Cosmological Models with Fractional Derivatives and Fractional Action Functional
aa r X i v : . [ g r- q c ] N ov COSMOLOGICAL MODELS WITH FRACTIONAL DERIVATIVES ANDFRACTIONAL ACTION FUNCTIONALV. K. Shchigolev Department of Theoretical Physics, Faculty of Physics andEngineering, Ulyanovsk State University, Ulyanovsk 432000, Russia
Abstract
Cosmological models of a scalar field with dynamical equations containing fractional derivativesor derived from the Einstein-Hilbert action of fractional order, are constructed. A number ofexact solutions to those equations of fractional cosmological models in both cases is given.PACS: 98.80.-k,02.40,11.10.EfKeywords: Fractional Derivative and Integral, Cosmology.
As is observed [1], there has been in the last decade active interest to application of fractionalderivatives in various fields of physics, where they play essential and sometimes leading role inunderstanding the complex classical and quantum systems. Surprisingly, the achievements of thefractional-differential calculus in the theory of gravitation and cosmology are very insufficient andprobably no more then thirty or so publications are devoted to this subject (see, for example,[2]-[7]). At the same time, some essential problems in modern theoretical cosmology [8, 9] areknown that lead to serious modification of the possible sources of the accelerated expansion ofthe Universe (or the gravitational theory itself). Under these circumstances, the interest to thefractional-differential modifications of cosmological models that appeared recently is fully justified.In this paper, we provide a review of the main publications devoted to the fractional differentialapproach in cosmology (known to the author). We also offer a novel approach to obtaining themodified Friedmann equations consistent with main principles and allowing, in our opinion, toavoid the conflict between the integer dimension of (pseudo) Riemann space - time of GR andfractional order of derivatives in the modified equations. As is mentioned in [5], there are twodifferent methods of approaching what fractional derivative cosmology could be. The simplestis the Last Step Modification (LSM) method, in which the Einstein’s field equations for a givenconfiguration are replaced with analogous fractional field equations. In other words, ∂ t → D αt afterthe field equations for a specific geometry have been derived. The fundamentalist methodologyis the First Step Modification (FSM), in which one starts by constructing fractional derivativegeometry. The intensively developed approach to modification of the main cosmological equationsand non-conservative systems of Lagrangian dynamics on the basis of a variational principle forthe action of a fractional order (Fractional Action-Like Variational Approach FALVA) developedin [2]-[4] represents one of the possible version of intermediate modification (Intermediate StepApproach, ISA) mentioned in [5].Our work is based in its main part on ISA and FALVA, because the equations of standardcosmology can be obtained from a variational principle for the Einstein-Hilbert action, in whichthe variation is made both over the scale factor and a laps function N , or, as is shown in [10],the last can be replaced by a condition of a time scale invariancy, and in any case it is easyto generalize masters equations to the case of fractional derivatives. We obtain the fractional-differential analogue of the Friedmann equations including a scalar field as a source of gravitationin frameworks of ISA and on the basis of FALVA. Until now in cosmology with fractional derivatives,there exist more questions than answers to such of them as: How should such models be obtained?How should the basic equations of these models be written down? What is the meaning of thesesolutions for the modified models in aspect of the modern problems of cosmology? In our work,we try to answer some of these questions. E-mail: [email protected] Fractional integrals and derivatives
There are no less than two dozens of definitions of the fractional derivative [1]. In the physicalapplications of the fractional differential calculus, more often one deals with Riemann-Liouvillederivative (RLD), Caputo derivative (CD), and some others.Such derivatives are defined by means of analytical continuation of the Cauchy formula for themultiple integral of integer order as a single integral with a power-law core into the field of realorder µ > c I µx f ( x ) = 1Γ( µ ) x Z c f ( t )( x − t ) µ − dt. (1)The Riemann-Liouville derivative of fractional order α ≥ f ( x ) is defined as the integerorder derivative of the fractional-order integral (1): D αx f ( x ) ≡ D nx c I n − αx f ( x ) = 1Γ( n − α ) d n dx n x Z c f ( t )( x − t ) α − n +1 dt (2)where D nx ≡ d n /dx n , n = [ α ] + 1. This definition corresponds to the so-called left derivative,frequently denoted as c D αx f ( x ). For the limit α = 1, this definition gives df ( x ) /dx . For example,the left RLD of x k for α ≤ , c = 0 equals: D αx x k = Γ( k + 1)Γ( k + 1 − α ) x k − α . (3)For α = 1, one has the usual result: D x x k = kx k − . The interesting feature of RLD is that RLDof non-zero constant C does not equal zero, but for α ≤ D αx C = C x − α / Γ(1 − α ).The right RLD is defined similarly to (2) on the interval [ c, d ]: x D αd f ( x ) = 1Γ( n − α ) (cid:18) − ddx (cid:19) n d Z x f ( t )( t − x ) α − n +1 dt (4)The Caputo derivative is the integral transform of a regular derivative, and it is defined bymoving the integer-order derivative in the Riemann-Liouville definition (2) inside the integral toact on the function f ( t ): C D αx f ( x ) ≡ c I n − αx D nx f ( x ) = 1Γ( n − α ) x Z c d n f ( t ) dt n ( x − t ) α − n +1 dt. (5)So, for α ≤ n = 1 c = 0, we have the following expression: C D αx x k = Γ( k )Γ( k + 1 − α ) kx k − α , which coincides with RLD D αx x k . In general, RLD does not coincide with CD [1]: C D αx f ( x ) = D αx f ( x ) + n X j =0 j − α ) f ( j ) ( c +)( x − c ) α − j . RLD and CD are not defined for power function x k with k = − /x can be obtained as the Weyl derivative defined as D α f ( x ) = ( − n − Γ( n − α ) ∞ Z x d n f ( t ) dt n ( t − x ) n − α dt, D α x − = − x − (1+ α ) Γ(1 + α ) for α ≤ n = 1).The left and the right CD are defined similarly to those for RLD, i.e., due to replacing thelimits c → x x → d in definition (5) consequently.One needs to be aware that according to the formulas of addition of orders, the followingholds([1], p.161): D αx D βx f ( x ) = D α + βx f ( x ) − n X j =1 D β − jx f ( c +) ( x − c ) − α − j Γ(1 − α − j ) , that is D αx D βx f ( x ) = D α + βx f ( x ), if only not all derivatives D β − jx f ( c +) at the beginning of theinterval are equal to zero. That is why D αx D αx f ( x ) = D αx f ( x ) in the general case. Generalizingthe Laplace operator in the equation for Newtonian gravitational potential, the author of [5]wrongly doubles the order of the repeated fractional derivative. The authors of [11] have avoidedthis mistake, having written down the Laplacian ∆ α as:∆ α u = 1 r α D αr ( r α D αr u ) + Γ ( α + 1) r α sin α θ ∂∂θ (sin α θ ∂u∂θ ) + Γ ( α + 1) r α sin α θ ∂ u∂φ . However their research did not concern cosmological problems.Let us note one more property of the fractional derivative expressed in modification of theLeibniz rule ([1],p.162): D αx [ f ( x ) g ( x )] = ∞ X k =0 α + 1 k !Γ( α − k + 1) D αx D α − kx f ( x ) D kx g ( x ) , (6)which becomes the usual rule as α = n . It can be represented as the integral over the order offractional derivative: D αx [ f ( x ) g ( x )] = ∞ Z −∞ Γ( α + 1)Γ( µ + 1)Γ( α + 1 − µ ) D α − µx f ( x ) D µx g ( x ) dµ. Later these rules of fractional differentiation will give us an essential modification to the cosmo-logical models with fractional derivatives.At last we want to note, that RLD can be expressed with the help of the integer derivative atthe initial point of the interval and CD ([1], p. 163) due to definition (5): D αx f ( x ) = n − X k =0 ( x − c ) k − α Γ(1 + k − α ) f ( k ) ( c +) + C D αx f ( x ) . Presumably, for the first time such models within LSM were considered in [2]. To avoid aconflict between the occurrence of fractional derivatives in the Friedmann equations and classicaldefinition of a tensor in the Einstein equation, based on the integer order derivative in tensor lawof transformation, the quoted author has repeated the well known derivation of the Friedmannequations for a dust from the classical approach (see [3], [12]) and than has replaced all integerderivatives with its fractional analog. For the spatially flat Universe, the Friedmann equationswith a cosmological term are written down in [2] as follows:( D αt a ( t )) = ( A ( Gρ ) α + B (Λ c ) α ) a ,D αt ( D αt a ( t )) = − ( A ( Gρ ) α − B (Λ c ) α ) a, (7)where a ( t ) is a scale factor of the Friedmann-Robertson-Walker (FRW) line element, ds = dt − a ( t )( dr + ξ ( r ) d Ω ) , (8)3here ξ ( r ) = sin r, r, sinh r for the sign of space curvature k = +1 , , −
1, consequently. Theoccurrence of α - degrees in the right-hand side of equations (7) is caused, probably, by dimensionalreasons. Then the author of the quoted work obtained the following solution to this set in a staticcase a = a = constant : ( Gρ ) α = C t α , (Λ c ) α = C t α . The conclusion is made, that the density of matter and cosmological constant decrease as 1 /t if G and speed of light in vacuum c remains constant, but if the density of matter and Λ remainconstant, then the following holds: G ∼ /t and c ∼ /t . The solution of equations (7), mentionedas an illustration of the method of [2], in the form a ( t ) = a E ,α ( Ct α ), where E α,β ( t ) = ∞ X k =0 t k Γ( αk + β )is the so-called Mittag-Leffler two-parametric function [1], is not confirmed by any calculations.It is interesting to note that for k = 0 and in the absence of matter and the cosmologicalconstant, that is if ρ = 0 and Λ = 0, equations (7) are reduced to D αt a ( t ) = 0 , D αt ( D αt a ( t )) = 0 , (9)which for α = 1 gives the obvious result: a ( t ) = a = constant , and interval (8) is reduced to thoseof the Minkowski space. If α = 1, then the substitution of the zero constant (that is the derivative D αt a ( t )) from the first equation of (9) into the second one results in identity irrespectively to thedefinition of fractional derivative: RLD or CD. The solution of the first equation in (9) for CD, aswell as in the case of integer derivative, is equal to constant, but for RLD its partial solution iszero, and the general solution depends on time: a ( t ) ∼ t α − for 0 < α ≤ k + ( D αt a ) ] = κρa , a D αt p = D αt [ a ( ρ + p )] , (10)where we use ρ for the energy density and p for the pressure. The second equation representsthe energy-momentum conservation law ( T ij ; i = 0) for the perfect fluid in space-time (8) with theinteger-order derivatives replaced by its fractional analogues. Assuming further k = 0 and thepower-law dependence of a, p and ρ on the time: a = Ct n , p = At m , ρ = Bt r , with the help offormulas (3) and (10) we can find: m = r = − α, B = 3 κ Γ( n + 1) Γ( n + 1 − α ) , and then from the equation of state p = ( γ − ρ and the second equation in (10), the followingequation for degree n in the law of evolution of the scale factor is obtained:( γ − γ Γ(1 − α )Γ(1 − α ) = Γ(3 n − α + 1)Γ(3 n − α + 1) . In the same spirit of naive approach, as it was named by the author of [5], the article [6] isprepared. In it, the Riemann curvature tensor and, as a consequence, the Einstein tensor aredefined by the unusual Christoffel symbols containing fractional (of order 0 < α ≤
1) derivativesof metrics coefficients, Γ µνλ ( α ) = 12 g µν ( ∂ αν g ρλ + ∂ αλ g ρν + ∂ αρ g νλ ) , (11)where ∂ αν is a fractional derivative (2) or (5) with respect to x ν . The quoted author writes downthe Einstein equation, R µν ( α ) − g µν R ( α ) = 8 πGc T µν ( α ) , d x µ dτ + Γ µνλ ( α ) dx ν dτ dx λ dτ = 0 , but does not give any solutions, and only demonstrates that linearized equations in the limit α → dx i = D αj x i dx j isundertaken for the flat two-dimensional space.There were much more advanced and proved results concerning FSM formalism and statedby S. Vacaru [13] (see the bibliography therein). In those works, the results of construction ofthe fractional theory of gravitation for the space - time of fractional (not integer) dimension areobtained. The author sees one of the simplest motivation for application of fractional differentialcalculus in the theory of gravitation in an opportunity to avoid singularities of the curvaturetensor of physical meaning due to the completely different geometrical and physical solutions ofthe fundamental equations. Besides, it is noted that models of fractional order are more adapted tothe description of processes with memory, branching and hereditarity, than those of integer order.The result of the application of the method developed by the author of nonholonomic deformationsto cosmology was the construction of new classes of cosmological models [14]. However, the lastdoes not concern to the models under consideration, and it will not be considered here.Let us now mention papers [3, 4], where the approach to the dynamical field theories in generaland to the theory of gravitation is developed on the basis of the variational principle, formulatedby the author, for the action of fractional order (FALVA). In this approach concerning ISA, theintegral of action S L [ q ] for the Lagrangian L ( τ, q ( τ ) , ˙ q ( τ )) is written down as the fractional integral(1): S L [ q i ] = 1Γ( α ) t Z t L ( τ, q i ( τ ) , ˙ q i ( τ ))( t − τ ) α − dτ, (12)being at fixed t the Stieltjes integral with integrating function g t ( τ ) = 1Γ(1 + α ) [ t α − ( t − τ ) α ],having the following scale property: g µt ( µτ ) = µ α g t ( τ ) , µ > . Then q i ( τ ) satisfies the fractional (or modified) Euler-Lagrange equation: ∂L∂q i − ddτ (cid:16) ∂L∂ ˙ q i (cid:17) = 1 − αt − τ ∂L∂ ˙ q i ≡ F i , i = 1 , , ..., n ; τ ∈ (0 , t ) , where a dot above the appropriate function stands for the first derivative with respect to time τ , F i is the modified decaying force of ”friction”, that is the general expression for non-conservativeforce. In the article [15], time τ is treated as the intrinsic (proper) time, and t is the observer time.The author of [3, 4], [16] states that at τ → ∞ we have F i = 0, and provides some examples ofapplication FALVA to the Riedmann geometry and perturbed cosmological models. Every timethe quoted author does not apply FALVA directly to the gravitational action S G [ a ( t )], where a ( t )is the scale factor in the FRW model (8), but tries to take into account influence of the fractionalorder action (12) on the Friedmann equations through perturbed (and time-dependent) classicalgravitational constant. Considering Lagrangian L = g ij ( x, ˙ x ) ˙ x i ˙ x j , the author obtains the modifiedgeodesic equation: ¨ x i + α − T ˙ x i + Γ ijk ˙ x j ˙ x k = 0 , (13)where Γ ijk are the standard Christoffel symbols, and T = t − τ . The second term here is interpretedas a dissipative force, which infinitely increases as τ → t for α = 1 and under condition of fixing5uture time t . Considering equation (13) in the Newtonian approximation, the author establishedthe time variation of Newton’s gravitational constant (i.e., tried to realize the Dirac hypothesis[17])and introduced the perturbation of the gravitational constant ∆ G = 3(1 − α )4 πρT ˙ aa . Only then, theeffective gravitational constant G eff = G +∆ G is substituted to the standard Friedmann equations:1 a ( ˙ a + k ) = 8 πG eff ρ + Λ3 , (14)¨ aa = − πG eff ρ + 3 p ) + Λ3 , (15)where the cosmological constant Λ equals zero or Λ = ( β/t )( ˙ a/a ), as it is made in [18], where G eff = G = const . Then in [18], [19] the solutions of equations (14), (15) are investigatedfor various equations of state ( p = γρ ), which could make sense if these equations or the way oneobtains them were enough justified. It would be natural to expect of the author of FALVA the directapplications of FALVA to construction of the modified Friedmann equations from the fractionalfunctional S G [ a ( t )] that leads to other results, as it will be shown below, without necessity touse equation (13) for the mentioned above interpretation of the extra terms in modified equation.Moreover, with the help of canonical parameter s = g t ( τ ), equation (13) is reduced to:¨ x i + Γ ijk ˙ x j ˙ x k = 0 , (16)where the over dots stand for derivatives with respects to s . Actually, the last means that equations(13) could be obtained without applying FALVA but simply by replacement of the parameter s = g t ( τ ) in equation (16). By the way, it is possible to address precisely the same remark to [20],where the modification of cosmology is undertaken on the basis of the periodic weight function g t ( τ ) in action: S L [ q i ] = t Z t L ( τ, q i ( τ ) , ˙ q i ( τ )) exp( − χ sin( βτ )) dτ, (17)which is named fractional in [20], though such name could be applied to functional (12) only,meaning its origination from fractional integral (1). As it follows from (17), in this case the weightfunction is defined by dg t ( τ ) dτ = exp( − χ sin( βτ )), and replacement s = g t ( τ ) in (16) is immediatelyresulted in equation obtained in [20] by variation of (17):¨ x i − βχ cos( βτ ) ˙ x i + Γ ijk ˙ x j ˙ x k = 0 , (18)again with treatment of the second term in (18) as the perturbation of gravitational constant ∆ G =3 χβ cos( βτ ) H πρ , where the Hubble parameter H = ˙ aa . Certainly, it would be possible to experimentfurther with various weight functions to solve those or other problems of the gravitational theoryand cosmology but it is reasonable before all to return to the framework stated at the beginningof our paper and to apply FALVA directly to the gravitational field of the Universe. By the way,it was possible to understand the purpose that El-Nabulsi declared while formulating FALVA (see,for example, [21], [22] just so). First we consider the naive (or LSM) approach to the fractional derivative cosmological modelsof a scalar field. For what follows, it is useful to reproduce the derivation of the gravitation andscalar field equations from the variational principle for the Einstein-Hilbert action inspired byADM formalism in cosmology (see, e.g. , [23]). The Einstein-Hilbert action-like functional forFRW model of the Universe ds = N ( t ) dt − a ( t )( dr + ξ ( r ) d Ω ) , (19)6here N is a laps function, filled with a real homogeneous scalar field φ ( t ), is as follows: S EH = S G + S Sc = 38 πG Z dtN (cid:18) − a ˙ a N + ka − Λ a (cid:19) + Z dtN a ˙ φ N − V ( φ ) ! , (20)where V ( φ ) is a potential of the field. By variation over a ( t ), φ ( t ) and N ( t ) (with the subsequentchoice of the gauge N = 1) in the action (20), one obtains the following standard Friedmann andscalar field equations: ¨ φ + 3 ˙ aa ˙ φ + dV ( φ ) dφ = 0 , (21)2 ¨ aa + ˙ a a + ka = − πG ˙ φ − V ( φ ) ! + Λ , (22)˙ a a + ka = 8 πG ˙ φ V ( φ ) ! + Λ3 . (23)On the other hand, equation (23) could be derived with preliminary gauge N = 1 in (19), that iswith proper time and representation of the FRW space-time interval as (8), proceeding from timeinvariancy of the action (20), as it was made in [10]. As a matter of fact, both approaches areequivalent, because (23) was derived from the Euler-Lagrange equation for N ( t ), that is ∂L∂N = 0,which just expresses noted time scale invariancy. Besides, instead of equation (22) one frequentlyuses the following equation: ¨ aa = − πG (cid:16) ˙ φ − V ( φ ) (cid:17) + Λ3 , (24)which turns out from equations (22), (23).Considering LSM, it is necessary to make substitution of fractional derivative of the scale factorand the scalar field instead of integer derivatives in equations (21), (23) and (24). Therefore therequired set of equations becomes as follows: D αt ( D αt φ ) + 3 (cid:18) D αt aa (cid:19) D αt φ + dV ( φ ) dφ = 0 , (25)( D αt a ) + k = 8 πG (cid:18)
12 ( D αt φ ) + V ( φ ) (cid:19) a + Λ3 a , (26) D αt ( D αt a ) = − πG (cid:16) ( D αt φ ) − V ( φ ) (cid:17) a + Λ3 a . (27)While among three equations (21) - (23) of standard cosmology only two equations are inde-pendent, and the third one can be derived as the differential consequence of two others, due tothe modified Leibniz rule for the fractional derivative (6), all equations of (25) - (27) are generallyindependent. The last means that one has either to solve equations (25) - (27) for a ( t ) , φ ( t ) and V ( φ ( t )) with constant G and Λ or to admit the dependence of G and/or Λ on time when thepotential V ( φ ) is given. This circumstance allows us to doubt the acceptability of models offeredearlier in the frameworks of the naive approach, which are submitted by the modified equations(7) and (10).The solution of nonlinear fractional equations (25) - (27) is rather problematic. Even theirclassical prototype (21) - (23) does not always have exact solutions. Unfortunately, the theory ofequations in fractional derivatives is incomparably more difficult and less advanced in comparisonwith the theory of integer-order differential equations (see, e.g., [1], [24]). Nevertheless, we wouldlike to give an example of an exact solution to (25) - (27) demonstrating the fact of existenceof some solutions. Let us consider the spatially flat model of the Universe without cosmologicalconstant ( k = Λ = 0). We assume the dependence of the scale factor, scalar field, and potential ontime as follows : a = a t n , φ = φ t m , V ( φ ( t )) = V t r . Due to the modified rules of differentiation(3) for the given functions, it follows from (25) - (27) that:7 = n, r = 2( n − α ) , (28) φ = a (4 πG ) − (cid:18) − Γ( n + 1 − α ) Γ( n + 1)Γ( n + 1 − α ) (cid:19) , (29) V = a (8 πG ) − Γ( n + 1) Γ( n + 1 − α ) (cid:18) n + 1 − α ) Γ( n + 1)Γ( n + 1 − α ) (cid:19) . (30) V ( φ ) = V n − α ) φ n (cid:18) φφ (cid:19) n − αn . (31)The power n in the dependence of scale factor on time is connected to the order of fractionalderivative α by the relationship:2(5 n − α ) Γ( n + 1)Γ( n + 1 − α ) = (cid:16) n + α ± p n − nα + α (cid:17) Γ( n + 1 − α )Γ( n + 1 − α ) , (32)in which both parameters appear in the arguments of Γ of -function, that essentially complicatesfurther general analysis of the model. Nevertheless, as the scalar field in equation (29) is real, therelationship (32) is possible if n ∈ (cid:18) α, α (cid:19) , where 0 < α < α ) analogous: S EH ≡ Z L EH dt = Z dtN (cid:20) πG (cid:18) − a ( D αt a ) N + ka − Λ a (cid:19) + a (cid:18) ( D αt φ ) N − V ( φ ) (cid:19)(cid:21) . (33)Variational problem with fractional derivatives for functions q j ( t ) in the action S [ q j ]( t ) = d Z c L ( q j ( t ) , c D αt q j ( t ) , t D βd q j ( t )) dt on the interval [ c, d ] yields the modified Euler-Lagrange equations[1],[25]: ∂L∂q j + t D αd (cid:18) ∂L∂ ( c D αt q j ) (cid:19) + c D βt ∂L∂ ( t D βd q j ) ! = 0 . (34)In our case q j ( t ) = φ ( t ) , N ( t ) and a ( t ). Therefore for L EH from (33) and (34), we arrive at thefollowing basic equations of the model: t D α ( a D αt φ ) − a dV ( φ ) dφ = 0 , (35)( D αt a ) + k = 8 πG (cid:18)
12 ( D αt φ ) + V ( φ ) (cid:19) a + Λ3 a , (36)2 t D α ( a D αt a ) + ( D αt a ) − k = 8 πG (cid:16) ( D αt φ ) − V ( φ ) (cid:17) a − Λ a , (37)where we denote D αt ≡ D αt and t D α ∞ ≡ t D α . If in the action S [ q j ]( t ) the derivatives are CDs, then(34) remains practically without changes, with the only exception of the derivatives of Lagrangianin the brackets are CDs [26]. Hence if the derivatives D αt a and D αt φ in (33) are determined as CDsthey also should be understood as CDs in equations (35),(37) but the repeated right derivatives(see definition (4)) in the first terms of (35) and (37) do not undergo redefinition. Let us notealso, that in the case of integer order derivative ( α = 1) due to the obvious expressions D t = ddt , t D = − ddt , the set of the fractional equations (35 -37) coincides with the classical one (21 -23).8 Cosmological models of scalar field with fractional action
We consider now the cosmological model of a scalar field, which follows from the variationalprinciple for the fractional action (12). This approach is in the framework of the intermediateapproach (ISA) and frequently, as it was mentioned above, is referred to FALVA. Here, it isnecessary to pay attention to the following peculiar properties of computation to avoid mistakesin obtaining the master equations of the model under consideration. The first term in the action(20) is obtained as the result of integration by parts of the first term in R √− g , which dependson the second derivative ¨ a . Actually this integration removes the derivative of the laps function˙ N from the action. It is a well known procedure. But now, if one makes simply fractional,as (12), generalization of the Einstein-Hilbert action on the basis of expression (20), the resultwill be incorrect, and some terms will be omitted. Due to the stated above reason, such terms areabsent in all quoted articles by El-Nabulsi, unfortunately also containing several other inaccuracies.Therefore, we use the modified Einstein-Hilbert action S αEH ≡ t Z L αEH dτ as the following fractionalintegral: S αEH = 1Γ( α ) t Z N " πG a ¨ aN + a ˙ a N − a ˙ a ˙ NN + ka − Λ a ! + a ǫ ˙ φ N − V ( φ ) ! ( t − τ ) α − dτ , (38)where all functions in L αEH depend on the intrinsic time τ , and ǫ = +1 , − q i = φ, a and N with the subsequentchoice of the gauge N = 1, we obtain the following Euler-Poisson equations, ∂L αEH ∂q i − ddτ (cid:18) ∂L αEH ∂ ˙ q i (cid:19) + d dτ (cid:18) ∂L αEH ∂ ¨ q i (cid:19) = 0 , for our model [27]: ¨ φ + 3 (cid:18) ˙ aa + 1 − α t (cid:19) ˙ φ + ǫ dV ( φ ) dφ = 0 , (39)¨ aa + 1 − α t (cid:18) ˙ aa (cid:19) + (1 − α )(2 − α )2 t = − πG (cid:16) ǫ ˙ φ − V ( φ ) (cid:17) + Λ3 , (40) (cid:18) ˙ aa (cid:19) + 1 − αt (cid:18) ˙ aa (cid:19) + ka = 8 πG ǫ ˙ φ V ( φ ) ! + Λ3 , (41)where and below the dots above functions designate the derivatives of the appropriate order withrespect to time t obtained by t − τ = T → t . Precisely the same equations one can obtain fromgeneralization of the Euler-Lagrange equations in FALVA for the Lagrangian with higher (second)derivative (see [28]) or from the previous integration by parts as mentioned above. In the last case,one has to take into account the weight function ( t − τ ) α − in the action integral (38) requiringthe limit lim t → ( a ˙ a/t ) to be finite.It is easy to verify that the set of equations (39) - (41) is represented by three independentequations, against two independent equations in the classical case (21) - (23), that is connected withviolation of the energy - momentum conservation law T µν ; ν = 0. Therefore in the set of equations(39) - (41) for three unknown functions ( a ( t ) , φ ( t ) and V ( t )), potential is not an arbitrary functionin general. One can rewrite equations (39) - (41) in terms of effective energy density ρ ( t ) andpressure p ( t ), taking into account the well known expressions: ρ = ǫ
12 ˙ φ + V ( φ ) , p = ǫ
12 ˙ φ − V ( φ ) . (42)As a result, it follows from (39)-(41) and (42) that:˙ ρ + 3 (cid:18) ˙ aa + 1 − α t (cid:19) ( ρ + p ) = 0 , (43)9 aa + 1 − α t (cid:18) ˙ aa (cid:19) + (1 − α )(2 − α )2 t = − πG ρ + 3 p ) + Λ3 , (44) (cid:18) ˙ aa (cid:19) + 1 − αt (cid:18) ˙ aa (cid:19) + ka = 8 πG ρ + Λ3 , (45)It is easy to integrate equation (43) for the perfect fluid with equation of state p = γρ : ρ = ρ a γ ) t (1 + γ )(1 − α ) . (46)The interesting fact is that the cosmological Λ-term in the action (38) can depend on time, but inequations (39) - (41), and also in (43) - (45), the only change appears in the dependence Λ = Λ( t ).Therefore the expression (46) is also valid.Let us provide an example of an exact solution for the flat model ( k = 0) and for the quasi-vacuum state of matter: γ = −
1. From (46) it follows that ρ ( t ) = ρ = constant , as well as inthe standard model. Then, the remaining equations of (43) - (45) for the Hubble parameter andΛ-term can be copied as follows:˙ H − − α t H + (1 − α )(2 − α )2 t = 0 , (47) H + 1 − αt H = 8 πG ρ + Λ3 . (48)From equation (47), it is easy to find that the Hubble parameter varies with time as H = C α t + H t − α , (49)where C α = (1 − α )(2 − α )(3 − α ) , H is a constant of integration. From (49), the following dependenceof the scale factor on time t appears: a = a tC α exp − α H t − α , (50)while the cosmological Λ-term changes with time asΛ = 3 H t − α + 3 H (1 − α )(7 − α )(3 − α ) t − (1 + α )2 + 3(1 − α ) (2 − α )(5 − α )(3 − α ) t − πGρ . (51)It is obvious that at the proceeding to the standard model in the limit α →
1, the obtainedsolutions (49) - (51) reduce to the known exponential expansion of the Universe : a = a eH t, H = H , Λ = 3 H − πGρ .We consider now the dynamics of the flat model of the Universe ( k = 0) filled by a scalar field.It is convenient to rewrite equations (39) - (41) in terms of the Hubble parameter in the followingform: ¨ φ + 3 (cid:18) H + 1 − α t (cid:19) ˙ φ + ǫ dV ( φ ) dφ = 0 , (52)˙ H − − α t H + (1 − α )(2 − α )2 t = − πGǫ ˙ φ , (53) H + 1 − αt H = 8 πG ǫ ˙ φ V ( φ ) ! + Λ3 . (54)It is easy to prove that the given set of the independent equations can contain some arbitrariness ina choice of unknown functions, for instance H ( t ) or V ( t ), only if the cosmological Λ-term depends10n time. However, it is possible to proceed from some dependence Λ( t ). Let us consider oneexample of an exact solution, assuming that the Hubble parameter and the scalar field depend ontime as follows: H = Ct , ˙ φ = Bt , (55)where
C, B are the constants to be defined. We take into account definitions (42) and the equationof state p = γρ for convenience of representation of solutions in terms of physically clear parametersand fractional order of the action α . Using substitution (55) in (52) - (54) and considering (42),we obtain the following result: a = a t ǫ (cid:18) − γ γ (cid:19) + α , φ = 1 λ ln t + φ , V ( φ ) = ǫ λ (cid:18) − γ γ (cid:19) e − λ ( φ − φ ) , (56)where the following notation is used: λ = λ ( α, γ ) = ± r πG − α (cid:20) − γ γ − ǫ (cid:18) − α − α (cid:19)(cid:21) − / ,C = ǫ (cid:18) − γ γ (cid:19) + α , B = λ − . It is easy to see that the baratropic index γ is not arbitrary but it should satisfy the followinginequality: 1 − γ γ > ǫ (cid:18) − α − α (cid:19) , and also the inequality which follows from a weak power condition ρ > γ > −
1. Thus theΛ-term depends on time t as follows: Λ = Λ /t , whereΛ = (1 − α ) (cid:20) (1 − γ ) γ ) − ǫ (1 − γ )6(1 + γ ) (9 − α ) − (cid:21) . The above mentioned solutions demonstrate the existence of exact solutions to the model, whichcan be of interest in aspect of modelling of processes occurring in the Universe. Let us investigatemore generally the properties of the masters equations for the spatially flat cosmological model(52) - (54). After simple manipulation, this set of equations can be reorganized as follows:˙ H + 3 H − − α ) t H − (1 − α )(2 − α ) t = t ˙Λ1 − α , (57)4 πGǫ ˙ φ = 3 H − − α )2 t H − − α )(2 − α )2 t − t ˙Λ1 − α , (58)8 πG V ( t ) + Λ = 3(7 − α )2 t H + 3(1 − α )(2 − α )2 t . (59)The term t ˙Λ1 − α in the right-hand side of equation (57) is obtained by dividing the both sides ofthis equality by (1 − α ). Therefore equation (57) is valid for all α ∈ (0; 1) but in the classical limit α → constant . The latter is the only consequence of α = 1, andamong three equations (57) - (59) it is necessary to solve only two of them with ˙Λ = 0, that allowsus to define one of the function among H ( t ) , φ ( t ) or V ( φ ) arbitrarily.The situation is essentially different for α = 1. Now we have three independent equations forfour functions: H ( t ) , φ ( t ) , V ( φ ) and Λ( t ), and specification of Λ( t ) allows us to find the dependence H ( t ) from equation (57). After that, it is possible to find φ ( t ) and V ( t ) from equations (58) and(59). It is important that the evolutionary, i.e. containing derivative over time, equation (57) forthe Hubble parameter experiences the influence of the parameter Λ( t ) only. Thus, the behaviourof the field and its potential becomes secondary, not determining an expansion dynamics of theUniverse directly, and can be simply found from the equations (58), (59). In some sense, the11calar field here plays a role of a latent parameter. Indeed, the system behaviour is determinedby the field behaviour but the Hubble parameter and the scale factor are determined not directlyby the scalar field but only through the Λ( t )-term. Using some phenomenological expression forthe cosmological parameter, it is possible to find the dependence of expansion on time, and thenthe scalar field that provided it. There is a certain sense in that, as we know practically nothingabout a scalar field that evoked cosmological inflation and the present accelerated expansion of theUniverse.We consider one example of the mentioned approach, assuming Λ = constant . In this case,equation (57) contains the only free parameter: α , and can be easily solved, that gives the Hubbleparameter as: H = 16 t (cid:18) − α − w α − c tw α c tw α (cid:19) , w α = p α − α + 105 , c > . (60)One can notice, that w α > α ∈ (0; 1). Integrating equation (60), we obtain the followingexpression for the scale factor: a = a n t (9 − α − w α ) / (cid:0) c tw α (cid:1)o / . (61)Substitution of the Hubble parameter from equation (60) in (58), (59) gives us expressions for φ ( t )and V ( t ). Due to the definition (42) and equations ( 58), (59) the energy density and the pressureof the scalar field (Λ = 0) are as follows: ρ = 196 πGt (cid:18) − α − w α − c tw α c tw α (cid:19) (cid:18) − α − w α − c tw α c tw α (cid:19) ,p = 196 πGt (cid:20)(cid:18) − α − w α − c tw α c tw α (cid:19) (cid:18) α − − w α − c tw α c tw α (cid:19) − − α )(2 − α ) (cid:21) . From these expressions, it follows that the barotropic index γ = ρ/p is not constant, and theequation of state changes with time. It is possible to show that at the initial moment of time, γ (0) strongly depends on the fractional order α , but at t → ∞ it practically loses this dependence,slowly approaching − / α → βt H , where β is apositive constant, then equation (ref 56) is reduced into the following one:(1 − α − β ) ˙ H + 3(1 − α ) H − [2(4 − α )(1 − α ) − β ] t H − (1 − α ) (2 − α ) t = 0 , differing from the same equation in the case β = 0, considered above, only by the constant multi-pliers. Therefore its solution is structurally similar to (60), (61). In our article, we have given the critical analysis of the main results and approaches known bynow that are directed towards the application in cosmology of the ideas of fractional differentiationand integration. 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