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the `` teleparallel '' equivalent of general relativity ( tegr ) @xcite is an equivalent formulation of classical gravity , in which instead of using the torsionless levi - civita connection one uses the curvatureless weitzenbck one . the dynamical objects are the four linearly independent vierbeins ( these are _ parallel _ vector fields represented by the appellation `` teleparallel '' ) . the advantage of this framework is that the torsion tensor is formed solely from products of first derivatives of the tetrad . as described in @xcite , the lagrangian density @xmath0 can be constructed from this torsion tensor under the assumptions of invariance under general coordinate transformations , global lorentz transformations , and the parity operation , along with requiring the lagrangian density to be second order in the torsion tensor . thus , apart from possible conceptual differences , tegr is completely equivalent and indistiguishable form general relativity ( gr ) at the level of equations , both background and perturbation ones . on the other hand , in general relativity one can add the quintessence scalar field in order to acquire a dynamical dark energy sector , a scenario that exhibits a very interesting cosmological behavior and has gained a huge amount of research @xcite . amongst others , one can generalize it by including a nonminimal coupling between the quintessence field and gravity @xcite , or more generally extend it to the scalar - tensor paradigm @xcite . one can also use a phantom instead of a canonical field @xcite , or the combination of both these fields in a unified scenario called quintom @xcite .    in this letter we are interested in formulating `` teleparallel dark energy '' , adding a canonical scalar field in tegr . in the minimal coupling case the resulting theory is identical to the ordinary quintessence , both at the background and perturbation levels . however , when the nonminimal coupling is switched on , teleparallel dark energy is different from its gr counterpart , and the cosmological behavior of such a new scenario proves to be very interesting . let us review very briefly the quintessence paradigm in general relativity . in such a scenario the dark energy sector is attributed to a homogeneous scalar field @xmath1 , and the action is given by  @xcite @xmath2 \label{action},\ ] ] with @xmath3 , @xmath4 , @xmath5 the scalar - field potential , @xmath6 the non - minimal coupling parameter , @xmath7 the ricci scalar , and @xmath8 the matter lagrangian . note the difference in the metric signature that exists amongst the various works in the literature and the corresponding sign changes in the action , since a change in the metric signature leads @xmath9 , @xmath10 and @xmath11 to change sign , while @xmath7 and the energy - momentum tensor remain unaffected @xcite . in this letter we use the signature @xmath12 in all sections , just to be closer to the literature of teleparallel gravity . thus , under this convention , the conformal value of @xmath6 is @xmath13 .    in the case of a flat friedmann - robertson - walker ( frw ) background metric @xmath14 where @xmath15 is the cosmic time , @xmath16 are the comoving spatial coordinates and @xmath17 is the scale factor , the friedmann equations write as @xmath18 where @xmath19 is the hubble parameter , a dot denotes differentiation with respect to @xmath15 , and @xmath20 and @xmath21 are the matter energy density and pressure , respectively , following the standard evolution equation @xmath22 , with @xmath23 the matter equation - of - state parameter . additionally , we have introduced the energy density and pressure of the nonminimally coupled scalar field , given by @xcite : @xmath24 where a prime denotes derivative with respect to @xmath1 . we note that the above relations have been simplified by using the useful expression @xmath25 in the frw geometry . as mentioned above , in such a scenario , dark energy is attributed to the scalar field , and thus its equation - of - state parameter reads : @xmath26 finally , the equations close by considering the evolution equation for the scalar field @xcite : @xmath27 which can alternatively be written in the standard form @xmath28 . we now briefly review tegr . the notation is as follows : greek indices @xmath29 ... and capital latin indices @xmath30 ... run over all coordinate and tangent space - time 0 , 1 , 2 , 3 , while lower case latin indices ( from the middle of the alphabet ) @xmath31 and lower case latin indices ( from the beginning of the alphabet ) @xmath32 ... run over spatial and tangent space coordinates 1 , 2 , 3 , respectively .    as stated in _ introduction _ , the dynamical variable of `` teleparallel '' gravity is the vierbein field @xmath33 . this forms an orthonormal basis for the tangent space at each point @xmath34 of the manifold , that is @xmath35 , where @xmath36 . furthermore , the vector @xmath37 can be analyzed with the use of its components @xmath38 in a coordinate basis , that is @xmath39 . in such a construction , the metric tensor is obtained from the dual vierbein as @xmath40 contrary to gr , which uses the torsionless levi - civita connection , in tegr ones takes the curvatureless weitzenbck connection @xcite , whose torsion tensor reads @xmath41 where @xmath42 . moreover , the contorsion tensor , which equals to the difference between weitzenbck and levi - civita connections , is defined as @xmath43 and we also define @xmath44 .    in the present formalism all the information concerning the gravitational field is included in the torsion tensor @xmath45 . using the above quantities one can extract the form of the `` teleparallel lagrangian '' , which is nothing else than the torsion scalar , namely @xcite :    in summary , the simplest action in a universe governed by teleparalel gravity is @xmath46,\end{aligned}\ ] ] where @xmath47 ( one could also include a cosmological constant ) . variation with respect to the vierbein fields gives equation of motion @xmath48 where @xmath49 stands for the usual energy - momentum tensor . these equations are exactly the same as those of gr for every geometry choice . in particular , for the frw background metric ( [ frwmetric ] ) , the vierbein choice of the form @xmath50 is an exact solution  @xcite of the field equation in eq . ( [ eom ] ) , which does not generate a divergent energy for the whole space - time . furthermore , it is easily seen that the corresponding friedmann equations are identical to the gr ones , both at the background and perturbation levels @xcite . let us now construct teleparalell dark energy . this will be done by adding a scalar field in the equivalent , teleparallel , formulation of gr . thus , the action will simply read : @xmath51 . \label{action2}\ ] ] we emphasize that in the above action a nonminimal coupling between the scalar field and gravity is allowed . although in the nonminimal case one could use alternative torsion scalars , we prefer to keep the standard one for simplicity . we also note that the action in ( [ action2 ] ) with the torsion formulation of gr is similar to the standard nonminimal quintessence where the scalar field couples to the ricci scalar .      in this scenario , similar to the standard quintessence , dark energy is attributed to the scalar field , and thus its equation - of - state parameter ( @xmath56 ) is defined to be the same as that in ( [ eos ] ) , but @xmath57 and @xmath58 are now given by ( [ telerho ] ) and ( [ telep ] ) , respectively . finally , one can see that the scalar field evolution ( [ fieldevol2 ] ) leads to the standard relation @xmath59 . we now explore the cosmological implications of the scenario at hand . firstly , we immediately observe that in the case of the minimal coupling , teleparallel dark energy coincides with quintessence ( see ( [ rhophi])-([pphi ] ) and ( [ telerho])-([telep ] ) ) , and one can verify that at the level of perturbations too . this is expected since , concerning the gravitational sector , tegr is identical with gr , and in the minimal case one just adds a distinct scalar sector , thus making no difference whether it is added in either of the two theories . however , things are different if we switch on the nonminimal coupling . in this case the additional scalar sector is coupled to gravity , with the curvature scalar in gr and with the torsion scalar in tegr , and thus the resulting coupled equations do not coincide . clearly , teleparallel dark energy , under the nonminimal coupling , is a different theory . let us proceed in presenting some basic and general features of the nonminimal coupling of the scalar - torsion theory . apart from the straightforward results that dark energy possesses a dynamical nature as well as it can drive the universe acceleration , the most interesting and direct consequence of the dark energy density and pressure relations ( [ telerho])-([telep ] ) is that the dark energy equation - of - state parameter can lie in the quintessence regime ( @xmath60 ) , in the phantom regime ( @xmath61 ) , or exhibit the phantom - divide crossing during cosmological evolution . this is a radical difference with the quintessence scenario and reveals the capabilities of the construction .    in order to present the above features in a more transparent way , we evolve numerically the cosmological system for dust matter ( @xmath62 ) , using the redshift @xmath63 as the independent variable , imposing the present scale factor @xmath64 to be equal to 1 , the dark energy density @xmath65 at present to be @xmath66 and its initial value to be @xmath67 . finally , concerning the scalar field potential we use the exponential ansatz of the form @xmath68 .    in fig [ plot ] we depict the @xmath56-evolution for three realizations of the scenario at hand . in the case of the black - solid curve the teleparallel dark energy behaves like quintessence , in the red - dashed curve it behaves like a phantom , while in blue - dotted curve the dark energy exhibits the phantom - divide crossing during the evolution . note that the crossing behavior in fig . [ plot ] is the one favored by the observational data , in contrast with viable @xmath69-gravity models where it is the opposite one @xcite . we remark that in the above graphs we focus on their qualitative features , and in particular we maintain the same potential just to stress that in principle one can obtain the various behaviors with the same potential . clearly , one could be quantitatively more accurate and impose the observational @xmath70 as an input , reconstructing the corresponding potential . however in the present work we desire to remain as general as possible . in the present scenario of `` teleparallel '' dark energy we have added a scalar field to the teleparallel equivalent to general relativity ( tegr ) , allowing for a nonminimal coupling between the field and gravity . in the minimally - coupled case the cosmological equations coincide with those of the standard quintessence . however when the nonminimal coupling is switched on the resulting theory exhibits different behavior . in particular , although the scalar field is canonical , one can obtain a dark energy sector being quintessence - like , phantom - like , or experiencing the phantom - divide crossing during evolution , a behavior that is much richer comparing to general relativity ( gr ) with a scalar field . moreover , the fact that the phantom regime can be described without the need of phantom fields , which have ambiguous quantum behavior @xcite , is a significant advantage . the physical reason for the aforementioned difference , despite the equivalence of pure gr and pure tegr , is that while in gr one couples the scalar field with the only suitable gravitational scalar , namely the ricci scalar @xmath7 , in the later one couples the scalar field with the only suitable gravitational scalar , namely the torsion scalar @xmath0 . the richness of the resulting theory comparing to gr quintessence is additionally manifested in the fact that , although in the later one can perform a conformal transformation and transit to an `` equivalent '' , minimally - coupled , theory with transformed field and potential @xcite , in the former such a transformation does not exist since one obtains extra terms depending on the torsion tensor itself , as can be easily verified transforming the vierbeins as @xmath71 ( one applies in our case the similar analysis of @xcite of the case of @xmath72 scenarios ) . thus , teleparallel dark energy can not be transformed to an `` equivalent '' minimally coupled form , which is known to be able to describe only the quintessence regime , and this indicates its richer structure . such an absence of conformal transformation exists in other cosmological scenarios too , for example in scalar - field models with non - minimal derivative couplings , where it is also known that the resulting theories possess a richer structure @xcite . the addition of a scalar field to tegr was inspired by the corresponding procedure in gr . however , although in gr one can alternatively and equivalently generalize the action to @xmath69 , freeing himself of the need to add the scalar field , in the teleparallel formulation of gr the generalization to @xmath72 @xcite seems to spoil the local lorentz invariance for all functions apart from the linear one @xcite . however , at the background level no new degrees of freedom are present , while at linear perturbation the new vector degree of freedom only satisfies constraint equations @xcite . similarly , in our generalization of tegr , in the case of non - minimal coupling , a lorentz - violating term appears ( the last term in the left hand side of ( [ eom2 ] ) ) , despite the fact that the theory is linear in @xmath0 . however , no new degree of freedom will appear at the background level on which we focus on this work . clearly , going beyond background evolution and examine whether the lorenz violations do indeed appear under cosmological geometries and scales ( we have checked that at the low - 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using the `` teleparallel '' equivalent of general relativity as the gravitational sector , which is based on torsion instead of curvature , we add a canonical scalar field , allowing for a nonminimal coupling with gravity . although the minimal case is completely equivalent to standard quintessence , the nonminimal scenario has a richer structure , exhibiting quintessence - like or phantom - like behavior , or experiencing the phantom - divide crossing . 
 the richer structure is manifested in the absence of a conformal transformation to an equivalent minimally - coupled model .