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Non-minimal torsion-matter coupling extension of f(T ) gravity

Tiberiu Harko,1, ∗ Francisco S. N. Lobo,2, † G. Otalora,3, ‡ and Emmanuel N. Saridakis4, 5, §

1Department of Mathematics, University College London,

Gower Street, London WC1E 6BT, United Kingdom2Centro de Astronomia e Astrofısica da Universidade de Lisboa,

Campo Grande, Edificıo C8, 1749-016 Lisboa, Portugal3Departamento de Fısica, ICE, Universidade Federal de Juiz de Fora, Caixa Postal 36036-330, Minas Gerais, Brazil

4Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece5Instituto de Fısica, Pontificia Universidad de Catolica de Valparaıso, Casilla 4950, Valparaıso, Chile

We construct an extension of f(T ) gravity with the inclusion of a non-minimal torsion-mattercoupling in the action. The resulting theory is a novel gravitational modification, since it is differentfrom both f(T ) gravity, as well as from the non-minimal curvature-matter-coupled theory. Thecosmological application of this new theory proves to be very interesting. In particular, we obtain aneffective dark energy sector whose equation-of-state parameter can be quintessence or phantom-like,or exhibit the phantom-divide crossing, while for a large range of the model parameters the Universeresults in a de Sitter, dark-energy-dominated, accelerating phase. Additionally, we can obtain early-time inflationary solutions too, and thus provide a unified description of the cosmological history.

PACS numbers: 04.50.Kd, 98.80.-k, 95.36.+x

I. INTRODUCTION

The recent observational advances in cosmology haveprovided a large number of high precision cosmologicaldata, which have posed new challenges for the under-standing of the basic physical properties of the Universe,and of the gravitational interaction that dominates itsdynamics and evolution. The observation of the accel-erated expansion of the Universe [1] has raised the fun-damental issue of the cause of this acceleration, whichis usually attributed to a mysterious and yet not di-rectly detected dominant component of the Universe,called dark energy [2]. In this context, the latest releasedPlanck satellite data of the 2.7 degree Cosmic MicrowaveBackground (CMB) full sky survey [3] have generally con-firmed the standard Λ Cold Dark Matter (ΛCDM) cos-mological model. On the other hand, the measurement ofthe tensor modes from large angle CMB B-mode polar-isation by BICEP2 [4], implying a tensor-to-scalar ratior = 0.2+0.07

−0.05, has provided a very convincing evidence forthe inflationary scenario, since the generation of gravi-tational wave fluctuations is a generic prediction of theearly de Sitter exponential expansion. However, the BI-CEP2 result is in tension with Planck limits on standardinflationary models [5], and thus alternative explanationsmay be required. In principle, magnetic fields generatedduring inflation can produce the required B-mode, for asuitable range of energy scales of inflation [5]. Moreover,the existence of the fluctuations of cosmological birefrin-gence can give rise to CMB B-mode polarization that fits

∗Electronic address: t.harko@ucl.ac.uk†Electronic address: flobo@cii.fc.ul.pt‡Electronic address: gotalora@fisica.ufjf.br§Electronic address: Emmanuel Saridakis@baylor.edu

BICEP2 data with r < 0.11, and no running of the scalarspectral index [6].

The above major observational advances require somegood theoretical explanations, with the role of giving afirm foundation to cosmology, and the underlying theoryof gravity. However, up to now, no convincing theoreticalmodel, supported by observational evidence that couldclearly explain the nature of dark energy, has been pro-posed. Moreover, not only the recent accelerated expan-sion of the Universe, but also observations at the galac-tic (galaxy rotation curves) and at extra-galactic scale(virial mass discrepancy in galaxy clusters) [7] suggestthe existence of another mysterious and yet undetectedmajor component of the Universe, the so-called dark mat-ter. From all these observations one can conclude thatthe standard general relativistic gravitational field equa-tions, obtained from the classic Einstein-Hilbert actionS =

∫

(R/2 + Lm)√−gd4x, where R is the scalar curva-

ture, and Lm is the matter Lagrangian density, in whichmatter is minimally coupled to the geometry, cannot givean appropriate quantitative description of the Universeat astrophysical scales going beyond the boundary of theSolar System. To explain dark energy and dark matterin a cosmological context requires the ad hoc introduc-tion of the dark matter and dark energy components inthe total energy-momentum tensor of the Universe, inaddition to the ordinary baryonic matter.

From an historical point of view, in going beyondthe Einstein-Hilbert action, the first steps were takenin the direction of generalizing the geometric part ofthe standard gravitational action. An extension of theEinstein-Hilbert action, in which the Ricci scalar in-variant R is substituted with an arbitrary function ofthe scalar invariant, f(R), has been extensively ex-plored in the literature [8]. Such a modification ofthe gravitational action can explain the late accelera-tion of the Universe, and may also provide a geomet-

2

ric explanation for dark matter, which can be describedas a manifestation of geometry itself [9]. Furthermore,quadratic Lagrangians, constructed from second ordercurvature invariants such as R2, RµνR

µν , RαβµνRαβµν ,

εαβµνRαβγδRγδµν , CαβµνC

αβµν , etc., have also been con-sidered as candidates for a more general gravitationalactions [10], which can successfully explain dark mat-ter and the late-time cosmic acceleration. Alternatively,the interest for extra-dimensions, which goes back to theunified field theory of Kaluza and Klein, led to the de-velopment of the braneworld models [11]. In braneworldmodels, gravitational effects due to the extra dimensionsdominate at high energies, but important new effects,which can successfully explain both dark energy and darkmatter, also appear at low energies.

Most of the modifications of the Einstein-Hilbert La-grangian involve a change in the geometric part of the ac-tion only, and assume that the matter Lagrangian playsa subordinate and passive role, which is implemented bythe minimal coupling of matter to geometry. However, ageneral theoretical principle forbidding an arbitrary cou-pling between matter and geometry does not exist a pri-

ori. If theoretical models, in which matter is consideredon an equal footing with geometry, are allowed, the gravi-tational theories with many interesting and novel featurescan be constructed.

A theory with an explicit coupling between an ar-bitrary function of the scalar curvature and the La-grangian density of matter was proposed in [12]. Thegravitational action of the latter model is of the formS =

∫

{f1(R) + [1 + λf2(R)]Lm}√−gd4x. In thesemodels an extra force acting on massive test particlesarises, and the motion is no longer geodesic. More-over, in this framework, one can also explain dark mat-ter [13]. The early “linear” geometry-matter coupling[12] was extended in [14] and a maximal extension of theEinstein-Hilbert action with geometry-matter coupling,of the form S =

∫

d4x√−gf (R,Lm) was considered in

[15]. An alternative model to f(R,Lm) gravity is thef(R, T ) theory [16], where T is the trace of the matterenergy-momentum tensor Tµν , and the corresponding ac-tion given by S =

∫

[f (R, T ) /2 + Lm]√−gd4x. The de-

pendence of the gravitational action on T may be due tothe presence of quantum effects (conformal anomaly), orof some exotic imperfect fluids. When the trace of theenergy-momentum tensor T is zero, T = 0, which is thecase of the electromagnetic radiation, the field equationsof f(R, T ) theory reduce to those of f(R) gravity.

However, the f(R,Lm) or f(R, T ) gravitational mod-els are not the most general Lagrangians with non-minimal geometry-matter couplings. One could furtherobtain interesting gravity models by introducing a termof the form RµνT

µν in the Lagrangian [17, 18]. Such cou-plings appear in Einstein-Born-Infeld theories [19], whenone expands the square root in the Lagrangian. Thepresence of the RµνT

µν coupling term has the advantageof entailing a non-minimal coupling of geometry to theelectromagnetic field.

All the above gravitational modifications are based onthe Einstein-Hilbert action, namely on the curvature de-scription of gravity. However, an interesting and richclass of modified gravity can arise if one modifies theaction of the equivalent torsional formulation of Gen-eral Relativity. As it is well known, Einstein also con-structed the “Teleparallel Equivalent of General Rela-tivity” (TEGR) [20–24], replacing the torsion-less Levi-Civita connection by the curvature-less Weitzenbock one,and using the vierbein instead of the metric as the fun-damental field. In this formulation, instead of the cur-vature (Riemann) tensor one has the torsion tensor, andthe Lagrangian of the theory, namely the torsion scalarT , is constructed by contractions of the torsion tensor.Thus, if one desires to modify gravity in this formula-tion, the simplest thing is to extend T to an arbitraryfunctionf(T ) [25, 26]. An interesting aspect of this ex-tension is that although TEGR coincides with GeneralRelativity at the level of equations, f(T ) is different thanf(R), that is they belong to different modification classes.Additionally, although in f(R) theory the field equationsare fourth order, in f(T ) gravity they are second order,which is a great advantage. f(T ) gravity models hasbeen extensively applied to cosmology, and amongst oth-ers it is able to explain the late-time accelerating ex-pansion of the Universe without the need of dark energy[26–28]. Furthermore, following these lines, and inspiredby the higher-curvature modifications of General Relativ-ity, one can construct gravitational modifications basedon higher-order torsion invariants, such is the f(T, TG)gravity [29], which also proves to have interesting cosmo-logical implications.

Another gravitational modification based on theteleparallel formulation is the generalization of TEGRto the case of a Weyl-Cartan space-time, in which theWeitzenbock condition of the vanishing of the curvatureis also imposed (Weyl-Cartan-Weitzenbock (WCW) grav-ity), with the addition of a kinetic term for the torsionin the gravitational action [30]. In this framework thelate-time universe acceleration can be naturally obtained,determined by the intrinsic geometry of the space-time.A further extension of the WCW gravity, in which theWeitzenbock condition in a Weyl-Cartan geometry is in-serted into the gravitational action via a Lagrange mul-tiplier, was analyzed in [31]. In the weak field limitthe gravitational potential explicitly depends on the La-grange multiplier and on the Weyl vector, leading to aninteresting cosmological behavior.

In the work, we are interested in proposing a novelgravitational modification based on the torsional for-mulation, by allowing the possibility of a non-minimaltorsion-matter coupling in the gravitational action. Inparticular, for the torsion-matter coupling we adopt the“linear” model introduced in the case of f(R) gravity in[12]. Hence, the gravitational field can be described interms of two arbitrary functions of the torsion scalar T ,namely f1(T ) and f2(T ), with the function f2(T ) linearlycoupled to the matter Lagrangian. This new coupling in-

3

duces a supplementary term [1 + λf2(T )]Lm in the stan-dard f(T ) action, with λ an arbitrary coupling constant.When λ = 0, the model reduces to the usual f(T ) grav-ity. We investigate in detail the cosmological implicationsof the torsion-matter coupling for two particular choicesof the functions f1(T ) and f2(T ). For both choices theUniverse evolution is in agreement with the observed be-havior, and moreover it ends in a de Sitter type vacuumstate, with zero matter energy density. The details ofthe transition depend on the numerical values of the freeparameters that appear in the functions f1(T ) and f2(T ).The paper is organized as follows. In Section II we

briefly describe the basics of the f(T ) gravity model.The field equations of the f(T ) theory with linear non-minimal torsion-matter coupling are obtained in Sec-tion III. The cosmological implications of the theory areanalyzed in Section IV. Finally, we conclude and discussour results in Section V.

II. f(T ) GRAVITY AND COSMOLOGY

In this Section, we briefly review the f(T ) gravita-tional paradigm. We use the notation where Greek in-dices run over the coordinate space-time and Latin in-dices run over the tangent space-time. As we mentionedin the Introduction, the dynamical variables are the vier-beins fields eA(x

µ), which at each point xµ of the man-ifold form an orthonormal basis for the tangent space,that is eA · eB = ηAB, with ηAB = diag(1,−1,−1,−1).Additionally, they can be expressed in terms of the com-ponents eµA in the coordinate basis as eA = eµA∂µ. Hence,the metric is obtained from the dual vierbein through

gµν(x) = ηAB eAµ (x) eBν (x). (1)

In this formulation, instead of the Levi-Civita connection

one uses the Weitzenbock one:w

Γλ

νµ ≡ eλA ∂µeAν [32], and

thus instead of curvature we acquire the torsion tensor

T λµν =

w

Γλ

νµ −w

Γλ

µν = eλA (∂µeAν − ∂νe

Aµ ). (2)

It proves convenient to define the contorsion tensor

Kµνρ ≡ − 1

2

(

T µνρ − T νµ

ρ − Tρµν)

, as well as the ten-

sor Sρµν ≡ 1

2

(

Kµνρ + δµρ Tαν

α − δνρ Tαµα

)

. Using these

one can write down the teleparallel Lagrangian (torsionscalar) [21–24, 33]

T ≡ 1

4T ρµνTρµν +

1

2T ρµνTνµρ − Tρµ

ρT νµν , (3)

which used in the action and varied in terms of the vier-beins gives rise to the same equations with General Rel-ativity. That is why such a theory is called “TeleparallelEquivalent of General Relativity” (TEGR).One can be based on the above torsional formulation of

General Relativity, in order to construct classes of modi-fied gravity. The simplest one is to extend T to a function

T + f(T ), that is writing an action of the form1

S =1

16πG

∫

d4xe [T + f(T )] , (4)

where e = det(eAµ ) =√−g, G is the gravitational con-

stant, and we have used units where the speed of lightis c = 1. Note that TEGR and thus General Relativityis restored when f(T ) = 0. Moreover, we stress that al-though TEGR coincides with General Relativity at thelevel of equations, f(T ) is different than f(R).Let us now proceed to the cosmological application of

f(T ) gravity. Introducing additionally the matter sectorthe total action becomes

S =1

16πG

∫

d4xe [T + f(T ) + Lm] , (5)

where the matter Lagrangian is assumed to correspondto a perfect fluid with energy density ρm and pressure pm(for simplicity we neglect the radiation sector, althoughits inclusion is straightforward). Varying the action (5)with respect to the vierbeins we obtain the field equations

(1 + f ′)[

e−1∂µ(eeρASρ

νµ)− eλATρµλSρ

µν]

+eρASρνµ∂µTf

′′ +1

4eνA[T + f ] = 4πGeρA

em

T ρν , (6)

where f ′ = ∂f/∂T , f ′′ = ∂2f/∂T 2, andem

T ρν denotes

the usual energy-momentum tensor.Proceeding forward, we impose the standard homoge-

neous and isotropic geometry, that is we consider

eAµ = diag(1, a(t), a(t), a(t)), (7)

which corresponds to a flat Friedmann-Robertson-Walker(FRW) universe with metric

ds2 = dt2 − a2(t) δijdxidxj , (8)

where a(t) is the scale factor.In summary, inserting the vierbein ansantz (7) into the

equations of motion (6) we extract the modified Fried-mann equations as

H2 =8πG

3ρm − f

6− 2H2f ′ (9)

H = − 4πG(ρm + pm)

1 + f ′ − 12H2f ′′ , (10)

with H ≡ a/a the Hubble parameter, and dots denotingderivatives with respect to t. Note that we have also usedthe relation

T = −6H2, (11)

which arises immediately for an FRW geometry using Eq.(3).

1 An alternative simple extension of TEGR is to allow for a non-minimal scalar-torsion coupling, as in [34].

4

III. f(T ) GRAVITY WITH NON-MINIMAL

TORSION-MATTER COUPLING

Having presented the f(T ) modified gravity in the pre-vious section, in this section we extend it, allowing for anon-minimal coupling between the torsion scalar and thematter Lagrangian. In particular, we consider the action

S =1

16πG

∫

d4x e {T + f1(T ) + [1 + λ f2(T )] Lm} ,(12)

where fi(T ) (with i = 1, 2) are arbitrary functions of thetorsion scalar T and λ is a coupling constant with unitsof mass−2. Varying the action with respect to the tetradeAρ yields the field equations

(1 + f ′1 + λf ′

2Lm)[

e−1∂µ(eeαASα

ρµ)− eαATµναSµ

νρ]

+(f ′′1 + λf ′′

2 Lm) ∂µTeαASα

ρµ + eρA

(

f1 + T

4

)

−1

4λf ′

2 ∂µTeαA

em

S αρµ + λf ′

2 eαASα

ρµ ∂µLm

= 4πG (1 + λf2) eαA

em

T αρ, (13)

where we have definedem

S Aρµ =

∂Lm

∂∂µeAρ, (14)

and the prime denotes differentiation with respect to thetorsion scalar. As expected Eq. (13) reduces to Eq. (6)when λ = 0.Since the Lagrangian density of a perfect fluid is the en-

ergy scalar, representing the energy in a local rest framefor the fluid, a possible “natural choice” for the mat-ter Lagrangian density is Lm/(16πG) = −ρm [35, 36].

In this case, we haveem

S Aρµ = 0, and also the usual

form of the energy momentum tensor for the perfect fluidem

T µν = (ρm + pm)uµuν − pmgµν .In summary, inserting the flat FRW vierbein choice

(7) and the above matter Lagrangian density, into thefield equations (13), we obtain the modified Friedmannequations

H2 =8πG

3

[

1 + λ(

f2 + 12H2f ′2

)]

ρm−1

6

(

f1 + 12H2f ′1

)

,

(15)

H = − 4πG (ρm + pm)[

1 + λ(

f2 + 12H2f ′2

)]

1 + f ′1 − 12H2f ′′

1 − 16πGλρm (f ′2 − 12H2f ′′

2 ).

(16)In the limit λ = 0, f1(T ) ≡ f(T ), and f2(T ) ≡ 0,Eqs. (15) and (16) reduce to Eqs. (9) and (10), re-spectively. The generalized Friedmann equations can berewritten as

3H2 = 8πG (ρDE + ρm) , (17)

2H + 3H2 = −8πG (pDE + pm) , (18)where the effective energy density and effective pressureof the dark energy sector are defined as

ρDE := − 1

16πG

(

f1 + 12H2f ′1

)

+ λρm(

f2 + 12H2f ′2

)

,

(19)

pDE := (ρm + pm)×[

1 + λ(

f2 + 12H2f ′2

)

1 + f ′1 − 12H2f ′′

1 − 16πGλρm (f ′2 − 12H2f ′′

2 )− 1

]

+1

16πG

(

f1 + 12H2f ′1

)

− λρm(

f2 + 12H2f ′2

)

. (20)

Furthermore, we can define the dark-energy equation-of-state parameter in the standard form

wDE :=pDE

ρDE. (21)

One can easily verify that the above affective dark energydensity and pressure satisfy the usual evolution equation

ρDE + ρm + 3H (ρDE + ρm + pDE + pm) = 0. (22)

Finally, we can introduce the deceleration parameter q,given by

q =d

dt

(

1

H

)

− 1, (23)

whose sign indicates the decelerating/accelerating natureof the cosmological expansion. Cosmological models withq < 0 are accelerating, while those having q > 0 experi-ence a decelerating evolution.

IV. COSMOLOGICAL IMPLICATIONS

Since we have extracted the basic background equa-tions of motion of the f(T ) gravity model with a non-minimal matter-torsion coupling, we are now able toinvestigate its phenomenological implications. Due tothe relation (11), for convenience in the following wewill change the T -dependence to the H-dependence inthe involved expressions, so that f1(T ) ≡ f1(H), andf2(T ) ≡ f2(H). For the derivatives of the functionsf1(T ) and f2(T ) we obtain f ′

i(H) ≡ f ′i(T )|T→−6H2 , and

f ′′i (H) ≡ f ′′

i (T )|T→−6H2 , i = 1, 2, respectively. Finally,in the following we fully adopt the natural system of unitsby taking 8πG = c = 1.

The basic cosmological equations describing the timeevolution of the non-minimally-coupled f(T ) gravity aregiven by Eqs. (15) and (16). From Eq. (15) we can ex-press the matter density as

ρm(t) =3H2 +

[

f1(H) + 12H2 f ′1(T )|T→−6H2

]

/2

1 + λ[

f2(H) + 12H2 f ′2(T )|T→−6H2

] .

(24)By substituting the matter density ρm into Eq. (16) weobtain the basic equation describing the cosmological dy-namics in non-minimally matter-coupled f(T ) gravity as

5

2H = −

{

1 + λ[

f2(H) + 12H2 f ′2(T )|T→−6H2

]}

{

[

f1(H)+12H2 f ′

1(T )|T→−6H2

]

/2+3H2

1+λ[

f2(H)+12H2 f ′

2(T )|T→−6H2

] + pm

}

1 + f ′1(T )|T→−6H2 − 12H2 f ′′

1 (T )|T→−6H2 −2λ

{[

f1(H)+12H2 f ′

1(T )|T→−6H2

]

/2+3H2}[

f ′

2(T )|T→−6H2−12H2 f ′′

2 (T )|T→−6H2

]

1+λ[

f2(H)+12H2 f ′

2(T )|T→−6H2

]

.

(25)

Once the functions f1(T ) and f2(T ) are fixed, Eqs. (24)and (25) become a system of two ordinary differentialequations for three unknowns, (H, ρm, pm). In order toclose the system of equations, the matter equation ofstate pm = pm (ρm) must also be given. Finally, thedeceleration parameter is given by

q = − H

H2− 1, (26)

while the dark energy equation-of-state parameter can beexpressed as

wDE = − 2H

ρDE−ρm + pm

ρDE−1 = −2H + 3H2 + pm

3H2 − ρm. (27)

In the following we will investigate the system ofEqs. (24) and (25), for different functional forms of f1(T )and f2(T ).

A. f1(T ) = −Λ+ α1T2 and f2(T ) = β1T

2

As a first example, we examine the case where f1(T ) =−Λ + α1T

2 and f2(T ) = β1T2, where α1 and β1 are

constants, since these are the first non-trivial correc-tions to TEGR, that is to General Relativity. As wementioned above, it proves convenient to express the in-volved functions in terms of H . In particular, in terms ofH the functional dependencies of f1 and f2 are givenby f1(H) = −Λ + αH4 and f2(H) = βH4, respec-tively, with α = 36α1, β = 36β1. For the derivativesof the functions f1 and f2 we obtain f ′

1(H) = −αH2/3,f ′2(H) = −βH2/3, f ′′

1 (H) = α/18, f ′′2 (H) = β/18. More-

over, we restrict our analysis to the case of dust matter,that is we take pm = 0.In this case the gravitational field equations (24) and

(25) become

ρm(t) =3αH4 − 6H2 + Λ

6βλH4 − 2, (28)

and

H(t) =

(

3αH4 − 6H2 + Λ) (

3βλH4 − 1)

4H2 (α+ βλΛ − 3βλH2)− 4, (29)

respectively. The time variation of the Hubble functionH , of the scale factor a, of the matter energy density ρm,and of the deceleration parameter q, obtained by numer-ically integrating Eqs. (28) and (29) are represented, for

0 5 10 15 20 25

0.05

0.06

0.07

0.08

0.09

0.10

t

HH

tLFIG. 1: Time variation of the Hubble function H(t) forthe non-minimally matter coupled f(T ) gravity model withf1(T ) = −Λ + α1T

2 and f2(T ) = β1T2, or equivalently

f1(H) = −Λ + αH4 and f2(H) = βH4 with α = 36α1,β = 36β1, for five different choices of the parameters Λ, α, β,and λ: Λ = 0.01, α = 0.16, β = 0.1, and λ = 1 (solid curve),Λ = 0.02, α = 0.18, β = 0.3, and λ = 1.2 (dotted curve),Λ = 0.03, α = 0.20, β = 0.35, and λ = 1.4 (short-dashedcurve), Λ = 0.04, α = 0.30, β = 0.45, and λ = 1.6 (dashedcurve), and Λ = 0.05, α = 0.40, β = 0.55, and λ = 1.8 (long-dashed curve), respectively. The initial value for H used tonumerically integrate Eq. (29) is H(0) = 0.1.

different numerical values of the free parameters Λ, α, βand λ, in Figs. 1-5.

As depicted in Fig. 1, the Hubble function is a mono-tonically decreasing function of time for all t > 0. In thelimit of large times the Hubble function tends to a con-stant value, limt→∞ H(t) = h0 = constant. Hence, forthe considered range of values of the free parameters, inthe f(T ) model with torsion-matter coupling, the Uni-verse ends its evolution in an accelerating, de Sitter-typephase. The scale factor a, shown in Fig. 2, indicates amonotonically time increase of the size of the Universe,and hence an expansionary behavior. The matter energydensity, depicted in Fig. 3, tends progressively to zero.Furthermore, the deceleration parameter q, presented inFig. 4, indicates a large variety of dynamical behaviors ofthe f(T ) model with matter-torsion coupling. In partic-ular, for some values of the free parameters the Universestarts its evolution in the matter-dominated phase froma decelerating phase, and ends in a de Sitter-type accel-erated behavior. Other values of the parameters produceUniverse models starting from a marginally acceleratingphase (q = 0), and ending in a de Sitter state. Finally, for

6

0 5 10 15 20 25

0.002

0.004

0.006

0.008

0.010

t

aH

tL

FIG. 2: Time variation of of the scale factor a(t) for the non-minimally matter coupled f(T ) gravity model with f1(T ) =−Λ+α1T

2 and f2(T ) = β1T2, or equivalently f1(H) = −Λ+

αH4 and f2(H) = βH4 with α = 36α1, β = 36β1, for fivedifferent choices of the parameters Λ, α, β, and λ: Λ = 0.01,α = 0.16, β = 0.1, and λ = 1 (solid curve), Λ = 0.02, α =0.18, β = 0.3, and λ = 1.2 (dotted curve), Λ = 0.03, α = 0.20,β = 0.35, and λ = 1.4 (short-dashed curve), Λ = 0.04, α =0.30, β = 0.45, and λ = 1.6 (dashed curve), and Λ = 0.05, α =0.40, β = 0.55, and λ = 1.8 (long-dashed curve), respectively.The initial values for a and a used to numerically integrateEq. (29) are a(0) = 10−3 and a(0) = 10−4, respectively.

0 5 10 15 20 25

0.000

0.005

0.010

0.015

0.020

0.025

t

ΡmH

tL

FIG. 3: Time variation of the matter energy density ρm(t)for the non-minimally matter coupled f(T ) gravity modelwith f1(T ) = −Λ + α1T

2 and f2(T ) = β1T2, or equiva-

lently f1(H) = −Λ+ αH4 and f2(H) = βH4 with α = 36α1,β = 36β1, for five different choices of the parametersΛ, α, β,and λ: Λ = 0.01, α = 0.16, β = 0.1, and λ = 1 (solid curve),Λ = 0.02, α = 0.18, β = 0.3, and λ = 1.2 (dotted curve),Λ = 0.03, α = 0.20, β = 0.35, and λ = 1.4 (short-dashedcurve), Λ = 0.04, α = 0.30, β = 0.45, and λ = 1.6 (dashedcurve), and Λ = 0.05, α = 0.40, β = 0.55, and λ = 1.8 (long-dashed curve), respectively. The initial value for H used tonumerically integrate Eq. (29) is H(0) = 0.1.

other parameter choices at the beginning of the matterdominated phase the Universe is already in an accelerat-ing phase, that is with q < 0. Lastly, as depicted in Fig. 5,for these specific choices of the parameters, the dark en-ergy equation-of-state parameter wDE is very close to thevalue −1, to which it rigorously tends in the large time

0 5 10 15 20 25

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

t

qH

tL

FIG. 4: Time variation of the deceleration parameter q(t)for the non-minimally matter coupled f(T ) gravity modelwith f1(T ) = −Λ + α1T

2 and f2(T ) = β1T2, or equiva-

lently f1(H) = −Λ+ αH4 and f2(H) = βH4 with α = 36α1,β = 36β1, for five different choices of the parameters Λ, α, β,and λ: Λ = 0.01, α = 0.16, β = 0.1, and λ = 1 (solid curve),Λ = 0.02, α = 0.18, β = 0.3, and λ = 1.2 (dotted curve),Λ = 0.03, α = 0.20, β = 0.35, and λ = 1.4 (short-dashedcurve), Λ = 0.04, α = 0.30, β = 0.45, and λ = 1.6 (dashedcurve), and Λ = 0.05, α = 0.40, β = 0.55, and λ = 1.8 (long-dashed curve), respectively. The initial value for H used tonumerically integrate Eq. (29) is H(0) = 0.1.

0 5 10 15 20 25

-1.0000

-0.9995

-0.9990

-0.9985

-0.9980

t

wH

tL

FIG. 5: Time variation of the dark energy equation-of-stateparameter wDE for the non-minimally matter coupled f(T )gravity model with f1(T ) = −Λ+α1T

2 and f2(T ) = β1T2, or

equivalently f1(H) = −Λ+ αH4 and f2(H) = βH4 with α =36α1, β = 36β1, for five different choices of the parameters Λ,α, β, and λ: Λ = 0.01, α = 0.16, β = 0.1, and λ = 1 (solidcurve), Λ = 0.02, α = 0.18, β = 0.3, and λ = 1.2 (dottedcurve), Λ = 0.03, α = 0.20, β = 0.35, and λ = 1.4 (shortdashed curve), Λ = 0.04, α = 0.30, β = 0.45, and λ = 1.6(dashed curve), and Λ = 0.05, α = 0.40, β = 0.55, and λ = 1.8(long dashed curve), respectively. The initial value for H usedto numerically integrate Eq. (29) is H(0) = 0.1.

limits. This is an advantage, since in this model the effec-tive torsion-matter coupling can successfully mimic thecosmological constant, in agreement with observations.

After the above numerical elaboration, we examinewhether we can obtain analytical expressions in variouslimits. In particular, we analyze the properties of the

7

equations in the limit of small and large H(t), respec-tively.

1. The limit of small H

In the limit of small H(t), that is at the late phases ofthe cosmological evolution, Eq. (29) becomes

H =1

4

(

αΛ + βλΛ2 − 6)

H2 +Λ

4(30)

yielding the following solution

H(t) =√

H1 tan

[

tan−1

(

H0√H1

)

+Λ

4√H1

(t− t0)

]

,

(31)with H1 = Λ/ [Λ(α+ βλΛ)− 6], and where we have usedthe initial condition H (t0) = H0. Thus, the scale factorevolves according to

a(t) = a0 cos− 4H1

Λ

[

tan−1

(

H0√H1

)

+Λ

4√H1

(t− t0)

]

,

(32)where a0 is an arbitrary constant of integration.Similarly, the deceleration parameter (26) becomes

q(t) = −Λ csc2

[

tan−1(

H0√H1

)

+ Λ(t−t0)

4√H1

]

4H1− 1. (33)

Additionally, the matter energy-density (28) can be ap-proximated as

ρm(t) = 3H(t)2 − Λ

2, (34)

and using Eq. (31) its explicit time dependence acquiresthe form

ρm(t) = 3H1 tan2

[

tan−1

(

H0√H1

)

+Λ (t− t0)

4√H1

]

− Λ

2.

(35)Finally, the dark-energy equation-of-state parameter

from (27), becomes

wDE(t) = −1− (α+ βλΛ)H(t)2, (36)

that is

wDE(t) = −1− (α + βλΛ)H1 ×

× tan2[

tan−1

(

H0√H1

)

+Λ (t− t0)

4√H1

]

.(37)

Interestingly enough, we observe that according to theparameter values, wDE can be either above or be-low −1, that is the effective dark-energy sector can bequintessence-like or phantom-like. This feature, which isexpected to happen in modified gravity [37], is an addi-tional advantage of the scenario at hand.Lastly, we mention that in the large-time limit the

Hubble function (31) becomes almost constant, implyingthat a de Sitter-type evolution is possible in the frame-work of the present model.

2. The limit of large H

In the limit of large H , corresponding to the earlyphases of the cosmological evolution, in the first order ap-proximation the differential equation (29) describing thecosmological dynamics of the Hubble function becomes

H = −3

4αH4, (38)

with the general solution given by

H(t) =22/3H0

[4 + 9H30α (t− t0)]

1/3. (39)

The behavior of the scale factor can then be describedby the equation

a(t) = exp

{

1

3 3√2α

[

4

H30

+ 9α (t− t0)

]2/3}

, (40)

that is, it is determined solely by the parameter α. Sim-ilarly, the deceleration parameter (26) is given by

q(t) =3αH2

0

[8 + 18αH30 (t− t0)]

2/3− 1. (41)

Moreover, for the matter energy density (28) we obtain

ρm(t) =α

2βλ, (42)

showing that during the time interval for which thisapproximation is valid the energy density of the mat-ter is approximately constant. Finally, the dark-energyequation-of-state parameter from (27), becomes

wDE(t) = −α

2H(t)2 = −α

2

24/3H20

[4 + 9H30α (t− t0)]

2/3. (43)

Again, we mention that according to the parameterchoice, wDE can be either above or below −1, that isthe effective dark-energy sector can be quintessence-likeor phantom-like.

B. f1(T ) = −Λ and f2(T ) = α1T + β1T2

As a second example, we examine the case wheref1(T ) = −Λ and f2(T ) = α1T + β1T

2, where Λ > 0,α1 and β1 are constants, since this scenario is also thefirst non-trivial correction to TEGR, that is to GeneralRelativity. Equivalently, we impose f1(H) = −Λ andf2(H) = αH2 + βH4, with α = −6α1 and β = 36β1.For the derivatives of the functions f1(T ) and f2(T ) weobtain f ′

1(T ) = f ′′1 (T ) = 0, f ′

2(H) = −α/6− βH2/3, andf ′′2 (H) = β/18. The basic evolution equations (24) and(25) in this case become

ρm(t) =Λ− 6H2

2αλH2 + 6βλH4 − 2, (44)

8

0 5 10 15 20 25

0.05

0.10

0.15

0.20

t

HH

tL

FIG. 6: Time variation of the Hubble function H(t) forthe non-minimally matter coupled f(T ) gravity model withf1(T ) = −Λ and f2(T ) = α1T + β1T

2, or equivalentlyf1(H) = −Λ and f2(H) = αH2 + βH4 with α = −6α1,β = 36β1, for five different choices of the parameters Λ, α, β,and λ: Λ = 0.01, α = 0.16, β = 0.1, and λ = 1 (solid curve),Λ = 0.02, α = 0.18, β = 0.3, and λ = 1.2 (dotted curve),Λ = 0.03, α = 0.20, β = 0.35, and λ = 1.4 (short-dashedcurve), Λ = 0.04, α = 0.30, β = 0.45, and λ = 1.6 (dashedcurve), and Λ = 0.05, α = 0.40, β = 0.55, and λ = 1.8 (long-dashed curve), respectively. The initial value for H used tonumerically integrate Eq. (45) is H(0) = 0.2.

and

H(t) =3(

Λ − 6H2) (

αλH2 + 3βλH4 − 1)

2 (αλΛ + 6βλH2 (Λ− 3H2)− 6), (45)

respectively. The time variation of the Hubble functionH , of the scale factor a, of the matter density ρm, andof the deceleration parameter, obtained by numericallyelaborating the system of Eqs. (44) and (45) for differentvalues of the free parameters and assuming the matterto be dust (wm = 0), are presented in Figs. 6-10, respec-tively.The Hubble function, presented in Fig. 6, decreases

monotonically in time, and tends to a constant valuein the large-time limit. Therefore, for all the parame-ter choices the Universe ends in a de Sitter phase. Thetime variation of the scale factor, depicted in Fig. (7),indicates that all considered models are expanding. Thematter energy density, shown in Fig. 8, monotonicallydecreases in time as expected. In the large-time limit theUniverse ends in a vacuum state, with negligible matterdensity, and thus being completely dominated by the ef-fective dark energy sector. The deceleration parameter,presented in Fig. 9 indicates a very strong dependenceof the dynamical behavior of the Universe on the modelparameters. For the considered values, in all cases at thebeginning of the matter-dominated phase, the Universeis in a decelerating phase, with q > 0. After a finite timeta, determined by the condition q (ta) = 0, the Universeenters in an accelerated phase, with q(t) < 0, ∀t > ta.Similarly to the model of the previous subsection, theUniverse always ends in a de Sitter phase, with q = −1.

0 5 10 15 20 25

0.002

0.004

0.006

0.008

0.010

0.012

t

aH

tL

FIG. 7: Time variation of the scale factor a(t) of the Universefor the non-minimally matter coupled f(T ) gravity modelwith f1(T ) = −Λ and f2(T ) = α1T + β1T

2, or equivalentlyf1(H) = −Λ and f2(H) = αH2 + βH4 with α = −6α1,β = 36β1, for five different choices of the parameters Λ, α, β,and λ: Λ = 0.01, α = 0.16, β = 0.1, and λ = 1 (solid curve),Λ = 0.02, α = 0.18, β = 0.3, and λ = 1.2 (dotted curve),Λ = 0.03, α = 0.20, β = 0.35, and λ = 1.4 (short-dashedcurve), Λ = 0.04, α = 0.30, β = 0.45, and λ = 1.6 (dashedcurve), and Λ = 0.05, α = 0.40, β = 0.55, and λ = 1.8 (long-dashed curve), respectively. The initial values for a and a

used to numerically integrate Eq. (45) are a(0) = 10−3 anda(0) = 2× 10−4, respectively.

0 5 10 15 20 25

0.00

0.02

0.04

0.06

0.08

0.10

t

ΡmH

tL

FIG. 8: Time variation of the matter energy density ρm(t)of the Universe for the non-minimally matter coupled f(T )gravity model with f1(T ) = −Λ and f2(T ) = α1T + β1T

2, orequivalently f1(H) = −Λ and f2(H) = αH2 + βH4 with α =−6α1, β = 36β1, for five different choices of the parametersΛ, α, β, and λ: Λ = 0.01, α = 0.16, β = 0.1, and λ = 1 (solidcurve), Λ = 0.02, α = 0.18, β = 0.3, and λ = 1.2 (dottedcurve), Λ = 0.03, α = 0.20, β = 0.35, and λ = 1.4 (shortdashed curve), Λ = 0.04, α = 0.30, β = 0.45, and λ = 1.6(dashed curve), and Λ = 0.05, α = 0.40, β = 0.55, and λ = 1.8(long dashed curve), respectively. The initial value for H usedto numerically integrate Eq. (45) is H(0) = 0.2.

Finally, as depicted in Fig. 10, in the large time limit thedark energy equation-of-state parameter wDE tends tothe value −1, namely limt→∞ wDE(t) = −1, thus show-ing that this choice of the functions f1(T ) and f2(T )can also successfully mimic an effective cosmological con-

9

0 5 10 15 20 25

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

t

qH

tL

FIG. 9: Time variation of the deceleration parameter q(t)of the Universe for the non-minimally matter coupled f(T )gravity model with f1(T ) = −Λ and f2(T ) = α1T + β1T

2, orequivalently f1(H) = −Λ and f2(H) = αH2 + βH4 with α =−6α1, β = 36β1, for five different choices of the parametersΛ, α, β, and λ: Λ = 0.01, α = 0.16, β = 0.1, and λ = 1 (solidcurve), Λ = 0.02, α = 0.18, β = 0.3, and λ = 1.2 (dottedcurve), Λ = 0.03, α = 0.20, β = 0.35, and λ = 1.4 (shortdashed curve), Λ = 0.04, α = 0.30, β = 0.45, and λ = 1.6(dashed curve), and Λ = 0.05, α = 0.40, β = 0.55, and λ = 1.8(long dashed curve), respectively. The initial value for H usedto numerically integrate Eq. (45) is H(0) = 0.2.

0 5 10 15 20 25

-1.20

-1.15

-1.10

-1.05

-1.00

t

wH

tL

FIG. 10: Time variation of the dark energy equation of stateparameter wDE of the Universe for the non-minimally mattercoupled f(T ) gravity model with f1(T ) = −Λ and f2(T ) =α1T +β1T

2, or equivalently f1(H) = −Λ and f2(H) = αH2+βH4 with α = −6α1, β = 36β1, for five different choices of theparameters Λ, α, β, and λ: Λ = 0.01, α = 0.16, β = 0.1, andλ = 1 (solid curve), Λ = 0.02, α = 0.18, β = 0.3, and λ = 1.2(dotted curve), Λ = 0.03, α = 0.20, β = 0.35, and λ = 1.4(short dashed curve), Λ = 0.04, α = 0.30, β = 0.45, andλ = 1.6 (dashed curve), and Λ = 0.05, α = 0.40, β = 0.55, andλ = 1.8 (long dashed curve), respectively. The initial valuefor H used to numerically integrate Eq. (45) is H(0) = 0.2.

stant. Note however, that for these specific parameterchoices wDE lies in the phantom regime, which is an ad-vantage of the scenario at hand, revealing its capabilities.After this numerical elaboration, we examine whether

we can obtain analytical expressions in various limits. Inparticular, we examine the properties of the equations in

the limit of small and large H(t), respectively.

1. The limit of small H

In the limit of small values of the Hubble function H ,that is at late times, Eq. (45) can be approximated as

H =3Λ

2 (6− αλΛ)+

3(

α2λ2Λ2 + 6βλΛ2 − 36)

2 (6− αλΛ)2 H2, (46)

with the general solution given by

H(t) =√

H2 tan

[

tan−1

(

H0√H2

)

+3Λ (t− t0)

2 (6− αλΛ)√H2

]

(47)with H2 = Λ(6 − αλΛ)/

(

α2λ2Λ2 + 6βλΛ2 − 36)

, andwhere we have used the initial condition H (t0) = H0.The scale factor then becomes

a(t) = a0 cos− 2H2(6−αλΛ)

3Λ ×

×[

tan−1

(

H0√H2

)

+3Λ (t− t0)

2 (6− αλΛ)√H2

]

,(48)

where a0 is an arbitrary constant of integration, whilethe deceleration parameter (26) behaves as

q(t) = −3Λ csc2

[

tan−1(

H0√H2

)

+ 3Λ(t−t0)

2(6−αλΛ)√H2

]

2 (6− αλΛ)H2− 1.

(49)Note that in the limit of large t the deceleration param-eter tends to −1, limt→∞ q(t) = −1. Additionally, thematter density (44) becomes

ρm(t) =

(

3− αλΛ

2

)

H(t)2 − Λ

2, (50)

and using (47) we acquire

ρm(t) = −Λ

2+

(

3− αλΛ

2

)

H2 ×

× tan2[

tan−1

(

H0√H2

)

+3Λ (t− t0)

2 (6− αλΛ)√H2

]

.(51)

Finally, the dark-energy equation-of-state parameterfrom (27), becomes

wDE(t) = −1 +αλΛ

αλΛ− 6−

18(

α2λ2Λ+ 2βλΛ− 6αλ)

(6− αλΛ)2H

2

= −1 +αλΛ

αλΛ− 6−

18(

α2λ2Λ+ 2βλΛ− 6αλ)

(6− αλΛ)2H2

× tan2

[

tan−1

(

H0√H2

)

+3Λ (t− t0)

2 (6− αλΛ)√H2

]

.(52)

which can lie both in the quintessence as well as in thephantom regime, depending on the specific choices of thefree parameters of the model, namely on α, β, λ and Λ,respectively.

10

2. The limit of large H

In the opposite limit of large H , that is at early times,at first order approximation Eq. (45) becomes

H − 3

2H2 = 0, (53)

with H (t0) = H0, and thus the general solution is givenby

H(t) =2H0

2− 3H0 (t− t0). (54)

The scale factor then reads

a(t) = a0

[

2

2− 3H0 (t− t0)

]2/3

, (55)

while the deceleration parameter is obtained as q =−5/2, that is the universe at early times always startswith acceleration, which corresponds to an inflationarystage. Finally, for the time variation of the matter en-ergy density in the large-H regime we find ρm(t) ≈ 0,which is consistent with the interpretation of this stageas inflationary.We mention that the above expressions for H , a, q and

ρm at first order approximation, are independent on thefree parameters of the model α, β, λ and Λ, respectively,and are determined only by the initial value of H at t =t0.

V. DISCUSSIONS AND FINAL REMARKS

In the present paper, we have considered an exten-sion of the f(T ) gravity model by introducing a non-minimal coupling between torsion and matter. The ge-ometric part of the action was extended through the in-troduction of two independent functions of the torsionscalar T , namely f1(T ) and f2(T ), respectively, with thefunction f2(T ) being non-minimally coupled to the mat-ter Lagrangian Lm. The resulting gravitational modelpresents some formal analogies with the non-minimallygeometry-matter coupling introduced in [12]. However,the resulting equations, as well as its physical and geo-metrical interpretations, are very different. The theory ofnon-minimal torsion-matter coupling is therefore a novelclass of gravitational modification.From the physical point of view, in this theory, matter

is not just a passive component in the space-time con-tinuum, but it plays an active role in the overall gravi-tational dynamics, which is strongly modified due to thesupplementary interaction between matter and geometry.Moreover, the major advantage of the f(T )-type models,namely that the field equations are second order, is notmodified by the torsion-matter coupling.As an application of the non-minimal torsion-matter

coupling scenario we have considered the dynamical evo-lution of a flat FRW universe. We have investigated

the time dependence of the cosmologically relevant phys-ical parameters, for two different choices of the functionsf1(T ) and f2(T ), corresponding to the simplest depar-tures from General Relativity. In these specific mod-els the dynamics of the Universe is determined by thefree parameters which appear in the functions f1(T ) andf2(T ), as well as by the matter-torsion coupling constant.Depending on the numerical values of these parameters alarge number of cosmological behaviors can be obtained.In our analysis we have considered the matter dominatedphase of the Universe evolution, that is, we neglected thematter pressure. More general models with pm can beeasily constructed and analyzed.We restricted our analysis in expanding evolutions, al-

though contracting or bouncing solutions can be easilyobtained as well. We have found a universe evolution inagreement with observations, that is a matter-dominatedera followed by an accelerating phase. Additionally, theeffective dark-energy equation-of-state parameter can liein the quintessence or phantom regime, which reveals thecapabilities of the scenario. Furthermore, a general andcommon property of the considered models is that theyall end in a de Sitter phase, with zero matter density, thatis to complete dark-energy domination. Finally, thesemodels also accept solutions with almost constant Hub-ble function, which can describe the inflationary regime.Thus, the scenario of non-minimal torsion-matter cou-pling can offer a unified description of the universe evo-lution, from its inflationary to the late-time acceleratedphases.Apart from the exact numerical elaboration, we have

extracted approximate analytical expressions in the limitof a small Hubble parameter, that is corresponding to thelarge-time limit, as well as for large Hubble parameters,that is corresponding to the beginning of the cosmologi-cal expansion. These expressions verify the above phys-ical features that were extracted through the numericalanalysis.In conclusion, based on the torsional formulation of

gravity, we have proposed a novel modified gravitationalscenario which contains an arbitrary coupling betweenthe torsion scalar and the matter Lagrangian. The cos-mological implications of this theory proves to be veryinteresting. However, in order for the present scenarioto be considered as a good candidate for the descriptionof Nature, additional investigations should be performed,such as the detailed comparison with cosmological obser-vations, the complete perturbation analysis, etc. Thesenecessary studies lie beyond the scope of the present workand are left for future projects.

Acknowledgments

FSNL acknowledges financial support of the Fundacaopara a Ciencia e Tecnologia through an InvestigadorFCT Research contract, with reference IF/00859/2012,funded by FCT/MCTES (Portugal), and grants

11

CERN/FP/123615/2011 and CERN/FP/123618/2011.GO would like to thank CAPES and FAPEMIG for fi-nancial support. The research of ENS is implementedwithin the framework of the Operational Program “Edu-

cation and Lifelong Learning” (Actions Beneficiary: Gen-eral Secretariat for Research and Technology), and is co-financed by the European Social Fund (ESF) and theGreek State.

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