( F Ric ) ∂y λ y µ , one can write thescalar curvature in Finsler geometry as S = g µν Ric µν ,and explicitly in the following way S = − " ¨ RR + ˙ R R − λ − r R (B7)An immediate observation shows that if we set λ = 1,then we find that S is half of the Ricci scalar curvaturein FRW universe. The modified Einstein field equationsin Finsler space-time ( G µν ≡ R ic µν − g µν S = 8 π F GT µν )yield G tt = 3 ˙ R R ! + λ − r R = 8 π F GT tt (B8) G rr = 2 ¨ RR + ˙ R R + λ − r R = − π F GT rr (B9) G θθ = G φφ = 2 ¨ RR + ˙ R R = − π F GT θθ = − π F GT φφ (B10)As in Akbar-Zadeh [17], with the modified Einsteintensor ( G µν = Ric µν − g µν S ),( G µν − πGT µν ) | M = 0 (B11)where ′ | ′ M represents this gravitational field equation re-stricted to the base space manifold M of the Finslerianlength element F . Here fibre coordinates are supposedto be velocities of the cosmic components. Regarding thevalidity of gravitational field equations [12] it has beenargued that these equations can be derivable from thegravitational dynamics for Finsler spacetime based onan action integral approach on the unit tangent bundle[18]. Of course, this derivation of the equation (B5) fromthe gravitational field equation in Finsler spacetime asgiven by Pfeifer et. al is only approximate, where thecurvature tensor not belonging to the base space of thetangent bundle has been neglected. But the strong pointis that the equation (B5) is constructed by the geomet-rical invariant (Ricci tensor) which is insensitive to theconnections. The equations for the present case are equa-tions (B2), (B3) and (B4). These equations are the sameas those which can be derived from the equivalent Rie-mannian spacetime with the with the metric tensor (7).Thus we see that although the equation (B5) is “approx-imately” valid as it is restricted to the base space of theFinslerian length element F, this equation seems to beexact for the present case of FLRW spacetime with Fins-lerian perturbation β = b µ b µ . [1] S. S. Chern, Z. Shen, “Riemann-Finsler Geometry”,World Scientific, Singapore (2004); S. S. Chern, “FinslerGeometry Is Just Riemannian Geometry without theQuadratic Restriction”, Not. Amer. Math. Soc. , 1403 (2008), gr-qc/0612157.[6] Z. Chang and X. Li, Phys. Lett. B. , 453 (2008),arXiv:0806.2184.[7] Z. Chang and X. Li, Phys. Lett. B. , 173 (2009),arXiv:0901.1023.[8] S. Vacaru, Int. J. Geom. Methods. Mod. Phys. , 473(2008), arXiv: 0801.4958.[9] A. P. Kouretsis, M. Stathakopoulos and P. C. Stavrinos,Phys. Rev. D. , 104011 (2009), arXiv: 0810.3267v3.[10] S. Basilakos, A. P. Kouretsis, E. N. Saridakis and P.Stavrinos, Phys. Rev. D. , 123510 (2013). [11] S. Basilakos and P. Stavrinos, Phys. Rev. D. , 4 (2013).[12] X. Li and Z. Chang, Phys. Rev. D. , 064049 (2014),arXiv:1401.6363.[13] X. Li, S. Wang and Z. Chang, Comm. Theo. Phys. ,781 (2014), arXiv:1309.1758.[14] F. Rahaman, N. Paul, S. S. De, S. Ray and Md.A. K. Jafry, Eur. Phys. J. C , 564 (2015),arXiv:1506.02501[gr-qc].[15] F. Rahaman, N. Paul, A. Banerjee, S. S. De, S. Rayand A. A. Usmani, Eur. Phys. J. C , 246 (2016),arXiv:1607.04329 [gr-qc].[16] G. F. B Riemann, Uber die Hypothesen welche der Ge-ometrie zu Grunde liegen, Habilitation thesisd, Yniver-sity of Gottingen (1854).[17] H. Akbar-Zadeh, Acad. Roy. Belg. Bull. Cl. Sci. , 281(1988).[18] C. Pfeifer and M. N. R. Wohlfarth, Phys. Rev. D ,064009 (2012).[19] S. Carloni, P. K. S. Dunsby and D. M. Solomons, Class.Quant. Grav. , 1913 (2006), arXiv:gr-qc/0510130. [20] R. Brandenberger and C. Vafa, Nucl. Phys. B. , 391(1988).[21] G. F. R. Ellis and R. Maartens, Class. Quant. Grav. ,223 (2004), arXiv:gr-qc/0211082.[22] G. F. R. Ellis, J. Murugan and C. G. Tsagas, Class.Quant. Grav. , 233 (2004), arXiv:gr-qc/0307112.[23] D. J. Mulryne, R. Tavakol, J. E. Lidsey and G.F. R. Ellis, Phys. Rev. D. , 123512 (2005),arXiv:astro-ph/0502589.[24] N. J. Nunes, Phys. Rev. D. , 103510 (2005),arXiv:astro-ph/0507683.[25] J. E. Lidsey and D.J. Mulryne, Phys. Rev. D. , 083508(2006), arXiv:hep-th/0601203.[26] A. Banerjee, T. Bandyopadhyay and S. Chakraborty,Grav. Cosmol. , 290 (2007), arXiv:0705.3933 [gr-qc].[27] A. Banerjee, T. Bandyopadhyay and S. Chakraborty,Gen. Relt. Grav.40