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the revolution in the understanding of the present universe has become possible due to the research in observational cosmology in the past two decade . recent observational data from plank collaboration @xcite , baryon oscillation spectroscopic survey ( boss ) @xcite and atacama cosmology telescope polarimeter ( actpol ) collaboration @xcite provides relevant experimental evidence about the acceleration of our universe . in the last century , modern cosmology reached a new vision to establish considerable advancements in the account of the current accelerated expanding universe . the two crucial observational groups including supernovae cosmology project and the high - redshift supernovae search team have provided the main evidence for the cosmic acceleration of the universe @xcite . the other cosmic observations like cosmic microwave background ( cmb ) fluctuations @xcite , large - scale structure ( lss ) @xcite , cosmic microwave radiation ( cmbr ) @xcite indicate that the present universe is undergoing an accelerated expansion . it is also believed that the universe changed with time from early deceleration phase to late time acceleration phase @xcite . now , a team of scientists led by professor subir sarkar @xcite has raised the question on this standard cosmological concept about the acceleration , which is also supported by shariff et al . + the two promising approaches confirmed by the cosmological research community for discussing the cosmic expansion of the universe are : the first one is the introduction of the most exotic entity of mysterious universe dubbed as dark energy , which has positive energy density and negative pressure . recently , from plank cosmological results and wilkson microwave anisotropic probe ( wamp ) 9 years results @xcite , it may be concluded that the universe embodied with 68.5% dark energy , 26.5% of dark matter and 5% of baryonic matter . dark energy can be represented either in terms of cosmological constant or by the help of equation of state parameter ( eos ) @xmath5 , where @xmath6 is the pressure and @xmath7 is the energy density . + the second approach to picturize the evolution of the universe is a modified version of einstein s field equations of general relativity , which can be established through the einstein - hilbert action principle . in this process , the matter lagrangian is replaced by an arbitrary function . thereafter , these modified theories become most attractive aspirant to observe the accelerated expansion of the universe as well as the effective causes related to dark energy . + in addition , it has been postulated that the standard einstein - hilbert action is modified by an arbitrary function @xmath8 , where @xmath1 is ricci scalar curvature . the @xmath8 gravity become an adequate theory to provide the gravitational alternative for dark energy and about the early inflation plus late - time cosmic acceleration of the universe @xcite . in 2007 , the @xmath8 gravity theory is restructured by merging the matter lagrangian density @xmath9 with initial arbitrary function of the ricci scalar @xmath1 @xcite . the unification of dark energy and early time inflation with late time acceleration from @xmath8 theory to all lorentz non - invariant theories is discussed by nojiri and odintsov @xcite . through continuation of this work of coupling , in 2011 , harko et al . @xcite proposed a new modified theory named as @xmath0 theory , where the gravitational part of the action still depends on the ricci scalar @xmath1 like @xmath8 theories and also a function of trace @xmath2 . it is suggested that due to the matter - energy coupling , the leading model of this theory depends on source term representing the variation of energy - momentum tensor . indefinitely many modified gravitational theories such as @xmath10 gravity , @xmath11 gravity , and @xmath12 gravity etc . were developed to achieve the accelerated expansion of the universe . myrzakulov et al . @xcite have investigated the inflation in @xmath13 theories of gravity where the scalar field is coupled with gravity . sebastiani and myrzakulov @xcite have briefly reviewed various @xmath8 gravity models for inflation , in particular , starobinsky - like inflation . after that , @xmath0 gravity become most prominent theory for investigating the fate of the late time accelerating expansion of the universe . a phase transition also occurred from matter - dominated era to an accelerated phase during the reconstruction of @xmath0 gravity theory @xcite . in the context of common perfect fluid matter , an axially symmetric cosmological model was constructed in the framework of @xmath0 gravity @xcite . in @xmath0 gravity theory , many cosmological models can be constructed changing choices in matter source . sahoo and mishra @xcite constructed kaluza - klein dark energy model by considering the wet dark fluid matter source . recently , moraes et al . @xcite derived the starobinksy model in @xmath0 gravity . + by considering the metric dependent lagrangian density @xmath9 , the respective field equation for @xmath0 gravity is formulated from the hilbert - einstein variational principle in the following manner : @xmath14where , @xmath15 is the usual matter lagrangian density of matter source , @xmath0 is an arbitrary function of ricci scalar @xmath1 and the trace @xmath2 of the energy - momentum tensor @xmath16 of the matter source , and @xmath17 is the determinant of the metric tensor @xmath18 . the energy - momentum tensor @xmath16 from lagrangian matter is defined in the form @xmath19and its trace is @xmath20.here , we have assumed that the matter lagrangian @xmath15 depends only on the metric tensor component @xmath18 rather than its derivatives . hence , we obtain @xmath21by varying the action @xmath22 in eq . ( 1 ) with respect to @xmath18 , the @xmath0 gravity field equations are obtained as @xmath23where , @xmath24 here , @xmath25 , @xmath26 , @xmath27 where @xmath28 is the covariant derivative.contracting eq . we get @xmath29where @xmath30.from eqs ( 4 ) and ( 6 ) , the @xmath0 gravity field equations takes the form @xmath31it is worth mentioning here that the physical nature of the matter field through @xmath32 is used to form the field equations of @xmath0 gravity . to construct different kinds of cosmological models according to the choice of matter source , harko et al . @xcite constructed three types of @xmath0 gravity as presented below @xmath33the individual field equations for each frames of @xmath0 gravity is given as + case - i : @xmath3 @xmath34 case - ii : @xmath4 @xmath35 most of the researchers constructed cosmological models in the presence of perfect fluid matter to analyse the accelerated expansion of the universe . recent observations provide the evidence for the accelerated expansion of the universe due to the presence of an unknown form of energy ( dark energy ) having negative pressure . on that account , we need to construct a cosmological model for expanding universe without considering dark energy and dark matter , while choosing most reliable matter component . it is believed that the cosmic viscosity acts as the dark energy candidate which may play an important role in causing the accelerated expansion of the universe by consuming negative effective pressure @xcite . at present , there is great interest in formulating the cosmological model with dissipative fluid matter rather than with dust ( a pressure - less distribution ) or with a perfect fluid matter , which gives more realistic model instead of others and most effective to pay attention to the dynamical background of homogeneous and isotropic universe . it is commonly accepted that , in the early phase of the universe , the matter behaved like a viscous fluid during the neutrino decoupling in radiation era @xcite . the nature of singularity occurred for perfect fluid can be modified through the dissipative mechanism of viscous fluid . this viscous fluid cosmological model helps to explain the matter distribution on the large entropy per baryon in the present universe . also , the phase transition and string creation are involved with viscous effects as per the grand unified theories ( gut ) . it has been found that the mixture of minimally coupled self - interacting scalar field can successfully derive an accelerated expansion of the universe , while the same mixture with perfect fluid unable to do it @xcite . hence , many cosmological models with viscous fluid in the early universe have been widely discussed in the literature @xcite . again , observations have been conducted to obtain the homogeneity and isotropic properties of the universe . it is believed that at the end of the inflationary era , the geometry of the universe was homogeneous and isotropic @xcite , where the flrw models played an important role is representing both spatially homogeneous and isotropic universe . but the theoretical argument and the anomalies found in cmb provide the evidence for the existence of an anisotropic phase , which is later called isotropic one . after the announcement of plank probe results @xcite , it is believed that the early universe may not have been exactly uniform . thus , the existence of inhomogeneous and anisotropic properties of the universe has gained popularity when it comes to constructing cosmological models under the supervision of anisotropic background . therefore , bianchi type models are very relevant for describing the early universe with the anisotropic background . due to some analytical difficulties in studying the inhomogeneous models , many researchers considered the bianchi type models for investigating the cosmic evolution of the early universe as they are homogeneous and anisotropic . there exist nine types ( @xmath36 ) of bianchi space - times in literature . here , we consider bianchi type - i space - time , as it is the simplest spatially homogeneous and anisotropic . it is also known as the immediate generalization of the flrw flat metric with different scale factors in each spatial direction . in some special cases , the bianchi type - i models include _ kasner metric _ , which helps to govern the dynamics near the singularity . the bianchi type - i cosmological models are more compatible with the simplest mathematical form which attracts various researcher to study different aspects . the nature of bianchi type - i cosmological model has been studied in the context of a viscous fluid . it has been observed that viscosity can cause the qualitative behaviour of solutions near the singularity without removing the total initial big bang singularity @xcite . recently , the bulk viscous matter content has been discussed along with cosmological constant in the framework of bianchi type - i space time @xcite . later , the spatially homogeneous anisotropic bianchi type - i model with bulk viscous fluid matter was studied with the assumption of constant deceleration parameter in the framework of @xmath0 gravity theory through the explanation of the energy conditions @xcite.this work is motivated by the aforesaid literature . the study is presented in different sections . section - i deals with the basic formalism of @xmath0 gravity field equations from einstein - hilbert action along with some essential literature review . in section - ii , we derive the exact solution of both the cases of @xmath0 gravity ( @xmath3 and @xmath4 ) for the spatially homogeneous anisotropic bianchi type - i space - time with the help of time varying deceleration parameter . the energy density , bulk viscous pressure , bulk viscous coefficient , the trace of matter , ricci scalar , the energy conditions are derived and their graphical representations are also discussed in this section . in section - iii , physical parameters of both the models are presented . the detail discussion of the figures are covered in section - iv . the last section contains the conclusion of the covered study . we consider the spatially homogeneous lrs bianchi type - i metric as @xmath37 where @xmath38 are functions of cosmic time @xmath39 only . + the energy momentum tensor for bulk viscus fluid is considered in the form @xmath40 where @xmath41 is the four velocity vector in co - moving coordinate system satisfying @xmath42 @xmath43 is the bulk viscous pressure which satisfies the linear equation of state @xmath44 , @xmath45 is the bulk viscous coefficient , @xmath46 is hubble s parameter , @xmath6 is pressure and @xmath7 is the energy density . + the trace of energy momentum tensor is given as @xmath47      the field equation ( 9 ) with @xmath48 , where @xmath49 is an arbitrary constant for the metric ( 11 ) is obtained as @xmath50 dot represent derivatives with respect to time @xmath51 + the deceleration parameter ( dp ) is defined as @xmath52 where , @xmath53 is the average scale factor . + we have three equations ( 15 - 17 ) involving four parameters as @xmath54 . in order to solve these equations , we assume the time varying deceleration parameter as @xmath55 where , @xmath56 is a constant . + the hubble s parameter is defined as @xmath57 , and from above equation we obtain @xmath58 where , the integrating constant is assumed as unity . integrating ( 20 ) we have found @xmath59 setting @xmath60 , where @xmath61 is the redshift , leads to relation @xmath62 the corresponding @xmath63 is obtained as @xmath64 the deceleration parameter describes the evolution of the universe . the cosmological models of the evolving universe transits from early decelerating phase ( @xmath65 ) to current accelerating phase ( @xmath66 ) . whereas , the models can be classified on the basis of the time dependence of dp . recent observations like sne ia @xcite and cmb anisotropy @xcite confirmed that the present universe is undergoing an accelerated phase of expansion and the value lies in between @xmath67 . fig-1 depicts the behavior of deceleration parameter with respect to redshift , in which the value of @xmath68 lies in specified range of accelerating phase . the transition phase of deceleration parameter from deceleration to acceleration occurs at some transition redshift @xmath69 which is completely decided by the parameter @xmath70 . in this work , we have considered three representative values of @xmath70 i.e. 1.4772 , 1.5 and 1.55 corresponding to @xmath71 , 0.5874 and 0.4706 respectively . the values of transition redshift @xmath69 for our model are agreeing with the observational data @xcite . the transition from deceleration to acceleration phase in @xmath0 gravity with polynomial function of @xmath2 is discussed by moraes et al . our first case clearly shows that the model is completely under accelerated phase which is conformity with observational data .     vs. redshift with different @xmath70.,width=283 ] the volume is defined as @xmath72 . using ( 21 ) in this relation , the values of the metric potentials @xmath38 are obtained as @xmath73 consequently metric ( 11 ) takes the form @xmath74 solving the field equations ( 15 - 17 ) , the values of @xmath7 and @xmath75 are obtained as @xmath76\ ] ] @xmath77\end{gathered}\ ] ] we can observe that the energy density remains positive throughout the evolution of the universe and is a decreasing function of cosmic time @xmath39 . it starts with a positive value and approaches to zero as @xmath78 . the bulk viscous pressure @xmath75 is an increasing function of time , which begins from a large negative value and tends to zero at present epoch . as per the observation , the negative pressure is due to de in the context of accelerated expansion of the universe . hence , the behavior of bulk viscous pressure in our model is agreed with this observation . + the coefficient of bulk viscosity @xmath45 and the pressure are obtained as @xmath79\end{gathered}\ ] ] @xmath80\ ] ]     vs. time with @xmath81 and different @xmath70.,width=245 ]     vs. time with @xmath81 and different @xmath70.,width=245 ]     vs. time with @xmath81 and different @xmath70.,width=245 ]    figure 4 , indicates that the bulk viscous coefficient @xmath45 is positive through out the universe and becomes finite as @xmath78 for our model . + the energy conditions are some alternative conditions for matter content of the theory . in general relativity ( gr ) , the role of these energy conditions is to prove the theorems about the existence of space - time singularity and black holes @xcite . the energy conditions are used in many approaches to understand the evolution of the universe . here , we discussed some of the most popular energy conditions for this model . the weak energy conditions ( wec ) , dominant energy conditions ( dec ) and strong energy conditions ( sec ) are given as @xmath82 these are used in various aspects through their own importance . for example , the dec gives an idea about the stability of matter source and imposes the dark energy along with equation of state parameter @xmath83 for lower bound @xmath84 , which may cause big rip @xcite . sec violation is a typical trait of a positive cosmological constant @xmath85 @xcite . finally , wec shows that the matter - energy is always non - negative . in @xmath0 gravity , the matter support of wormholes is examined by the behavior of energy conditions and the nature of solution for frw model with perfect fluid matter is studied through energy conditions @xcite . according to the predefined literature , one can consider the energy conditions to be useful to analyze the behavior of cosmological solutions throughout the universe . therefore , we dealt with some well - known energy conditions like wec , dec and sec to observe our solutions in both of the cases . figures(5 - 7 ) show the behavior of wec , dec , and sec with the proper choice of constants respectively . +     and @xmath70 with @xmath86.,width=245 ]     and @xmath70 with @xmath86.,width=245 ]     and @xmath70 with @xmath86.,width=245 ]    from figures 5 - 7 , it is observed that all the energy conditions are satisfied for this model . + the values of ricci scalar @xmath1 and the trace of matter source @xmath2 are obtained as @xmath87 = \biggl(6\beta-\frac{27}{2}\biggr)e^{2\beta t}(e^{\beta t}-1)^{-2}-6\beta e^{\beta t}(e^{\beta t}-1)^{-1}\ ] ] @xmath88\end{gathered}\ ] ] using the above equations , the function @xmath0 is obtained as @xmath89     versus @xmath39 and @xmath70 with @xmath90.,width=283 ]    fig . 8 shows the behaviour of the function @xmath0 for this model . in this case we assumed @xmath91 and @xmath92 , where @xmath93 is an arbitrary constant . the corresponding field equations in the general form is @xmath94 where @xmath95 . the set of field equations for the metric ( 11 ) are @xmath96 the above set of field equations ( 38 - 40 ) admits the same solutions ( 24 ) and ( 25 ) as obtained in the previous case . using the metric potentials from ( 24 ) and ( 25 ) , the values of energy density @xmath7 and bulk viscous pressure @xmath75 are @xmath97\ ] ] @xmath98\end{gathered}\ ] ] fig . 9 shows that the energy density @xmath7 is decreasing function of time and remains always positive . it converges to zero as @xmath78 . 10 depicts the variation of @xmath75 versus cosmic time @xmath39 . it is observed from the figure that bulk viscous pressure is increasing function of time from a large negative value and is approaching to zero at present time . + for this case , the values of bulk viscosity coefficient @xmath45 and effective pressure @xmath6 are @xmath99\end{gathered}\ ] ] @xmath100\ ] ]     vs. time with @xmath101 and different @xmath70,width=245 ]     vs. time with @xmath101 and different @xmath70,width=245 ]     vs. time with @xmath101 and different @xmath70,width=245 ]    as suggested in figure 11 , the bulk viscosity coefficient is constant throughout the universe as required . + the weak energy conditions ( wec ) , dominant energy conditions ( dec ) and strong energy conditions ( sec ) for this model are plotted below . +     and @xmath70 with @xmath101.,width=245 ]     and @xmath70 with @xmath101.,width=245 ]     and @xmath70 with @xmath101.,width=245 ]    figures 12 - 14 , show that the energy conditions are completely agreed with gr . + we can obtain the trace of matter @xmath2 for this model as @xmath102\end{gathered}\ ] ] the relation @xmath0 for the above case is obtained in the form @xmath103     versus @xmath39 and @xmath70 with @xmath101.,width=283 ]    fig . 15 shows the behaviour of @xmath0 for @xmath4 model . the rate of expansion of the universe with respect to time is defined by hubble s parameter as well as deceleration parameter . for detail kinematical descriptions of the cosmological expansions can be obtained by taking in account of some extended set of parameters having higher order time derivatives of the scale factor . + the spatial volume turns out to be @xmath104 the above equation indicates that in both the models the spatial volume is zero at initial time @xmath105 . it shows that the evolution of our universe starts with big bang scenario . it is further noted that from ( 21 ) the average scale factor becomes zero at the initial epoch . hence , both models have a point type singularity @xcite . the spatial volume increases with time . + the hubble s parameter @xmath46 , expansion scalar @xmath106 and shear scalar @xmath107 become @xmath108 @xmath109 @xmath110 from the above equations , we can observe that the hubble factor , scalar expansion and shear scalar diverge at @xmath105 and they become finite as @xmath78 . it is noted here that the isotropic condition @xmath111 becomes constant ( from early to late time ) , which shows that the model does not approach isotropy throughout the evolution of the universe . the anisotropy parameter @xmath112 the anisotropic parameter becomes constant for our models . from the above mentioned equation it can be observed that our models are expanding and accelerating universe which starts at a big bang singularity . + * jerk parameter : * + the jerk parameter is considered as one of the important quantity for describing the dynamics of the universe . the models close to @xmath85cdm can be described by the cosmic jerk parameter @xmath113 @xcite . for flat @xmath85cdm model the value of jerk is @xmath114 @xcite . jerk parameter is a dimensionless third derivative of scale factor @xmath53 with respect to cosmic time @xmath39 and is defined as @xmath115 the above expression can be written in terms of deceleration parameter as @xmath116 thus , the jerk parameter for our models is @xmath117     with different @xmath70.,width=283 ]    from the fig . 16 , it is clear that our value does not overlap with the value @xmath118 obtained from combination of three kinematical data sets : the gold sample data of type ia supernovae @xcite , the snia data obtained from the snls project @xcite , and the x - ray galaxy cluster distance measurements @xcite . we have plotted the jerk parameter for different values of @xmath70 in fig . . one can observe that the jerk parameter remains positive through out the universe and is equal to the @xmath85cdm model at @xmath119 for the considered values of @xmath70 . it is interesting to note that our model is close to @xmath85cdm model for the following set of values as presented in table - i . + * @xmath120 parameter : * + the state - finder pair @xmath121 is defined as @xcite @xmath122 the state - finder pair is a geometrical diagnostic parameter , which is constructed from a space - time metric directly , and it is more universal compared to physical variables , which depend on the properties of physical fields describing de , since physical variables are model dependent . for the flat @xmath85cdm model the state - finder pair obtained as @xmath123 @xcite . the values of the state - finder parameter for our model are @xmath124 @xmath125\ ] ] from the expressions of @xmath126 and @xmath127 parameters , we found that @xmath123 only when @xmath128 . the variation of @xmath70 and @xmath39 for @xmath123 is presented in table - i . for the set of values of @xmath129 our models represents @xmath85cdm models , which are presented in table - i . the @xmath120 trajectory of our models is presented in the fig . .variation of @xmath70 and @xmath39 for @xmath121 [ cols="^,^,^,^",options="header " , ]     from the above table - i , it is observed that at initial epoch @xmath105 the parameter @xmath126 becomes unity and @xmath127 becomes finite and diverges for @xmath130 .     vs. @xmath127.,width=283 ] it is well known that the standard @xmath85cdm ( cosmological constant @xmath85+cold dark matter ) model fits well to the currently available observational data . the most recent data from planck+bao+jla+@xmath131 yields the matter density parameter @xmath132 in the @xmath85cdm model @xcite . using this value , we find that the @xmath85cdm universe transits from deceleration to acceleration at the redshift @xmath133 . in our models , it corresponds to @xmath134 as may be seen in figure 1 , where we show the variation of deceleration parameter vs redshift for @xmath134 . thus the models present here have a transition from deceleration to acceleration with transition redshift @xmath69 satisfying the observational data . figure 2 and figure 9 depict the variation of energy density against time in presence of different parameters as presented in the figure for case - i and case - ii respectively . the figures indicate that , in both the cases energy density is positive valued and decreasing function of time . it is also approaching zero with the evolution of time . the variation of bulk viscous pressure and coefficient of bulk viscosity is presented in the figure 3 and figure 10 and figure 4 and figure 11 for case - i and case - ii respectively . from the figures , we noticed that bulk viscous pressure and coefficient of bulk viscosity are negative and positive valued function of time respectively . also , the coefficient of bulk viscosity decreases with the evolution of time and maintains a constant rate after @xmath135 the variation of energy conditions against time for both the cases is presented from figure 5 to figure 7 and figure 12 to figure 14 respectively . in both the cases , energy conditions ( wec , sec , dec ) are satisfied . all the physical parameters presented in both the cases follow the same quantitative behaviour as that of observational data . in this article , we have investigated the lrs bianchi type i cosmological model in presence of bulk viscosity in the framework of @xmath0 gravity . according to the choice of @xmath0 we have presented two cosmological models . the exact solutions of the modified einstein s field equations are obtained under the choice of deceleration parameter of the form ( 19 ) . the observations of both the models are as follows :    * both the models presented here are accelerating and the expanding universe models follow an exponential expansion . * energy density and coefficient of bulk viscosity are positive valued and decreasing function of time in both the cases and also @xmath136 when @xmath137 . * bulk viscosity pressure @xmath138 is negative valued in both the cases . * energy conditions ( sec , wec , dec ) are satisfied for both cases . * jerk parameter and state - 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the objective of this work enclosed with the study of spatially homogeneous anisotropic bianchi type - i universe in @xmath0 gravity ( where @xmath1 is the ricci scalar and @xmath2 is the trace of stress energy momentum tensor ) in two different cases viz . @xmath3 and @xmath4 with bulk viscosity matter content . in this study , we consider a time varying deceleration parameter , which generates an accelerating universe to obtain the exact solution of the field equations . 
 the physical and kinematical properties of both the models are discussed in detail for the future evolution of the universe . 
 we have explored the nature of wec , dec , sec and energy density for both the cases . 
 we have found that both the models , with bulk viscosity matter component , show an acceleration of the universe . 
 we have also shown that the cosmic jerk parameter is compatible with the three kinematical data sets .