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cosmological observations coming from type ia supernovae @xcite , cosmic microwave background radiation @xcite and the large scale structure @xcite , provide evidences that the universe is currently in an accelerating phase . this result is , in general , ascribed to the existence of a sort of dark energy ( de ) sector in the universe , an exotic energy source characterized by a negative pressure . at late times , the dark - energy sector eventually dominates over the cold dark matter ( cdm ) , and drives the universe to the observed accelerating expansion . the simplest candidate for de is the cosmological constant @xmath2 , which has an equation - of - state parameter @xmath3 . although this model is in agreement with current observations , it is plagued by some difficulties related to the small observational value of de density with respect to the expected one arising from quantum field theories ( the well known cosmological constant problem @xcite ) . moreover , the @xmath2cdm paradigm , where cold dark matter ( cdm ) is considered into the game , may also suffer from the age problem , as it was shown in @xcite , while the present data seem to slightly favor an evolving de with the equation - of - state parameter crossing @xmath3 from above to below in the near cosmological past @xcite .    over the past decade several de models have been proposed , such as quintessence @xcite , phantom @xcite , k - essence @xcite , tachyon @xcite , quintom @xcite , chaplygin gas @xcite , generalized chaplygin gas ( gcg ) @xcite , holographic de @xcite , new agegraphic de @xcite , ricci de @xcite etc . on the other hand , there are also numerous models that induce an effective dark energy which arises from modifications of the gravitational sector itself , such as @xmath4 gravity @xcite ( this class is very efficient in verifying observational and theoretical constraints and explain the universe acceleration and phantom crossing @xcite ) , or gravity with higher curvature invariants @xcite , by coupling the ricci scalar to a scalar field @xcite , by introducing a vector field contribution @xcite , or by using properties of gravity in higher dimensional spacetimes @xcite ( for a review see @xcite ) .    a possibility that can be explored to explain the accelerated phase of the universe is to consider a theory of gravity based on the weitzenbck connection , instead of the levi - civita one , which deduces that the gravitational field is described by the torsion instead of the curvature tensor . in such theories , the torsion tensor is achieved from products of first derivatives of tetrad fields , and hence no second derivatives appear . this _ teleparallel _ approach @xcite , is closely related to general relativity , except for `` boundary terms '' @xcite that involve total derivatives in the action , and thus one can construct the teleparallel equivalent of general relativity ( tegr ) , which is completely equivalent with general relativity at the level of equations but is based on torsion instead of curvature . teleparallel gravity possesses a number of attractive features related to geometrical and physical aspects @xcite . hence , one can start from tegr and construct various gravitational modifications based on torsion , with @xmath1 gravity being the most studied one @xcite . in particular , it may represent an alternative to inflationary models without the use of the inflaton , as well as to effective de models , in which the universe acceleration is driven by the extra torsion terms @xcite ( for a detailed review , see @xcite ) . the main advantage of @xmath1 gravity is that the field equations are 2nd - order ones , a property that makes these theories simpler if compared to the dynamical equations of other extended theories of gravity , such as @xmath4 gravity . the aim of this paper is to explore the implications of @xmath1 gravity to the formation of light elements in the early universe , i.e. to the big bang nucleosynthesis ( bbn ) . on the other hand , we want to explore the possibility to constrain @xmath1 gravity by bbn observatio nal data . bbn has occurred between the first fractions of second after the big bang , around @xmath5 sec , and a few hundreds of seconds after it , when the universe was hot and dense ( indeed bbn , together with cosmic microwave background radiation , provides the strong evidence about the high temperatures characterizing the primordial universe ) . it describes the sequence of nuclear reactions that yielded the synthesis of light elements @xcite , and therefore drove the observed universe . in general , from bbn physics , one may infer stringent constraints on a given cosmological model . hence , in this work , we shall confront various @xmath1 gravity models with bbn calculations based on current observational data on primordial abundance of @xmath0 , and we shall extract constraints on their free parameters . the layout of the paper is as follows . in section [ revmodel ] we review @xmath1 gravity and the related cosmological models . in section [ bbnanal ] we use bbn calculations in order to impose constraints on the free parameters of specific @xmath1 gravity models . conclusions are reported in section [ conclusions ] . finally , in the appendix we summarize the main notions of bbn physics . let us briefly review @xmath1 gravity , and apply it in a cosmological framework . in this formulation , the dynamical variable is the vierbein field @xmath6 , @xmath7 , which forms an orthonormal basis in the tangent space at each point @xmath8 of the manifold , i.e. @xmath9 , with @xmath10 the minkowsky metric with signature @xmath11 : @xmath12 . denoting with @xmath13 , @xmath14 the components of the vectors @xmath15 in a coordinate basis @xmath16 , one can write @xmath17 . as a convection , here we use the latin indices for the tangent space , and the greek indices for the coordinates on the manifold . the dual vierbein allows to obtain the metric tensor of the manifold , namely @xmath18 .    in teleparallel gravity , one adopts the curvatureless weitzenbck connection ( contrarily to general relativity which is based on the torsion - less levi - civita connection ) , which gives rise to the non - null torsion tensor : @xmath19 remarkably , the torsion tensor ( [ torsion ] ) encompasses all the information about the gravitational field . the lagrangian density is built using its contractions , and hence the teleparallel action is given by @xmath20 with @xmath21 , and where the torsion scalar @xmath22 reads as @xmath23 here , it is @xmath24 with @xmath25 the contorsion tensor which gives the difference between weitzenbck and levi - civita connections . finally , the variation of action ( [ action0 ] ) in terms of the vierbiens gives rise to the field equations , which coincide with those of general relativity . that is why the above theory is called the teleparallel equivalent of general relativity ( tegr ) . one can now start from tegr , and generalize action ( [ action0 ] ) in order to construct gravitational modifications based on torsion . the simplest scenario is to consider a lagrangian density that is a function of @xmath22 , namely @xmath26},\ ] ] that reduces to tegr as soon as @xmath27 . considering additionally a matter lagrangian @xmath28 , variation with respect to the vierbein gives the field equations @xcite @xmath29-     e_i^\lambda { t^\rho}_{\mu\lambda}{s_\rho}^{\nu\mu}[1+f']\nonumber\\     & &    + e^\rho_i { s_\rho}^{\,\,\mu\nu}(\partial_\mu t)f '' +      \frac{1}{4}e^\nu_i [ t+f]=4\pi g\,{e_i}^\rho\ , { \theta_\rho}^\nu\,,\end{aligned}\ ] ] where @xmath30 , @xmath31 and @xmath32 is the energy - momentum tensor for the matter sector .    in order to explore the cosmological implications of @xmath1 gravity , we focus on homogeneous and isotropic geometry , considering the usual choice for the vierbiens , namely @xmath33 which corresponds to a flat friedmann - robertson - walker ( frw ) background metric of the form @xmath34 where @xmath35 is the scale factor . equations ( [ torsion ] ) , ( [ lagrangian ] ) , ( [ s ] ) and ( [ contorsion ] ) allow to derive a relation between the torsion @xmath22 and the hubble parameter @xmath36 , namely @xmath37 hence , in the case of frw geometry , and assuming that the matter sector corresponds to a perfect fluid with energy density @xmath38 and pressure @xmath39 , the @xmath40 component of ( [ equations ] ) yields @xmath41+[t+f]=16\pi g\rho,\ ] ] while the @xmath42 component gives @xmath43-(t - f)=16\pi gp.\ ] ] the equations close by considering the equation of continuity for the matter sector , namely @xmath44 . one can rewrite ( [ friedmann ] ) and ( [ acceleration ] ) in the usual form @xmath45 @xmath46 where @xmath47,\\      p_t&=&\frac{1}{16\pi g}\ , \frac{f - t f'+ 2t^2f''}{1+f'+ 2{t}f '' } \ , , \label{pt}\end{aligned}\ ] ] are the effective energy density and pressure arising from torsional contributions . one can therefore define the effective torsional equation - of - state parameter as @xmath48 in these classes of theories , the above effective torsional terms are responsible for the accelerated phases of the early or / and late universe @xcite . let us present now three specific @xmath1 forms , which are the viable ones amongst the variety of @xmath1 models with two parameters out of which one is independent , i.e which pass the basic observational tests @xcite .    1 . the power - law model by bengochea and ferraro ( hereafter @xmath49cdm ) @xcite is characterized by the form @xmath50 where @xmath51 and @xmath52 are the two model parameters . inserting this @xmath1 form into friedmann equation ( [ friedmann ] ) at present , we acquire @xmath53 where @xmath54 is the matter density parameter at present , and @xmath55 is the current hubble parameter value . the best fit on the parameter @xmath52 is obtained taking the @xmath56 observational data , and it reads @xcite @xmath57 clearly , for @xmath58 the present scenario reduces to @xmath2cdm cosmology , namely @xmath59 , with @xmath60 . 2 .   the linder model ( hereafter @xmath61cdm ) @xcite arises from @xmath62 with @xmath63 and @xmath39 ( @xmath64 ) the two model parameters . in this case ( [ friedmann ] ) gives that @xmath65 the @xmath56 observational data imply that the best fit of @xmath64 is @xcite @xmath66 as we can see , for @xmath67 the present scenario reduces to @xmath2cdm cosmology . 3 .   motivated by exponential @xmath4 gravity @xcite , bamba et . al . introduced the following @xmath1 model ( hereafter @xmath68cdm ) @xcite : @xmath69 with @xmath63 and @xmath39 ( @xmath64 ) the two model parameters . in this case we acquire @xmath70 for this model , and using @xmath56 observational data , the best fit is found to be @xcite @xmath71 similarly to the previous case we can immediately see that @xmath68cdm model tends to @xmath2cdm cosmology for @xmath67 .    the above @xmath1 models are considered viable in literature because pass the basic observational tests @xcite . they are characterized by two free parameters . actually there are two more models with two free parameters , namely the logarithmic model @xcite , @xmath72 and the hyperbolic - tangent model @xcite , @xmath73 nevertheless since these two models do not possess @xmath2cdm cosmology as a limiting case and since they are in tension with observational data @xcite , in this work we do not consider them .    finally , let us note that one could also construct @xmath1 models with more than two parameters , for example , combining the above scenarios . however , considering many free parameters would be a significant disadvantage concerning the corresponding values of the information criteria . in the section , we examine the bbn in the framework of @xmath1 cosmology . as it is well known , bbn occurs during the radiation dominated era . the energy density of relativistic particles filling up the universe is given by @xmath74 , where @xmath75 is the effective number of degrees of freedom and @xmath76 the temperature ( in the appendix we review the main features related to the bbn physics ) . the neutron abundance is computed via the conversion rate of protons into neutrons , namely @xmath77 and its inverse @xmath78 . the relevant quantity is the total rate given by @xmath79 explicit calculations of eq . ( [ lambda ] ) lead to ( see ( [ lambdafinapp ] ) in the appendix ) @xmath80 where @xmath81 is the mass difference of neutron and proton , and @xmath82gev@xmath83 . the primordial mass fraction of @xmath84 can be estimated by making use of the relation @xcite @xmath85 here @xmath86 , with @xmath87 the time of the freeze - out of the weak interactions , @xmath88 the time of the freeze - out of the nucleosynthesis , @xmath89 the neutron mean lifetime given in ( [ rateproc3 ] ) , and @xmath90 is the neutron - to - proton equilibrium ratio . the function @xmath91 is interpreted as the fraction of neutrons that decay into protons during the interval @xmath92 $ ] . deviations from the fractional mass @xmath93 due to the variation of the freezing temperature @xmath94 are given by @xmath95      \frac{\delta { \cal t}_f}{{\cal t}_f}\,,\ ] ] where we have set @xmath96 since @xmath97 is fixed by the deuterium binding energy @xcite . the experimental estimations of the mass fraction @xmath93 of baryon converted to @xmath98 during the big bang nucleosynthesis are @xcite @xmath99 inserting these into ( [ deltayp ] ) one infers the upper bound on @xmath100 , namely @xmath101    during the bbn , at the radiation dominated era , the scale factor evolves as @xmath102 , where @xmath103 is cosmic time . the torsional energy density @xmath104 is treated as a perturbation to the radiation energy density @xmath38 . the relation between the cosmic time and the temperature is given by @xmath105 ( or @xmath106mev ) . furthermore , we use the entropy conservation @xmath107 . the expansion rate of the universe is derived from ( [ modfri ] ) , and can be rewritten in the form @xmath108 where @xmath109 ( @xmath110 is the expansion rate of the universe in general relativity ) . thus , from the relation @xmath111 , one derives the freeze - out temperature @xmath112 , with @xmath113 mev ( which follows from @xmath114 ) and @xmath115 from which , in the regime @xmath116 , one obtains : @xmath117 with @xmath118gev@xmath83 .    in what follows we shall investigate the bounds that arise from the bbn constraints , on the free parameters of the three @xmath1 models presented in the previous section . these constraint will be determined using eqs . ( [ deltat / tboundg ] ) and ( [ rhot ] ) . moreover , we shall use the numerical values @xmath119 where @xmath120 is the present value of cmb temperature . 1 .   @xmath121cdm model . + for the @xmath121cdm model of ( [ eq : ftmyrzabis ] ) relation ( [ rhot ] ) gives @xmath122 \nonumber \\     & = & \frac{3h_0 ^ 2}{8\pi g}\ , \omega_{m0}\left(\frac{{\cal t}}{{\cal t}_0}\right)^{4n } \label{rhoth } \ , ,   \end{aligned}\ ] ] and then ( [ deltat / tboundg ] ) yields @xmath123 in fig . [ deltatf1 ] we depict @xmath124 from ( [ deltat / tboundgfin ] ) vs @xmath52 , as well as the upper bound from ( [ deltat / tbound ] ) . as we can see , constraints from bbn require @xmath125 . remarkably , this bound is in agreement with the best fit for @xmath52 of ( [ bestfit1 ] ) , namely @xmath126 , that was obtained using @xmath56 observational data in @xcite . +    _ @xmath124 from ( [ deltat / tboundgfin ] ) vs @xmath52 ( thick line ) for the @xmath121cdm model of ( [ eq : ftmyrzabis ] ) , and the upper bound for @xmath124 from ( [ deltat / tbound ] ) ( dashed line ) . as we can see , constraints from bbn require @xmath125._,width=278,height=216 ] 2 .   @xmath127cdm model . + in the case of @xmath128cdm model of ( [ modf2 ] ) and @xmath129cdm model of ( [ modf3 ] ) , and for the purpose of this analysis , we can unified their investigation parameterizing them as @xmath130\,,\ ] ] with @xmath131 where @xmath132 for model @xmath128cdm and @xmath133 for model @xmath129cdm . inserting ( [ f12insieme ] ) into ( [ deltat / tboundg ] ) we acquire @xmath134e^{-p({\cal t}_f/{\cal t}_0)^{4 m } }      \ ! -\!\frac{1}{2}\right\}.\end{aligned}\ ] ] hence , using this relation we can calculate the value of @xmath135 for various values of @xmath136 that span the order of magnitude of the best fit values ( [ bestfit1 ] ) and ( [ bestfit2 ] ) that were obtained using @xmath56 observational data in @xcite , and we present our results in table [ tab1 ] . as we can see , in all cases the value of @xmath135 is well below the bbn bound ( [ deltat / tbound ] ) . hence , bbn can not impose constraints on the parameter values of @xmath128cdm and @xmath129cdm models . .@xmath135 from ( [ delta12insieme ] ) for different values of @xmath136 , for @xmath137 ( @xmath128cdm model ) and @xmath138 ( @xmath129cdm model ) . [ cols="^,^,^,^,^",options="header " , ] in this work we have investigated the implications of @xmath1 gravity to the formation of light elements in the early universe , i.e. to the bbn . in particular , we have examined the three most used and well studied viable @xmath1 models , namely the power law , the exponential and the square - root exponential , and we have confronted them with bbn calculations based on current observational data on primordial abundance of @xmath0 . hence , we were able to extract constraints on their free parameters . concerning the power - law @xmath1 model , the obtained constraint on the exponent @xmath52 , is @xmath125 . remarkably , this bound is in agreement with the constraints obtained using @xmath56 observational data @xcite . concerning the exponential and the square - root exponential , we showed that , for realistic regions of free parameters , they always satisfy the bbn bounds . this means that , in these cases , bbn can not impose strict constraints on the values of free parameters .    in summary , we showed that viable @xmath1 models , namely those that pass the basic observational tests , can also satisfy the bbn constraints . this feature acts as an additional advantage of @xmath1 gravity , which might be a successful candidate for describing the gravitational interaction . as discussed in @xcite , this kind of constraints could contribute in the debate of fixing the most realistic picture that can be based on curvature or torsion . this article is based upon work from cost action ca15117 `` cosmology and astrophysics network for theoretical advances and training actions '' ( cantata ) , supported by cost ( european cooperation in science and technology ) . in this appendix we briefly review the main features of big bang nucleosynthesis following @xcite . in the early universe , the primordial @xmath0 was formed at temperature @xmath139 mev . the energy and number density were formed by relativistic leptons ( electron , positron and neutrinos ) and photons . the rapid collisions maintain all these particles in thermal equilibrium . interactions of protons and neutrons were kept in thermal equilibrium by means of their interactions with leptons @xmath140    the neutron abundance is estimated by computing the conversion rate of protons into neutrons , i.e. @xmath141 , and its inverse @xmath78 . thus , the weak interaction rates ( at suitably high temperature ) are given by @xmath142 the rate @xmath143 is the sum of the rates associated to the processes ( [ proc1])-([proc3 ] ) , namely @xmath144 finally , the rate @xmath143 is related to the rate @xmath145 as @xmath146 , with @xmath147 the mass difference of neutron and proton .    during the freeze - out stage , one can use the following approximations @xcite : ( i ) the temperatures of particles are the same , i.e. @xmath148 . ( ii ) the temperature @xmath76 is lower than the typical energies @xmath149 that contribute to the integrals entering the definition of the rates ( one can therefore replace the fermi - dirac distribution with the boltzmann one , namely @xmath150 ) . ( iii ) the electron mass @xmath151 can be neglected with respect to the electron and neutrino energies ( @xmath152 ) . having these in mind , the interaction rate corresponding to the process ( [ proc1 ] ) is given by @xmath153 where @xmath154\ , , \label{wa}\\     { \cal p }   &    \equiv &   p_n+p_{\nu_e}-p_p - p_e\ , ,   \\   { \cal m } & = & \left(\frac{g_w}{8m_w}\right)^2 [ { \bar u}_p\omega^\mu u_n][{\bar u}_e\sigma_\mu v_{\nu_e}]\ , , \label{m } \\ \omega^\mu & \equiv & \gamma^\mu(c_v - c_a \gamma^5)\ , ,    \\ \sigma^\mu &    \equiv & \gamma^\mu(1-\gamma^5 ) \ , . \end{aligned}\ ] ] in ( [ m ] ) we have used the condition @xmath155 , where @xmath156 is the mass of the vector gauge boson @xmath157 , with @xmath158 the transferred momentum . from eq . 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we use bbn observational data on primordial abundance of @xmath0 to constrain @xmath1 gravity . 
 the three most studied viable @xmath1 models , namely the power law , the exponential and the square - root exponential are considered , and the bbn bounds are adopted in order to extract constraints on their free parameters . for the power - law model , 
 we find that the constraints are in agreement with those acquired using late - time cosmological data . for the exponential and the square - root exponential models , we show that for realiable regions of parameters space they always satisfy the bbn bounds . 
 we conclude that viable @xmath1 models can successfully satisfy the bbn constraints .