Modified Gravity in Space-Time with Curvature and Torsion: F(R, T) Gravity

Transcription

1 Modified Gravity in Space-Time with Curvature and Torsion: FR, T) Gravity Ratbay Myrzakulov Eurasian International Center for Theoretical Physics and Department of General & Theoretical Physics, Eurasian National University, Astana , Kazakhstan Abstract In this report, we consider a theory of gravity with a metric-dependent torsion namely the FR, T) gravity, where R is the curvature scalar and T is the torsion scalar. We study the geometric root of such theory. In particular we give the derivation of the model from the geometrical point of view. Then we present the more general form of FR, T) gravity with two arbitrary functions and give some of its particular cases. In particular, the usual FR) and FT) gravity theories are particular cases of the FR, T) gravity. In the cosmological context, we find that our new gravitational theory can describe the accelerated expansion of the Universe. Contents 1 Introduction 1 2 Preliminaries of FR), FG) and FT) gravities FR) gravity FT) gravity FG) gravity Geometrical roots of FR, T) gravity General case FRW case Lagrangian formulation of FR, T) gravity 7 5 Cosmological solutions Example Example Conclusion 12 1 Introduction In the last years the interest in modified gravity theories like FR) and FG)-gravity as alternatives to the ΛCDM Model grew up. Recently, a new modified gravity theory, namely the FT)-theory, has been proposed. This is a generalized version of the teleparallel gravity originally proposed by Einstein [13]-[24]. It also may describe the current cosmic acceleration without invoking dark energy. Unlike the framework of GR, where the Levi-Civita connection is used, in teleparallel gravity TG) the used connection is the Weitzenböck one. In principle, modification of gravity may contain a huge list of invariants and there is not any reason to restrict the gravitational rmyrzakulov@gmail.com; rmyrzakulov@csufresno.edu 1

2 theory to GR, TG, FR) gravity and/or FT) gravity. Indeed, several generalizations of these theories have been proposed see e.g. the quite recent review [9]). In this paper, we study some other generalizations of FR) and FT) gravity theories. At the beginning, we briefly review the formalism of FR) gravity and FT) gravity in Friedmann-Robertson-Walker FRW) universe. The flat FRW space-time is described by the metric ds 2 = dt 2 + a 2 t)dx 2 + dy 2 + dz 2 ), 1.1) where a = at) is the scale factor. The orthonormal tetrad components e i x µ ) are related to the metric through g µν = η ij e i µe j ν, 1.2) where the Latin indices i, j run over for the tangent space of the manifold, while the Greek letters µ, ν are the coordinate indices on the manifold, also running over FR) and FT) modified theories of gravity have been extensively explored and the possibility to construct viable models in their frameworks has been carefully analyzed in several papers see [9] for a recent review). For such theories, the physical motivations are principally related to the possibility to reach a more realistic representation of the gravitational fields near curvature singularities and to create some first order approximation for the quantum theory of gravitational fields. Recently, it has been registred a renaissance of FR) and FT) gravity theories in the attempt to explain the late-time accelerated expansion of the Universe [9]. PROBLEM: Construct such FR,T) gravity which contents FR) gravity and FT) gravity as particular cases. In the modern cosmology, in order to construct generalized) gravity theories, three quantities the curvature scalar, the Gauss Bonnet scalar and the torsion scalar are usually used about our notations see below): R s = g µν R µν, 1.3) G s = R 2 4R µν R µν + R µνρσ R µνρσ, 1.4) T s = S ρ µν T ρ µν. 1.5) In this paper, our aim is to replace these quantities with the other three variables in the form R = u + g µν R µν, 1.6) G = w + R 2 4R µν R µν + R µνρσ R µνρσ, 1.7) T = v + S ρ µν T ρ µν, 1.8) where u = ux i ;g ij, g ij, g ij,...;f j ), v = vx i ;g ij, g ij, g ij,...;g j ) and w = wx i ;g ij, g ij, g ij,...;h j ) are some functions to be defined. As a result, we obtain some generalizations of the known modified gravity theories. With the FRW metric ansatz the three variables 1.3)-1.5) become R s = 6Ḣ + 2H2 ), 1.9) G s = 24H 2 Ḣ + H2 ), 1.10) T s = 6H 2, 1.11) where H = lna) t. In the contrast, in this paper we will use the following three variables R = u + 6Ḣ + 2H2 ), 1.12) G = w + 24H 2 Ḣ + H2 ), 1.13) T = v 6H ) Finally we would like to note that we expect w = wu,v). This paper is organized as follows. In Sec. 2, we briefly review the formalism of FR) and FT)-gravity for FRW metric. In particular, 2

3 the corresponding Lagrangians are explicitly presented. In Sec. 3, we consider FR, T) theory, where R and T will be generalized with respect to the usual notions of curvature scalar and torsion scalar. Some reductions of FR, T) gravity are presented in Sec. 4. In Sec. 5, the specific model FR,T) = µr + νt is analized and in Sec. 6 the exact power-law solution is found; some cosmological implications of the model will be here discussed. The Bianchi type I version of FR,T) gravity is considered in Sec. 7. Sec. 8 is devoted to some generalizations of some modified gravity theories. Final conclusions and remarks are provided in Sec Preliminaries of FR), FG) and FT) gravities At the beginning, we present the basic equations of FR), FT) and FG) modified gravity theories. For simplicity we mainly work in the FRW spacetime. 2.1 FR) gravity The action of FR) theory is given by S R = d 4 xe[fr) + L m ], 2.1) where R is the curvature scalar. We work with the FRW metric 1.1). In this case R assumes the form R = R s = 6Ḣ + 2H2 ). 2.2) The action we rewrite as where the Lagrangian is given by S R = dtl R, 2.3) L R = a 3 F RF R ) 6F R aȧ 2 6F RR Ṙa 2 ȧ a 3 L m. 2.4) The corresponding field equations of FR) gravity read 2.2 FT) gravity 6ṘHF RR R 6H 2 )F R + F = ρ, 2.5) 2Ṙ2 F RRR + [ 4ṘH 2 R]F RR + [ 2H 2 4a 1 ä + R]F R F = p, 2.6) ρ + 3Hρ + p) = ) In the modified teleparallel gravity, the gravitational action is S T = d 4 xe[ft) + L m ], 2.8) where e = det e i µ) = g, and for convenience we use the units 16πG = = c = 1 throughout. The torsion scalar T is defined as T S ρ µν T ρ µν, 2.9) where T ρ µν e ρ i µ e i ν ν e i ) µ, 2.10) K µν ρ 1 2 T µν ρ T νµ ρ T ρ µν ), 2.11) S ρ µν 1 2 K µν ρ + δ ρt µ θν θ δρt ν θµ ) θ. 2.12) For a spatially flat FRW metric 1.1), as a consequence of equations 3.9) and 1.1), we have that the torsion scalar assumes the form T = T s = 6H ) 3

4 The action 3.8) can be written as where the point-like Lagrangian reads S T = dtl T, 2.14) The equations of FT) gravity look like L T = a 3 F F T T) 6F T aȧ 2 a 3 L m. 2.15) 12H 2 F T + F = ρ, 2.16) ) 48H 2 F TT Ḣ F T 12H 2 + 4Ḣ F = p, 2.17) ρ + 3Hρ + p) = ) 2.3 FG) gravity The action of FG) theory is given by S G = d 4 xe[fg) + L m ], 2.19) where the Gauss Bonnet scalar G for the FRW metric is G = G s = 24H 2 Ḣ + H2 ). 2.20) 3 Geometrical roots of FR, T) gravity We start from the M 43 - model about our notations, see e.g. Refs. [10]-[11]). This model is one of the representatives of FR,T) gravity. The action of the M 43 - model reads as S 43 = d 4 x g[fr,t) + L m ], R = R s = ɛ 1 g µν R µν, 3.1) T = T s = ɛ 2 S ρ µν T ρ µν, where L m is the matter Lagrangian, ɛ i = ±1 signature), R is the curvature scalar, T is the torsion scalar about our notation see below). In this section we try to give one of the possible geometric formulations of M 43 - model. Note that we have different cases related with the signature: 1) ɛ 1 = 1,ɛ 2 = 1; 2) ɛ 1 = 1,ɛ 2 = 1; 3) ɛ 1 = 1,ɛ 2 = 1; 4) ɛ 1 = 1,ɛ 2 = 1. Also note that M 43 - model is a particular case of M 37 - model having the form S 37 = d 4 x g[fr,t) + L m ], where R = u + R s = u + ɛ 1 g µν R µν, 3.2) T = v + T s = v + ɛ 2 S ρ µν T ρ µν, are the standard forms of the curvature and torsion scalars. 3.1 General case R s = ɛ 1 g µν R µν, T s = ɛ 2 S ρ µν T ρ µν 3.3) To understand the geometry of the M 43 - model, we consider some spacetime with the curvature and torsion so that its connection G λ µν is a sum of the curvature and torsion parts. In this paper, the Greek alphabets µ, ν, ρ,... = 0,1,2,3) are related to spacetime, and the Latin alphabets 4

5 i,j,k,... = 0,1,2,3) denote indices, which are raised and lowered with the Minkowski metric η ij = diag 1,+1,+1,+1). For our spacetime the connection G λ µν has the form G λ µν = e i λ µ e i ν + e j λ e i νω j iµ = Γ λ µν + K λ µν. 3.4) Here Γ j iµ is the Levi-Civita connection and Kj iµ is the contorsion. Let the metric has the form ds 2 = g ij dx i dx j. 3.5) Then the orthonormal tetrad components e i x µ ) are related to the metric through so that the orthonormal condition reads as Here η ij = diag 1,1,1,1), where we used the relation The quantities Γ j iµ and Kj iµ are defined as g µν = η ij e i µe j ν, 3.6) η ij = g µν e µ i eν j. 3.7) e i µe µ j = δi j. 3.8) Γ l jk = 1 2 glr { k g rj + j g rk r g jk } 3.9) and K λ µν = 1 2 T λ µν + T µν λ + T νµ λ ), 3.10) respectively. Here the components of the torsion tensor are given by The curvature R ρ σµν is defined as T λ µν = e i λ T i µν = G λ µν G λ νµ, 3.11) T i µν = µ e i ν ν e i µ + G i jµe j ν G i jνe j µ. 3.12) R ρ σµν = e i ρ e j σr i jµν = µ G ρ σν ν G ρ σµ + G ρ λµg λ σν G ρ λνg λ σµ = R ρ σµν + µ K ρ σν ν K ρ σµ + K ρ λµk λ σν K ρ λνk λ σµ +Γ ρ λµ Kλ σν Γ ρ λν Kλ σµ + Γ λ σνk ρ λµ Γ λ σµk ρ λν, 3.13) where the Riemann curvature of the Levi-Civita connection is defined in the standard way R ρ σµν = µ Γ ρ σν ν Γ ρ σµ + Γ ρ λµ Γλ σν Γ ρ λν Γλ σµ. 3.14) Now we introduce two important quantities namely the curvature R) and torsion T) scalars as R = g ij R ij, 3.15) T = S ρ µν Tµν, ρ 3.16) where S ρ µν = 1 2 Kρ µν + δ µ ρt θ θν δ ν ρt θ θµ ). 3.17) Then the M 43 - model we write in the form 3.1). To conclude this subsection, we note that in GR, it is postulated that T λ µν = 0 and such 4-dimensional spacetime manifolds with metric and without torsion are labelled as V 4. At the same time, it is a general convention to call U 4, the manifolds endowed with metric and torsion. 5

6 3.2 FRW case From here we work with the spatially flat FRW metric ds 2 = dt 2 + a 2 t)dx 2 + dy 2 + dz 2 ), 3.18) where at) is the scale factor. In this case, the non-vanishing components of the Levi-Civita connection are Γ 0 00 = Γ 0 0i = Γ 0 i0 = Γ i 00 = Γ i jk = 0, Γ 0 ij = a 2 Hδ ij, 3.19) Γ i jo = Γ i 0j = Hδ i j, where H = lna) t and i,j,k,... = 1,2,3. Now let us calculate the components of torsion tensor. Its non-vanishing components are given by: T 110 = T 220 = T 330 = a 2 h, T 123 = T 231 = T 312 = 2a 3 f, 3.20) where h and f are some real functions see e.g. Refs. [12]). Note that the indices of the torsion tensor are raised and lowered with respect to the metric, that is Now we can find the contortion components. We get T ijk = g kl T ij l. 3.21) K 1 10 = K 2 20 = K 3 30 = 0, K 1 01 = K 2 02 = K 3 03 = h, K 0 11 = K 0 22 = K 0 22 = a 2 h, 3.22) K 1 23 K 1 32 = K 2 31 = K 3 12 = af, = K 2 13 = K 3 21 = af. The non-vanishing components of the curvature R ρ σµν are given by R = R = R = a 2 Ḣ + H2 + Hh + ḣ), R = R = R = 2a 3 fh + h), R = R = R = ahf + f), R = R = R = a 2 [H + h) 2 f 2 ]. 3.23) Similarly, we write the non-vanishing components of the Ricci curvature tensor R µν as R 00 = 3Ḣ 3ḣ 3H2 3Hh, R 11 = R 22 = R 33 = a 2 Ḣ + ḣ + 3H2 + 5Hh + 2h 2 f 2 ). 3.24) At the same time, the non-vanishing components of the tensor S µν ρ are given by S1 10 = 1 K δ 1 2 1Tθ θ0 S 10 1 = S 20 2 = S 30 S 23 1 = 1 2 δ1t 0 θ θν ) 1 = h + 2h) = h, 3.25) 2 3 = 2h, 3.26) K δ δ 3 1 S 23 1 = S 31 2 = S 21 3 = f 2a ) = f 2a, 3.27) 3.28) and T = T 1 10S T 2 20S T 3 30S T 23 1 S T 2 31S T 3 12S ) 6

7 Now we are ready to write the explicit forms of the curvature and torsion scalars. We have R = 6Ḣ + 2H2 ) + 6ḣ + 18Hh + 6h2 3f ) T = 6h 2 f 2 ). 3.31) So finally for the FRW metric, the M 43 - model takes the form S 43 = d 4 x g[fr,t) + L m ], R = 6Ḣ + 2H2 ) + 6ḣ + 18Hh + 6h2 3f 2, 3.32) T = 6h 2 f 2 ). It that is the M 43 - model) is one of geometrical realizations of FR,T) gravity in the sense that it was derived from the purely geometrical point of view. 4 Lagrangian formulation of FR, T) gravity Of course, we can work with the form 3.32) of FR,T) gravity. But a more interesting and general form of FR,T) gravity is the so-called M 37 - model. The action of the M 37 - gravity reads as [10] S 37 = d 4 x g[fr,t) + L m ], where R = u + R s = u + 6ɛ 1 Ḣ + 2H2 ), 4.1) T = v + T s = v + 6ɛ 2 H 2, R s = 6ɛ 1 Ḣ + 2H2 ), T s = 6ɛ 2 H ) So we can see that here instead of two functions h and f in 3.32), we introduced two new functions u and v. For example, for 3.32) these functions have the form u = 61 ɛ 1 )Ḣ + 2H2 ) + 6ḣ + 18Hh + 6h2 3f 2, 4.3) v = 6h 2 f 2 ɛ 2 H 2 ) 4.4) that again tells us that the M 43 - model is a particular case of M 37 - model [Note that if ɛ 1 = 1 = ɛ 2 we have u = 6ḣ + 18Hh + 6h2 3f 2, v = 6h 2 f 2 H 2 )]. But in general we think or assume) that u = ut,a,ȧ,ä,... a,...;f i ) and v = vt,a,ȧ,ä,... a,...;g i ), while f i and g i are some unknown functions related with the geometry of the spacetime. So below we will work with a more general form of FR,T) gravity namely the M 37 - gravity 4.1). Introducing the Lagrangian multipliers we now can rewrite the action 4.1) as S 37 = dta {FR,T) 3 ȧ λ [T 2 ] [ ä )] } v 6ɛ 2 a 2 γ R u 6ɛ 1 a + ȧ2 a 2 + L m, 4.5) where λ and γ are Lagrange multipliers. If we take the variations with respect to T and R of this action, we get λ = F T, γ = F R. 4.6) Therefore, the action 4.5) can be rewritten as S 37 = dta 3 { FR,T) F T [ T v 6ɛ 2 ȧ 2 Then the corresponding point-like Lagrangian reads a 2 ] F R [ R u 6ɛ 1 ä )] } a + ȧ2 a 2 + L m. 4.7) L 37 = a 3 [F T v)f T R u)f R + L m ] 6ɛ 1 F R ɛ 2 F T )aȧ 2 6ɛ 1 F RR Ṙ + F RT T)a 2 ȧ. 4.8) 7

8 As is well known, for our dynamical system, the Euler-Lagrange equations read as ) d L37 L 37 = 0, 4.9) dt Ṙ R ) d L37 dt T L 37 = 0, 4.10) T ) d L37 L 37 = ) dt ȧ a Hence, using the expressions we get L 37 Ṙ = 6ɛ 1F RR a 2 ȧ, 4.12) L 37 T L 37 ȧ respectively. Here = 6ɛ 1 F RT a 2 ȧ, 4.13) = 12ɛ 1 F R ɛ 2 F T )aȧ 6ɛ 1 F RR Ṙ + F RT T + F Rψ ψ)a 2 + a 3 F T vȧ + a 3 F R uȧ, 4.14) a 3 ȧ 2 ) F TT T v 6ɛ 2 a 2 ) a 3 F RR R u 6ɛ 1 ä a + ȧ2 a 2 ) = 0, 4.15) = 0, 4.16) U + B 2 F TT + B 1 F T + C 2 F RRT + C 1 F RTT + C 0 F RT + MF + 6ɛ 2 a 2 p = 0, 4.17) U = A 3 F RRR + A 2 F RR + A 1 F R, 4.18) A 3 = 6ɛ 1 Ṙ 2 a 2, 4.19) A 2 = 12ɛ 1 Ṙaȧ 6ɛ 1 Ra 2 + a 3 Ṙuȧ, 4.20) A 1 = 12ɛ 1 ȧ 2 + 6ɛ 1 aä + 3a 2 ȧuȧ + a 3 uȧ a 3 u a, 4.21) B 2 = 12ɛ 2 Taȧ + a 3 Tvȧ, 4.22) B 1 = 24ɛ 2 ȧ ɛ 2 aä + 3a 2 ȧvȧ + a 3 vȧ a 3 v a, 4.23) C 2 = 12ɛ 1 a 2 ṘT, 4.24) C 1 = 6ɛ 1 a 2 T 2, 4.25) C 0 = 12ɛ 1 Taȧ + 12ɛ 2 Ṙaȧ 6ɛ 1 a 2 T + a 3 Ṙvȧ + a 3 Tuȧ, 4.26) M = 3a ) If F RR 0, F TT 0, from Eqs. 4.17) and 4.17), it is easy to find that R = u + 6ɛ 1 Ḣ + 2H2 ), T = v + 6ɛ 2 H 2, 4.28) so that the relations 4.1) are recovered. Generally, these equations are the Euler constraints of the dynamics. Here a, R, T are the generalized coordinates of the configuration space. On the other hand, it is also well known that the total energy Hamiltonian) corresponding to Lagrangian L 37 is given by H 37 = L 37 ȧ ȧ + L 37 Ṙ + L 37 Ṙ T T L ) Hence using 4.12)-4.14) we obtain H 37 = [ 12ɛ 1 F R ɛ 2 F T )aȧ 6ɛ 1 F RR Ṙ + F RT T + F Rψ ψ)a 2 + a 3 F T vȧ + a 3 F R uȧ]ȧ 6ɛ 1 F RR a 2 ȧṙ 6ɛ 1F RT a 2 ȧ T [a 3 F TF T RF R + vf T + uf R + L m ) 6ɛ 1 F R ɛ 2 F T )aȧ 2 6ɛ 1 F RR Ṙ + F RT T + F Rψ ψ)a 2 ȧ]. 4.30) 8

9 Let us rewrite this formula as where H 37 = D 2 F RR + D 1 F R + JF RT + E 1 F T + KF + 2a 3 ρ, 4.31) D 2 = 6ɛ 1 Ṙa 2 ȧ, 4.32) D 1 = 6ɛ 1 aä + a 3 uȧȧ, 4.33) J = 6ɛ 1 a 2 ȧt, 4.34) E 1 = 12ɛ 2 aȧ 2 + a 3 vȧȧ, 4.35) K = a ) As usual we assume that the total energy H 37 = 0 Hamiltonian constraint). So finally we have the following equations of the M 37 - model [10]-[11]: D 2 F RR + D 1 F R + JF RT + E 1 F T + KF = 2a 3 ρ, U + B 2 F TT + B 1 F T + C 2 F RRT + C 1 F RTT + C 0 F RT + MF = 6a 2 p, 4.37) ρ + 3Hρ + p) = 0. It deserves to note that the M 37 - model 4.1) admits some interesting particular and physically important cases. Some particular cases are now presented. i) The M 44 - model. Let the function FR,T) be independent from the torsion scalar T: F = FR, T) = FR). Then the action 4.1) acquires the form S 44 = d 4 xe[fr) + L m ], 4.38) where R = u + R s = u + ɛ 1 g µν R µν, 4.39) is the curvature scalar. It is the M 44 - model. We work with the FRW metric. In this case R takes the form R = u + 6ɛ 1 Ḣ + 2H2 ). 4.40) The action can be rewritten as where the Lagrangian is given by S 44 = dtl 44, 4.41) L 44 = a 3 [F R u)f R + L m ] 6ɛ 1 F R aȧ 2 6ɛ 1 F RR Ṙa 2 ȧ. 4.42) The corresponding field equations of the M 44 - model read as Here and D 2 F RR + D 1 F R + KF = 2a 3 ρ, A 3 F RRR + A 2 F RR + A 1 F R + MF = 6a 2 p, 4.43) ρ + 3Hρ + p) = 0. D 2 = 6ɛ 1 Ṙa 2 ȧ, 4.44) D 1 = 6ɛ 1 a 2 ä + a 3 uȧȧ, 4.45) K = a ) A 3 = 6ɛ 1 Ṙ 2 a 2, 4.47) A 2 = 12ɛ 1 Ṙaȧ 6ɛ 1 Ra 2 + a 3 Ṙuȧ, 4.48) A 1 = 12ɛ 1 ȧ 2 + 6ɛ 1 aä + 3a 2 ȧuȧ + a 3 uȧ a 3 u a, 4.49) M = 3a ) 9

10 If u = 0 then we get the following equations of the standard FR s ) gravity after R = R s ): 6ṘHF RR R 6H 2 )F R + F = ρ, 4.51) 2Ṙ2 F RRR + [ 4ṘH 2 R]F RR + [ 2H 2 4a 1 ä + R]F R F = p, 4.52) ρ + 3Hρ + p) = ) ii) The M 45 - model. The action of the M 45 - model looks like S 45 = d 4 xe[ft) + L m ], 4.54) where e = det e i µ) = g and the torsion scalar T is defined as Here T = v + T s = v + ɛ 2 S ρ µν T ρ µν. 4.55) T ρ µν e ρ i µ e i ν ν e i ) µ, 4.56) K µν ρ 1 2 T µν ρ T νµ ρ T ρ µν ), 4.57) S ρ µν 1 2 K µν ρ + δ ρt µ θν θ δρt ν θµ ) θ. 4.58) For a spatially flat FRW metric 3.18), we have the torsion scalar in the form T = v + T s = v + 6ɛ 2 H ) The action 4.54) can be written as where the point-like Lagrangian reads S 45 = dtl 45, 4.60) L 45 = a 3 [F T v)f T + L m ] + 6ɛ 2 F T aȧ ) So finally we get the following equations of the M 45 - model: E 1 F T + KF = 2a 3 ρ, B 2 F TT + B 1 F T + MF = 6a 2 p, 4.62) ρ + 3Hρ + p) = 0. Here E 1 = 12ɛ 2 aȧ 2 + a 3 vȧȧ, 4.63) K = a ) and B 2 = 12ɛ 2 Taȧ + a 3 Tvȧ, 4.65) B 1 = 24ɛ 2 ȧ ɛ 2 aä + 3a 2 ȧvȧ + a 3 vȧ a 3 v a, 4.66) M = 3a ) If we put v = 0 then the M 45 - model reduces to the usual FT s ) gravity, where T s = 6ɛ 2 H 2. As is well-known the equations of FT s ) gravity are given by 12H 2 F T + F = ρ, 4.68) ) 48H 2 F TT Ḣ F T 12H 2 + 4Ḣ F = p, 4.69) ρ + 3Hρ + p) = 0, 4.70) where we must put T = T s. Finally we note that it is well-known that the standard FT s ) gravity is not local Lorentz invariant [33]. In this context, we have a very meager hope that the M 45 - model 4.54) is free from such problems. 10

11 5 Cosmological solutions In this section we investigate cosmological consequences of the FR, T) gravity. As example, we want to find some exact cosmological solutions of the M 37 - gravity model. Since its equations are very complicated we here consider the simplest case when FR,T) = µr + νt, 5.1) where µ and ν are some constants. Then equations 4.37) take the form where We can rewrite this system as µd 1 + νe 1 + KνT + µr) = 2a 3 ρ, µa 1 + νb 1 + MνT + µr) = 6a 2 p, 5.2) ρ + 3Hρ + p) = 0, D 1 = 6ɛ 1 a 2 ä + a 3 uȧȧ, 5.3) E 1 = 12ɛ 2 aȧ 2 + a 3 vȧȧ, 5.4) K = a 3, 5.5) A 1 = 12ɛ 1 ȧ 2 + 6ɛ 1 aä + 3a 2 ȧuȧ + a 3 uȧ a 3 u a, 5.6) B 1 = 24ɛ 2 ȧ ɛ 2 aä + 3a 2 ȧvȧ + a 3 vȧ a 3 v a, 5.7) M = 3a ) 3σH 2 0.5ȧzȧ z) = ρ, σ2ḣ + 3H2 ) + 0.5ȧzȧ z) aż ȧ z a ) = p, 5.9) ρ + 3Hρ + p) = 0, where z = µu + νv, σ = µɛ 1 νɛ 2. This system contents two independent equations for five unknown functions a,ρ,p,u,v). But in fact it contain 4 unknown functions H,ρ,p,z). The corresponding EoS parameter reads as ω = p ρ = 1 + 2σḢ aż ȧ z a ) 3σH 2 0.5ȧzȧ z). 5.10) Let us find some simplest cosmological solutions of the system 5.9). 5.1 Example 1 We start from the case σ = 0. In this case the system 5.9) takes the form At the same time the EoS parameter becomes Now we assume that where κ and l are some real constants. Then 0.5ȧzȧ z) = ρ, 0.5ȧzȧ z) aż ȧ z a ) = p, 5.11) ρ + 3Hρ + p) = 0. ω = p ρ = 1 aż ȧ z a ) 3ȧzȧ z). 5.12) z = κa l, 5.13) ω = 1 l ) This result tells us that in this case our model can describes the accelerated expansion of the Universe for some values of l. 11

12 5.2 Example 2 Now consider the de Sitter case that is H = H 0 = const so that a = a 0 e H0t. Then the system 5.9) reads as The EoS parameter takes the form 3σH ȧzȧ z) = ρ, 3σH ȧzȧ z) aż ȧ z a ) = p, 5.15) If z has the form 5.13) that is z = κe H0lt then we have ρ + 3Hρ + p) = 0. ω = p ρ = 1 + ażȧ z a ) 18σH 2 0 3ȧz ȧ z). 5.16) ω = p ρ = 1 If H 0 l > 0 then as t we get again lκ 18σH 2 0 e H0lt + 3κ. 5.17) ω = 1 l 3, 5.18) which is same as equation 5.14). This last equation tells us that our model can describes the accelerated expansion of the Universe e.g. for l 0. Also it corresponds to the phantom case if l > 0. Finally we present other forms of the generalized Friedmann equations in the system 5.9). Let us rewrite these equations in the standard form as Here 3H 2 = ρ + ρ z, 5.19) 2Ḣ + 3H2 ) = p + p z. 5.20) ρ z = 0.5σ 1 ȧzȧ z), 5.21) p z = 0.5σ 1 ȧzȧ z) 1 6σ aż ȧ z a ) 5.22) are the z or u v contributions to the energy density and pressure, respectively. 6 Conclusion In [30], Buchdahl proposed to replace the Einstein-Hilbert scalar Lagrangian R with a function of the scalar curvature. The resulting theory is nowadays known as FR) gravity. Almost 40 years later, Bengochea and Ferraro proposed to replace the TEGR that is the torsion scalar Lagrangian T with a function FT) of the torsion scalar, and studied its cosmological consequences [31]. This type of modified gravity is nowadays called as FT) gravity theory. These two gravity theories [that is FR) and FT)] are, in some sense, alternative ways to modify GR. From these results arises the natural question: how we can construct some modified gravity theory which unifies FR) and FT) theories? Examples of such unified curvature-torsion theories were proposed in [10]-[11]. Such type of modified gravity theory is called the FR,T) gravity. In this FR,T) gravity, the curvature scalar R and the torsion scalar T play the same role and are dynamical quantities. In this paper, we have shown that the FR,T) gravity can be derived from the geometrical point of view. In particular, we have proposed a new method to construct particular models of FR,T) gravity. As an example we have considered the M 43 - model, deriving its action in terms of the curvature and torsion scalars. Then in detail we have studied the M 37 - model and presented its action, Lagrangian and equations of motion for the FRW metric case. Finally we have shown that the last model can describes the accelerated expansion of the Universe. 12

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