Cosmological Evolution and Exact Solutions in a Fourth-order Theory of Gravity
aa r X i v : . [ g r- q c ] M a r Cosmological Evolution and Exact Solutions in a Fourth-order Theory of Gravity
Andronikos Paliathanasis
1, 2, ∗ Instituto de Ciencias F´ısicas y Matem´aticas, Universidad Austral de Chile, Valdivia, Chile Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa (Dated: July 9, 2018)A fourth-order theory of gravity is considered which in terms of dynamics has the same degreesof freedom and number of constraints as those of scalar-tensor theories. In addition it admits acanonical point-like Lagrangian description. We study the critical points of the theory and weshow that it can describe the matter epoch of the universe and that two accelerated phases can berecovered one of which describes a de Sitter universe. Finally for some models exact solutions arepresented.
PACS numbers: 98.80.-k, 95.35.+d, 95.36.+xKeywords: Cosmology; Modified theories of gravity; Dynamical evolution; Exact Solutions
1. INTRODUCTION
The origin of the late-time acceleration phase of the universe is an unsolved puzzle in modern cosmology [1–4]. Theeffects of the late-time acceleration have been attributed to the so-called dark energy. As dark energy is characterizedas the matter source which provides the missing terms in the field equations of Einstein’s General Relativity andwhich leads to solutions that describe the accelerating expansion of the universe. The proposed solutions for thenature of dark energy can be categorized into two big classes: (i) the dark energy models where an energy momentumtensor, which describes an exotic matter source [5–13], is introduced into Einstein’s General Relativity and/or (ii) theEinstein-Hilbert action is modified such that the new field equations provide additional terms which are assumed tocontribute to the acceleration of the universe; for instance see [14–27]. In the second approach, the dark energy has ageometrical origin and description [28]. Some recent cosmological constraints of modified theories of gravity are givenin [29–33] while some solar system tests can be found in [34–37].A special class of modified theories of gravity which has drawn attention over recent years comprises the f ( X )theories of gravity where X is an invariant, for instance the Ricci Scalar, R , of the underlying space. In the latter casethe theory is the well-studied f ( R )-gravity [15] which in general is a fourth-order gravitational theory. Furthermore,when it is a second-order theory General Relativity is recovered because function f has to be a linear function.Though the functional form of the theory which describes the universe it is unknown, however, various (toy) modelshave been proposed in the literature in order to describe different phenomena, for a review see [16]. As we mentionedabove, f ( R )-gravity is a fourth-theory and the dependent variables of the gravitational field equations are the onewhich follow from the line element which defines the spacetime.Like every fourth-order differential equation which can be written as a system of a two second-order differentialequation, f ( R )-gravity can be written as a second-order theory by introducing a new degree of freedom. Thatnew degree of freedom it is equivalent with that of Brans-Dicke scalar field, with zero Brans-Dicke parameter. Theequivalence of a modified theory with a scalar field it is not true for all the f ( X )-theories, but always it depends uponthe nature of the invariant(s) X .Another f theory of special interest is the f ( T ) teleparallel gravity [19] in which T is related to the antisymmetricconnection used in the theory [38]. In the f ( T )-gravity General Relativity, and specifically the teleparallel equivalenceof General Relativity, is recovered when the function f is linear [39]. Because T admits terms with first derivatives, f ( T )-gravity provides a theory in which the gravitational field equations are of second-order. However, f ( T )-gravityin general provides different properties from General Relativity [40, 41]. Moreover there is not any scalar-field/scalartensor description like in f ( R )-gravity and in terms of dynamics the terms which gives the dark energy descriptioncan be seen as extra constraints on the dynamical system.The existence of other invariants in the Action Integral provides different components in the modified gravitationalfield equations. Some other theories which have been proposed are the modified R + f ( G ) Gauss-Bonnet gravity[42, 43], the more general f ( R, G ) Gauss Bonnet gravity [44, 45], the f (cid:0) R, T ( m ) (cid:1) -gravity, where T ( m ) is the trace ofthe energy momentum tensor [46], and many others. In this work we are interested in the so-called f ( R, T ) gravity ∗ Electronic address: [email protected] [47], where R is the Ricci Scalar of the underlying space and T the invariant of teleparallel gravity. That theoryis equivalent with the proposed f ( T, B ) theory [48], where B is the boundary term which relates T and R , andspecifically B = 2 e − ν ∂ ν (cid:0) eT ρνρ (cid:1) so that R = − T + B, where T βµν is the curvatureless Weitzenb¨ock connection.For the dynamics of the field equations it is easy to see that f ( R, T ) is a fourth-order theory of gravity, but newconstraints follow from the T terms. However, in the limit in which f ,T T = 0 the theory is reduced to that of f ( R )-gravity and has the same number of constraints as the Brans-Dicke theory. However, that is not the unique case inwhich the theory has the same number of constraints, in the dynamics, as that of scalar-tensor theories. As we seebelow that property exists and for the f ( R, T ) ≡ T + F ( R + T ) theory, or in the equivalent description, for the f ( T, B ) ≡ T + F ( B ) theory of gravity or f ( R, B ) = R + F ( B ). This is the toy-model that we study in this work.The plan of the paper is as follows.In Section 2 we define our cosmological model and with the use of a Lagrange multiplier we derive the gravitationalfield equations. In order to study the general evolution of the field equations in Section 3 we study the criticalpoints for an arbitrary function F ( R + T ). We find it is possible for the theory to provide two accelerated eras, onestable and one unstable, which can be related with the early acceleration phase (inflation) and the late accelerationphase. Moreover we see that there exists a critical point in which the scalar field mimics the additional perfect fluidthat we consider exists and that point can describe the matter-dominated epoch of the universe. Furthermore, someclosed-form analytical solutions are presented in Section 4 while in Section 5 we draw our conclusions and discuss ourresults.
2. THE FIELD EQUATIONS
The theory that we are interesting is a special form of f ( R, T )-gravity in which R is the Ricci Scalar of theunderlying space and T the the invariant of Weitzenb¨ock connection. The two quantities are related by the expression R = − T + B , where B = 2 e − ν ∂ ν (cid:0) eT ρνρ (cid:1) (1)is the boundary term. We follow the notation of [48] and we find that for f ( T, R + T ) = f ( T, B ) gravity thegravitational field equations are16 πGe T aλ = 2 eh λa ( f ,B ) ; µν g µν − eh σa ( f ,B ) ; λ ; σ + eBh λa f ,B + 4( eS aµλ ) ,µ f T + 4 e h ( f ,B ) ,µ + ( f ,T ) ,µ i S aµλ − ef ,T T σµa S σλµ − ef h λa , (2)where T ρν is the energy-momentum tensor of the matter source, a comma denotes partial derivative, “;” denotescovariant derivative and e i = h µi ( x ) ∂ i is the vierbein field which defines the Weitzenb¨ock connection, ˆΓ λµν = h λa ∂ µ h aν ,where T βµν = ˆΓ βνµ − ˆΓ βµν = h βi ( ∂ µ h aν − ∂ ν h aµ ) . Moreover S βµν = ( K µνβ + δ µβ T θνθ − δ νβ T θµθ ) and K µνβ is the cotorsiontensor given by the expression K µνβ = −
12 ( T µνβ − T νµβ − T βµν ) (3)and equals the difference between the Levi-Civita connections in the holonomic and the nonholonomic frame . Finally e = det( e iµ ) = √− g .For the gravitational field equations (2) it is easy to see that, when f ,BB = 0, the field equations reduce to those of f ( T ) teleparallel gravity while, as it has been mentioned in [48], for f ( T, B ) = f ( − T + B ), f ( R )-gravity is recovered.Last but not least in general the field equations (2) are of fourth-order.We assume that the geometry which describe the universe is that of a spatially flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) spacetime with line element ds = − N ( t ) dt + a ( t ) (cid:0) dx + dy + dz (cid:1) , (4)where a ( t ) is the scale factor and N ( t ) is the lapse function. Moreover we consider the diagonal frame for the vierbeinto be, h iµ ( t ) = diag ( N ( t ) , a ( t ) , a ( t ) , a ( t )) (5) For more details on the covariant formulation of teleparallel gravity we refer the reader to [49] from which we calculate that T = − N (cid:18) ˙ aa (cid:19) , B = − N ¨ aa + 2 ˙ a a − ˙ a ˙ NaN ! . (6)For that frame and for the comoving observer, u λ = N − δ t , (cid:0) u λ u λ = − (cid:1) , the gravitational field equations are f − a ˙ f ,B aN + 6 f T ˙ a a N + 3 f ,B N ¨ aa − ˙ a ˙ NaN + 2 (cid:18) ˙ aa (cid:19) ! = ρ (7)and f a ˙ f ,T aN + (3 f ,B + 2 f ,T ) N ¨ aa − ˙ a ˙ NaN + 2 (cid:18) ˙ aa (cid:19) ! − ¨ f ,B N + ˙ f ,B ˙ NN = − p (8)where overdot denotes total derivative with respect to t and { ρ, p } are the energy density, ρ = T µν u µ u ν and thepressure p = T µν ( g µν + u µ u ν ) of the matter source. As we have already mentioned, f ( T, B )-gravity is a fourth-order theory. However, as in the case of f ( R )-gravityLagrange multipliers can be introduced in order to reduce the order of the differential equations. However, the lattermeans that the degrees of freedom are increased. Therefore from the definition of T and B , that is expression (6),and with the introduction of the Lagrange multipliers, λ and λ , the gravitational Action Integral becomes A = Z dt " f N a − λ T + 6 (cid:18) ˙ aN a (cid:19) ! − λ B + 6 N ¨ aa + 2 ˙ a a − ˙ a ˙ NaN !! , (9)where for simplicity we have assumed the vacuum case.Variation of the Action Integral above with respect to the variables, T and B , provides the definition of λ and λ from the expression δAδT = 0 , δAδB = 0 . Therefore we find that λ = N a f ,T and λ = N a f ,B . Hence the gravitational action becomes A = Z dt " ( f N a − N a f ,T T + 6 (cid:18) ˙ aN a (cid:19) ! − N a f ,B B + 6 N ¨ aa + 2 ˙ a a − ˙ a ˙ NaN !! , (10)from which by integration by parts we find the Lagrangian of the field equations to be L f ( T,B ) = − N a ˙ a f ,T + 6 N a ˙ a ˙ f ,B + N a ( f − T f ,T − Bf ,B ) . (11)Finally the field equations are given from the Euler-Lagrange equations of (11) with respect to the variables { N, a, T, B } , where ∂L∂N = 0, is the constraint equation.Without loss of generality we can assume the lapse function to be N ( t ) = 1. We define the new variable φ = f ,B .Thus the Lagrangian (11) takes the simpler form L f ( T,B ) = − a ˙ a f T + 6 a ˙ a ˙ φ − a V ( φ, T ) , (12)where now V ( φ, T ) = T f ,T + Bf ,B − f ( T, B ) . (13)The field equations in f ( T, B )-gravity are in general of fourth-order, except when f B is constant. By introducingthe field φ , the Lagrangian (12) describes the evolution of a dynamical system in the space of variables { a, φ, T } while,when f = f ( T − B ), we see that f ,T = − f ,B = − φ , which means that the Lagrangian of O’Hanlon gravity [50] isrecovered.Furthermore it is easy to see that (12) is a singular Lagrangian when f T T = 0, as the Lagrangian of the fieldequations is in f ( T )-gravity. f ( T, B ) = T + F ( B ) Inspired from the other modified theories of gravity, specifically from f ( R ), for which models of the form f ( R ) = R + F ( R ), or in f ( T ) with f ( T ) = T + F ( T ) have been proposed [19], here we select to work with the theory f ( T, B ) = T + F ( B ), which is exactly equivalent to the theories f ( R, B ) = R + F ( B ) or f ( R, T ) = T + F ( T + R ).The main characteristic of that selection is that the Lagrangian of the field equations (12) is a regular Lagrangianin the space of variables { a, φ } , as also is independent of T. Moreover for small values of the function, F ( B ), we arein a small deviation from General Relativity while, for F ,B ( B ) = 0, General Relativity is recovered.In our cosmological scenario we consider a perfect fluid with constant equation of state parameter p m = wρ m . For f = T + F ( B ) the gravitational field equations are3 H − H ˙ φ − V ( φ ) = ρ m , (14)˙ H + 3 H + 16 V ,φ = 0 , (15)and ¨ φ + 3 H + 12 V + 13 V ,φ − p m = 0 . (16)Moreover, we assume that the there is not any interaction in the Action integral of the matter source with thegravitation terms the Bianchi identity provides the conservation equation ˙ ρ m + 3 ( ρ m + p m ) H = 0 . (17)We observe that (14) is the constrain equation (7), while equation (15) describes the evolution of the Hubblefunction. The third equation (16) is the “Klein-Gordon” (-like) equation for the field φ which with the use of (15)gives the fourth-order equation. However while the fourth-order f ( R )-gravity is equivalent with a Brans-Dicke scalarfield, and specifically with the O’Hanlon theory [50], that is not true for our model where indeed the higher-orderderivatives are describing by the field φ , but it is not a canonical field. On the other hand the field equations are moreclose to that of a particle in the Generalized Uncertainty principle [13], where as the position of the particle now weconsider that of the scale factor a ( t ).In the following sections we study the general evolution of the field equations (14)-(17) and we search for analyticalsolutions of the field equations for specific forms of V ( φ ). Recall that now the partial differential equation (13) hasbeen reduced to the Clairaut first-order differential equation V ( F B ) = BF ,B − F ( B ) . (18)The latter has always two solutions, the linear solution F S ( B ) = F B + F ( F ), for arbitrary potential V ( φ ), asalso a singular solution which is given from the solution of the second-order differential equation dV ( F B ) d ( F B ) − B = 0 . Thelatter solution is the one in which we are interested, because for the linear solution F S we are in General Relativityin which F ( F ) plays the role of the cosmological constant.We continue with the study of the dynamics of the field equations (14)-(17). The Action Integral of the O’Hanlon theory it coincides with that of Brans-Dicke theory for zero Brans-Dicke parameter. However, thetheory has been introduced in order to produce a Yukawa type interaction in the gravitational potential [51]. The lhs of equations (14)-(16) can be calculated easily from (7)-(8) by assuming φ = f ,B , or from the action of the Euler-Lagrangeoperator on the Lagrangian (13).
3. COSMOLOGICAL EVOLUTION
In order to perform our analysis we assume that the matter source, ρ m , p m , is a perfect fluid with constant equationof state parameter w m , i.e., p m = w m ρ m . We define the new dimensionless variables x = ˙ φH , y = 16 V ( φ ) H , Ω m = ρ m H (19)in analogue to the scalar tensor theories [52–55]. Equation (14) provides us with the constraint equationΩ m = 1 − x − y (20)which holds for value of ( x, y ) where 0 ≤ Ω m ≤ N = ln a , and now in the new variables the field equations form the following system offirst-order differential equations dxdN = − y − x ) + λy (2 − x ) + 3 w m Ω m , (21) dydN = (6 − λ (2 y + x )) y (22)and dλdN = − xλ ¯Γ ( λ ) , (23)where λ = − V ,φ V , ¯Γ ( λ ) = V ,φφ ( V ,φ ) − . (24)Finally the equation of state parameter for the total fluid, w tot = − −
23 ˙ HH , is expressed as a function of x, y, λ asfollows w tot = 1 − λy. (25)We study the general evolution of the system (21)-(23) for the two cases: (a) λ = const , which means ¯Γ ( λ ) = 0,i.e. V ( φ ) = V e − λφ and (b) λ = 0. In case (a) the system (21)-(23) reduces to a two-dimensional system. Consider now that V ( φ ) = V e − λφ , which corresponds to the function F ( B ) = − Bλ (cid:18) ln (cid:18) − Bλ (cid:19) − (cid:19) . (26)For that potential, the fixed points of the dynamical system, (21)-(22), are the points: P A where ( x A , y A ) = (1 , P B with ( x B , y B ) = (cid:0) λ (1 + w ) , λ (1 − w ) (cid:1) and P C with coordinates ( x C , y C ) = (cid:0) − λ + 2 , λ − (cid:1) . Specifically foreach point we have: • Point P A corresponds to a universe without matter source; Ω m = 0 and the total equation of state parameteris w tot = 1, that is, the scalar field behaves like a stiff matter. The eigenvalues of the linearized system arecalculated to be e A = 3 (1 − w m ) , e A = 6 − λ. (27)For the range w m ∈ [ − , e A is always positive. Hence the point is unstable. TABLE I: Fixed points, cosmological parameters and stability for the dynamical system (23)-(23) with exponential potential
Point ( x , y ) Existence Ω m w tot Acceleration Stability P A (1 , λ ∈ R P B (cid:0) λ (1 + w m ) , λ (1 − w m ) (cid:1) λ ≥ (9+3 w )2 − w m )2 λ w m w m < − λ > λ B , P C (cid:0) − λ + 2 , λ − (cid:1) λ ∈ R ∗ (2 λ − λ < λ < (3 + w ) x -1.5 -1 -0.5 0 0.5 1 y -1-0.500.511.52 de Sitter Universe Ω m0 =1 Ω m0 =0 Phase Portrait for exponential potential λ =3 , w m =0 FIG. 1: Phase portrait for the potential V ( φ ) = exp ( − λφ ) with λ = − w m = 0. The stable point is a de Sitter point.The left and right thick lines y = 1 − x , and y = − x , are the borders in which 0 ≤ Ω m ≤ . The different lines describe differentinitial conditions ( x , y ). • At point P B the physical quantities are calculated to be Ω m = 1 − w m )2 λ and w tot = w m . The point is welldefined for λ ≥ (9+3 w )2 . In that point the field φ mimics the matter source as the analogy of the exponentialmodel in the minimally coupled cosmological scenario, which describes an accelerated universe for w m < − .The two eigenvalues of the linearized system are e B = 14 (cid:16) − − w m ) − √ p (1 − w m ) (75 + 21 w m − λ ) (cid:17) , (28) e B = 14 (cid:16) − − w m ) + √ p (1 − w m ) (75 + 21 w m − λ ) (cid:17) . (29)For values of w m in the range w m ∈ [ − , Re (cid:0) e B (cid:1) < λ ≥ (25 + 7 w m ) , λ B = (25 + 7 w m ), Re (cid:0) e B (cid:1) = Re (cid:0) e B (cid:1) < λ < λ B , the point is stable when λ B < λ < λ B in which λ B = (3 + w m ). Hence we conclude for λ > λ B the point P B is always stable. • Point P C describes a universe dominated by the field φ , where Ω m = 0 and equation of state for the total fluidis w tot = (2 λ − λ < and describes a de Sitter universe for λ = 3 . The eigenvalues of the linearized system are e C = − λ , e C = − − w m + 2 λ (30)which gives that point is stable when λ < λ < (3 + w m ). Furthermore if we consider that w m ∈ [0 , t w t o t ( z ) -1.2-1-0.8-0.6-0.4-0.200.20.40.6 Qualitative behavior of w tot for λ =3 FIG. 2: Qualitative evolution of the total equation of state parameter w tot for different initial conditions for the potential V ( φ ) = exp ( − λφ ) with λ = 3 and w m = 0. The solid line is for initial conditions ( x , y ) = (0 . , . , the dash-dash line for( x , y ) = (0 . , . , the dot-dot line for ( x , y ) = (0 . , .
3) and the dash-dot line for initial conditions ( x , y ) = (0 . , . . Consider now a general potential V ( φ ), which corresponds to a general function F ( B ) and in a general function¯Γ ( λ ). Now, if there exists a value λ = λ ∗ such as ¯Γ ( λ ) = 0, then from (21)-(22) we find the fixed points ¯ P A , ¯ P B and¯ P C . The cosmological variables are the same as those of Table I. However, the stability analysis is different. Thereare two more possibilities for which the system (21)-(23) admit stationary points, x = 0, or λ = 0, with λ ¯Γ ( λ ) welldefined.For x = 0, we find that the rhs of (21)-(22) vanishes at the point P D with coordinates ( x, y, λ ) = (0 , ,
3) while for λ = 0 the fixed points are P E , ( x, y, λ ) = (1 , , P A for λ = 0. Asfar as concerns the physical quantities at the point P D we have that Ω m = 0 and w tot = −
1, which means that P D isa de Sitter point. The stability of the points it follows • The eigenvalues of the linearized system around the point ¯ P A are¯ e A = 3 (1 − w m ) , ¯ e A = 6 − λ , ¯ e A = − λ ¯Γ ,λ ( λ ) (31)from which we can see that ¯ e A is always positive for w m ∈ [ − , • For the point ¯ P B the eigenvalues of the linearized system are¯ e B = 14 h − − w m ) − p B i , e ¯ B = 14 h − − w m ) + p B i (32)and ¯ e B = − w m ) λ ¯Γ ,λ ( λ ) , (33)where ∆ B = (1 − w ) (75 + 21 w m − λ ) . (34)Eigenvalue e B is negative only when λ ¯Γ ,λ ( λ ) >
0. Now, for ∆ B ≤ Re (cid:0) ¯ e B (cid:1) = Re (cid:0) ¯ e B (cid:1) < B >
0, ¯ e B < e B e B >
0, that gives 32 (1 − w m ) (9 + 3 w m − λ ) < , (35)from which we find that ¯ P B is stable when λ > (3 + w m ). TABLE II: Fixed points and cosmological parameters for the dynamical system (23)-(23) with arbitrary potential
Point ( x , y , λ ) Existence Ω m w tot Acceleration ¯P A (1 , , λ ) λ ∈ R , ¯Γ ( λ ) = 0 0 +1 No ¯P B (cid:0) λ (1 + w m ) , λ (1 − w m ) , λ (cid:1) λ ≥ (9+3 w )2 , ¯Γ ( λ ) = 0 1 − w m )2 λ w m w m < − P C (cid:0) − λ + 2 , λ − , λ (cid:1) λ ∈ R ∗ , ¯Γ ( λ ) = 0 0 (2 λ − λ ∗ < P D (0 , ,
3) Always 0 − P E (1 , ,
0) Always 0 +1 NoTABLE III: Eigenvalues and stability for the critical points of the dynamical system (23)-(23) with arbitrary potential
Point/Eigenv. e e e Stability ¯P A − w m ) 6 − λ − λ ¯Γ ,λ ( λ ) Unstable ¯P B (cid:2) − − w m ) − √ B (cid:3) (cid:2) − − w m ) + √ B (cid:3) − w m ) λ ¯Γ ,λ ( λ ) λ > (3 + w m ) P C − λ − − w m + 2 λ − − λ ) ¯Γ ,λ ( λ ) λ < (3 + w m ) , ( − λ ) ¯Γ ,λ ( λ ) > P D (cid:16) − − p − (cid:17) (cid:16) − p − (cid:17) − w m ) Re (cid:0) ¯Γ [3] (cid:1) > P E − w m ) 6 0 Unstable • For the point ¯ P C we find the eigenvalues¯ e C = − λ , ¯ e C = − − w m + 2 λ , ¯ e C = − − λ ) ¯Γ ,λ ( λ ) (36)from which the point is stable when λ < (3 + w m ) and ( − λ ) ¯Γ ,λ ( λ ) > • At the point P D the matrix of the linearized system has the following eigenvalues e D = 32 (cid:18) − − q − (cid:19) , e D = 32 (cid:18) − q − (cid:19) , e C = − w m ) (37)which is a stable de Sitter point when Re (cid:0) ¯Γ [3] (cid:1) >
0. Note that P D is a special point of P C when λ = 3.However, the eigenvalues of the linearized system are different. That means that in a model with running λ ,the two points P C and P D can exist. • Finally the last point P E provides always a positive eigenvalue, that is, the point is always unstable.We conclude that for a general potential a second stable de Sitter point exists which is stable for potentials in which¯Γ (3) >
0. In general two de Sitter phases are possible, the points P C and P D . The above results are collected inTables II and III.As a special example consider the potential V ( φ ) = V e − σφ + V , from which we have F ( B ) = − Bλ (cid:18) ln (cid:18) − Bλ (cid:19) − (cid:19) + V . (38)For that potential we have that φ = − σ ln (cid:16) λ ¯ V σ − λ (cid:17) , ¯ V = V V and ¯Γ ( λ ) = − − V (cid:0) − σλ (cid:1) . For the point P C wefind that λ = σ
1+ ¯ V . Hence P C is stable when λ <
32 (3 + w m ) and − ( − λ ) σλ ¯ V > . (39)Hence, if 3 < λ < (3 + w m ), then σ ¯ V < λ < σV >
0. On the other handpoint P D is stable when σ < − (cid:0) V (cid:1) for ¯ V > , (40) σ > − (cid:0) V (cid:1) for ¯ V < . (41)In the following section we proceed with the derivation of some analytical solutions for the field equations (14)-(17).
4. EXACT COSMOLOGICAL SOLUTIONS
We consider that in the field equations (14)-(17) the matter source corresponds to that of a dust fluid, i.e. w m = 0,and p m = 0. Hence (17) provides ρ m = ρ m a − . Furthermore for the potential, V ( φ ), we consider that V ( φ ) = V exp ( − φ ), which leads to a de Sitter universe and V ( φ ) = V exp ( − φ ) − , . According to the above this hastwo de Sitter phases, points P C and P D . V ( φ ) = V exp ( − φ ) For the potential V ( φ ) the Lagrangian of the field equations becomes L (cid:16) a, ˙ a, φ, ˙ φ (cid:17) = − a ˙ a + 6 a ˙ a ˙ φ − a V e − φ (42)so that the field equations are the Euler-Lagrange equations of (42) with respect to the variables { a, φ } , while thefirst modified Friedmann’s equations can be seen as the Hamiltonian function of (42), H = E, where now ρ m = 2 | E | . It is straightforward to see that (42) admits the two extra Noetherian conservation law which are I = ˙ φ − ˙ aa and I = t (cid:18) ˙ φ − ˙ aa (cid:19) − ( φ − ln a ) . (43)We perform the coordinate transformation a = u , φ = v − ln ( u ). In the new coordinates Lagrangian (42) iswritten L ( u, ˙ u, v, ˙ v ) = 2 ˙ u ˙ v − V e − v (44)and the field equations are taking the simple form2 ˙ u ˙ v + V e − v = 2 E, (45)¨ u − V e − v = 0 and ¨ v = 0 . (46)Finally the solution is given in a closed-form expression as follows u ( t ) = ¯ V v e − v t + u t + u , (47)where ¯ V = V e − v and E = u v . We have that the de Sitter phase is recovered when v <
0. From (47) it followsthat the scale factor has the form a ( t ) = a e βt + a t + a , (48)where the spacetime has a singularity at t = 0 when a = − a , that is, the scale factor becomes a ( t ) = a (cid:0) e βt − (cid:1) + a t .Moreover, in the vacuum solution in which u v = 0, we have two possibilities: u = 0 or v = 0. For the lattercase, that is β = 0, the solution is a ( t ) ≃ t , which corresponds to the solution of GR with a perfect fluid withequation of state parameter w = − .However, in the latter case for which u = 0 i.e., a = 0, the scale factor with a ( t →
0) = 0 is of the form a ( t ) = a (cid:0) e βt − (cid:1) . (49)Easily we have that t = β ln (cid:16) a a (cid:17) , from which we calculate the Hubble Function( H ( a )) = β β a a − + β a ) a − . (50)This means that the theory provides us with a cosmological constant term, a dust term and a stiff fluid, equivalentlywith that of the minimally coupled scalar field [60]. For the application of point symmetries in cosmological studies see [56–58] and references therein while a partial classification of Noetherpoint symmetries in f ( T, B ) can be found in [59] V ( φ ) = V exp ( − φ ) − As a second potential we consider the same as the above where now we include a cosmological constant term. It iseasy to see that in the coordinate system { u, v } the field equations become2 ˙ u ˙ v + V e − v − u = 2 E (51)and ¨ u − V e − v = 0 , ¨ v − Λ = 0 , (52)from which we have that the scale factor is expressed in terms of the error function, E ( t ), as a ( t ) = ¯ V
3Λ exp (cid:18) −
32 Λ t − v t (cid:19) + (53)+ √ π ¯ V (Λ t + v ) E √
62Λ (Λ t + v ) ! + u t + u , (54)where ¯ V = V e − v and v ( t ) = Λ2 t + v t + v .The reason that this is possible is that the Lagrangian of the field equations admits a Noetherian conservation lawwhich is not generated by point symmetries as in the potential V ( φ ) but from generalized symmetries. In particularthe Killing tensor of the minisuperspace provides a contact symmetry (see [61] and references therein).
5. CONCLUSIONS
In the context of modified theory of gravities we considered a gravitational theory in which the deviation fromGeneral Relativity is given by a function of the boundary term which relates the Ricci Scalar, R , and the invariant, T , of teleparallel gravity. The theory that we considered is a fourth-order theory and in the case of an isotropic andhomogeneous universe the field equations can be written as a (constraint) Hamiltonian system with two degrees offreedom. One degree of freedom corresponds to the scalar factor of the geometry and the second one is a field whichdescribes the higher-order derivatives, as in the case of f ( R )-gravity. The theory admits a constraint and it is theequation of motion which corresponds to the lapse function of the geometry.Though that theory is a fourth-order theory differs from f ( R )-gravity and the field which is introduced from theapplication of the Lagrange multipliers does not describe a scalar tensor theory. The minisuperspace Lagrangian isgiven by L (cid:16) a, ˙ a, φ, ˙ φ (cid:17) = − N a ˙ a + 6 N a ˙ a ˙ φ + N a V ( φ ) . (55)However, under the change a = Ae φ and N = e φ n , this Lagrangian becomes L (cid:16) A, ˙ A, φ, ˙ φ (cid:17) = − n A ˙ A + 32 n A ˙ φ + na (cid:0) e φ V ( φ ) (cid:1) (56)which is the Lagrangian of a canonical minimally coupled (phantom) scalar field with potential U ( φ ) = e φ V ( φ ). Itis easy to see that the transformation ( N, a ) → (cid:16) e φ n, Ae φ (cid:17) does not relate conformal equivalent theories, such as inthe scalar-tensor theories. However, the relation between the two Lagrangians, (55) and (56), is important becausethe analysis of [62] can be applied and it can be easily shown that the gravitational field equations (14)-(16) form anintegrable dynamical system.In order to study the effects which follow from the new terms in the dynamics of the field equations the criticalpoints were calculated. Every point corresponds to a physical state and the physical parameters were calculated. Theimportance of the existence of the points is that for families of initial conditions the evolution of the universe passesclosely to the physical states which are described from the points (unstable points) or at the end reach the solutionwhich is described by the critical point (stable point). In our analysis we found that for it is possible to have a theorywhich provides a matter era (unstable point) and two acceleration phases in which the one can be stable and the1other unstable. This is an interesting result and it is different from that of f ( R )-gravity. Moreover some closed-formsolutions were derived and the explicitly form of the FLRW spacetime was found.There are various open questions which have to be answered for that consideration, but the property that the onlyconstraint in the field equations is that of the “Hamiltonian” is essential because various methods can be applied,from the scalar field description, in order to study the theory. In a future work we would like to extend the presentanalysis in order to search for other kinds of cosmological solutions and extend the analysis of the critical points atthe infinite region. The existence of static-spherical solution is also of special interests. Acknowledgments
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