f(T,B) gravity with statistically fitting of H(z)

S H Shekh,N Myrzakulov,A Bouali and A Pradhan

1 Department of Mathematics,S.P.M.Science and Gilani Arts,Commerce College,Ghatanji,Yavatmal,Maharashtra 445301,India

2 L.N.Gumilyov Eurasian National University,Astana 010008,Kazakhstan

3 Ratbay Myrzakulov Eurasian International Center for Theoretical Physics,Astana,010009,Kazakhstan

4 Laboratory of Physics of Matter and Radiation,Mohammed I University,BP 717,Oujda,Morocco

5 Centre for Cosmology,Astrophysics and Space Science (CCASS),GLA University,Mathura-281 406,Uttar Pradesh,India

Abstract Some recent developments (accelerated expansion) in the Universe cannot be explained by the conventional formulation of general relativity.We apply the recently proposed f(T,B)gravity to investigate the accelerated expansion of the Universe.By parametrizing the Hubble parameter and estimating the best fit values of the model parameters b0,b1,and b2 imposed from Supernovae type Ia,Cosmic Microwave Background,Baryon Acoustic Oscillation,and Hubble data using the Markov Chain Monte Carlo method,we propose a method to determine the precise solutions to the field equations.We then observe that the model appears to be in good agreement with the observations.A change from the deceleration to the acceleration phase of the Universe is shown by the evolution of the deceleration parameter.In addition,we investigate the behavior of the statefinder analysis,equation of state (EoS) parameters,along with the energy conditions.Furthermore,to discuss other cosmological parameters,we consider some wellknown f(T,B) gravity models,specifically,f(T,B)=aTb+cBd.Lastly,we find that the considered f(T,B) gravity models predict that the present Universe is accelerating and the EoS parameter behaves like the ΛCDM model.

Keywords: isotropic homogeneous space-time,f(T,B) gravity,statistical fitting of H(z) data,cosmology

1.Introduction

Currently,one of the most exciting problems in astrophysics and cosmology is finding the physical mechanism responsible for the cosmic acceleration of our Universe in late time[1,2].Various models have been proposed to describe the nature of this phenomenon.One of these is the cosmological constant,which is considered the phenomenologically simplest possibility when cold dark matter constitutes the cosmological standard model known as the ΛCDM.A number of dark energy cosmological models without the cosmological constant have been proposed to explain cosmic acceleration.There are a quintessence (canonical scalar field) [3—5],a phantom(non-canonical scalar field)[6—8],fermion fields[9—12],tachyon fields [13,14],Chaplygin gas with a special equation of state (EoS) [15,16] and so on.All these models have been successfully investigated in the framework of Riemannian gravitational theories with a Levi-Civita connection,where space-time is mediated by curvature in general relativity (GR).Additionally,the scientific community has stimulated interest in modifications of the Einstein GR action,in order to include a higher-order curvature invariant with respect to the curvature.Theories of modified gravity have attracted much attention in the explanation of both early-time and late-time acceleration [17,18].Specific models R2,RabRaband RabcdRabcdcorrections are considered in the literature.These corrections to GR were found to be important and close to the Planck scale.Confrontation with observational data in the case of f(R)gravitational theory[19]was performed in [20,21].Additionally,comparisons with solar system data were performed in [22,23].Some other of the latest investigations in f(R) gravity are given in [24—27].

An alternative gravitation theory that describes gravitational interactions in terms of torsion is the known teleparallel equivalent of general relativity (TEGR) introduced by Einstein [28,29].This theory is conceptually different from the GR,at least at the level of the gravitational equations of motion.Linear frames and tetrads are two basic objects that will be fundamental in the construction of TEGR.In this theory,the Levi-Civita connection is replaced with a so-called Weitzenbock connection.In recent years,extending torsional gravity,namely f(T)gravitational theory has been extensively proposed and investigated in the literature [30,31].The f(T)theory of gravity construction can provide a theoretical interpretation of the early-time and late-time acceleration.The Lagrangian of f(T) is taken to be a nonlinear arbitrary function of the TEGR.Note that local Lorentz invariance in the formulation of f(T) gravity is strongly restricted.

Extended and modified gravities have attracted the attention of many cosmologists,because they provide a geometric and systematic approach to the explanation of cosmological observations.Recently,some authors proposed an interesting extension of modified f(T) gravity namelyf(T,T)gravity,where T is torsion scalar and is theT trace of the matter energy-momentum tensor [32].Inspired by the torsional formulation,the teleparallel equivalent of Gauss-Bonnet gravity has been constructed and is known as f(T,TG)[33].These gravity theories are tested by observational tests at cosmological scales.

Boundaries/boundary terms are fundamental concepts in many areas of theoretical physics.Our interest is another extension of the f(T) theory,in which the Lagrangian takes the form of f(T,B),where T and B are scalars (T is the torsion scalar and B is the boundary term) that characterize the equivalency with GR [34].Paper [35] investigated finding the exact solutions for spherically symmetric Lemaitre-Tolman-Bondi dust models for f(T,B) gravity.Paper [36] explored the cosmological evolution of the Universe with the Lagrange multiplier in teleparallel f(T,B)gravity.The authors performed dynamical analysis and investigated their stationary points.The f(T,B) teleparallel gravity in a five-dimensional brane scenario was studied and the gravitational perturbations were investigated in [37].Work [38] discussed classes of exact and perturbative spherically symmetric solutions in modified teleparallel f(T,B) gravity,while in [39] the authors examined the second law of thermodynamics in f(T,B) theory and using cosmological reconstruction technique,showed that some models of f(T,B) can mimic the de-Sitter,power-law and ΛCDM models.Analysis of the Tsallis holographic dark energy and energy conditions in teleparallel f(T,B) gravity theory were considered in [40,41].

In cosmology,Markov Chain Monte Carlo (MCMC)simulations [42,43] are used to find the best-fitting distance modulus for each cosmological model and for the most probable free parameter value with given certain physical constraints.In this study,we explore the parametrization of the Hubble parameter to obtain the scenario of an accelerating universe.The best fit values of model parameters were obtained from recent observational data: the Hubble datasets H(z),consisting of a list of 57 measurements that were compiled from the differential age method and others;the Type Ia supernovae sample called Pantheon datasets,consisting of 1048 points covering the redshift range 0.01 ＜z ＜2.26;and the Baryon Acoustic Oscillation fs(BAO) datasets,consisting of six points [44,45].Our analysis uses the combination of the H(z),Pantheon samples and BAO datasets to constrain the cosmological model.In order to recreate the shape of the f(T) alteration in a modelindependent manner,Cai et al [46] developed the Gaussian processes analysis for the case of f(T)gravity,utilising as the sole input the observational data sets of the Hubble function measurements H(z).Santos et al [47] have investigated the observational constraints on f(T) gravity from modelindependent data.A model-independent method with phantom dividing line crossing in Weyl-type f(Q,T) gravity has been recently investigated by Koussour [48].Mu et al[49]discussed the most recent cosmic observations,including Pantheon+SNe Ia samples,to reconstruct the modified gravity,which is defined by the modified factor μ in linear matter density perturbation theory,in a completely datadriven and model-independent manner.Model-independent constraints on modified gravity from current data and from the Euclid and SKA future surveys are investigated by Taddei et al[50].In the current study,model-independent constraints on the modified f(T,B) gravity have been examined using data from the H(z),Pantheon samples,and BAO datasets.

The structure of this article is as follows: We present a brief description of f(T,B)gravity in section 2.In section 3,a statistical fitting of H is given.Observational constraints are studied in section 4,while section 5 covers certain particular cosmographic parameters.Some specific cosmological models of f(T,B) gravity are investigated in section 6.Finally,we discuss our findings in section 7.

2.f(T,B) gravity with homogeneous space-time

Basically,the modification of the left side of the Einstein field equations (by some arbitrary function) are called modified theories of gravity,which are the probable access to describe the accelerating expansion of the Universe.A few inspirations to walk around the modified theories of gravity are f(R),f(T),f(R,T),f(G),f(R,G),f(T,B).Among these,one model which is established on the T and B or the combination of f(R)and f(T)gravity,namely f(T,B)gravity[34],is as follows:

By varying the action given in equation (1) with respect to the tetrad,the field equation is defined as:

In the standard cosmological principle,our universe is filled with perfect fluid.Consequently,the energy-momentum tensor for this perfect fluid is defined as the following form:

where p and ρ are the pressure and the energy density of the fluid,respectively.Note thatuν=(0,0,0,1) is the fourvelocity vector of the fluid with uνuν=1.The non-vanishing elements of the energy-momentum tensor are

Here,we explore the spatially,isotropic and homogeneous Friedmann-Robertson-Walker (FRW) line element as

where gijare the function of (t,x1,x2,x3) and t refers to the cosmological time.In four dimensional FRW space-time,from above equation we have

The above relations show that,for the FRW universe,all three metrics are equal(i.e g11=g22=g33=a2(t,x)).For the FRW metric (5),the (0-0) and the (i-i) components of Einstein’s gravitational equation (2) become the following forms:

The overhead dot denotes the derivative with respect to cosmological time t.

The torsion scalar for the FRW line element (5) is

The Ricci curvature scalar and torsion scalar are related as

The curvature scalar R is used in the creation of the entire conventional relativity theory.However,in f(T,B) gravity theory,it is made up of the torsion scalar T and the boundary term B.Be aware that the boundary term [51] makes these models unique.The boundary term B for the FRW metric(5)has

The curvature scalar R from (10) is obtained as

The standard first and second Friedman equations are

The parameters ρtotand ptotin f(T,B) gravity are found as

Using equations (7),(8) and (12)—(15),we find the effective isotropic pressure pdand effective energy density ρdtowards f(T,B) gravity as

where the quantities pdand ρdare the parts of the pressure and energy density respectively that appear from f(T,B) gravity.The expressions of ρdand pdgiven in equations(16)and(17)slightly differ from those in the equations given in(7)and(8)in view of the standard Friedmann equations provided in(12)and(13).Hence,we consider that equations(16)and(17)are the pressure and density for f(T,B) gravity.Now,we have explored the f(T,B) gravity model of the form

For the above-said model of f(T,B) gravity,the set of field equations (16) and (17) become

The next important parameter of the Universe is the equation of state(EoS).As we know,the equation of state parameter is associated with energy density and pressure,which classifies the expansion of the Universe.If the value of the equation of state parameter is exactly 1 then it represents the static fluid era.If it is 0 then the Universe represents the matterdominated era,while forω=the Universe represents the radiation-dominated era.Whereas if it lies in between -1 to 0,i.e.-1 ＜ω ＜0,then the Universe shows the quintessence era,while ω=-1 shows the cosmological constant,i.e.,the ΛCDM era and the phantom era is observed when ω ＜-1.From the above equations,we obtained the equation of state parameter ω as

3.Statistical fitting of H

Note that in general,regression equations express the relationship between two variables.One of our goals is to explore linear and nonlinear regression equations.In the present study,we examine the relationship between the regression equation with the redshift parameter z as a predictor and the Hubble parameter H.To obtain a suitable relation between the redshift z and Hubble parameter H,we have to get the best possible regression relation for our model.To find the best fit we need to calculate R2,whereThe term SST is the sum of squares total.This term mathematically indicates the sum of square deviations of the value of the dependent variables around their mean.And the SSR term stands for the regression sum of squares.This term mathematically indicates the sum of square differences between the sample mean of the prediction and the regression predictions.An application of the regression method presented in [52].Now we will try to study three models in the following forms:

Among the regression relations,whichever gives the highest value of R2close to 1,will be assumed to be the best model.In equations (22),z and H(z),H1(z),H2(z) represent the redshift and Hubble parameter,respectively.Also,b0represents the regression constant and b1,b2are the regression coefficients of the given models.The three models of regression represented in equations (22) are now fitted to the Supernovae type Ia,Cosmic Microwave Background (CMB),Baryon Acoustic Oscillation and Hubble data sets.The three regression models and the corresponding R2towards H(z),H1(z),H2(z)are given in table 1.All of the models have been proven to be almost equally efficient through the high value of R2.Four decimal places have been retained to understand the relative efficiency of each model.Although the three models have equal efficiency,the H(z) model is found to be more accurate than H1(z) and H2(z) (see table 1).

Hence,considering the highest value of R2,the first model of the form is considered to be the best model to establish the functional relationship between z and H(z).

The expression of H(z) with its present value H=0 is obtained as

The first derivative of H(z) is observed as

4.Observational constraints

We have briefly described the f(T,B) gravity and solved the field equation with a new parametrization of the Hubble parameter.In order to extract the best fit values,the considered form of H(z) was constrained by SNIa from Pantheon,CMB from Planck 2018,BAO and 36 data points from Hubble datasets using the MCMC approach.In what follows,we describe in detail the methodology adopted and data used in our analysis.Figures 1 and 2 represent the behavior of the distance modulus and Hubble parameter in terms of the redshift z.The results of our study are shown in the contour plots (twodimensional)with 1-σ and 2-σ errors(figure 3).In figure 3 the chains are run sufficiently to get convergent results.In addition,from the MCMC contour plots,one can notice that the posteriors are smooth with only one maximum.We have again constrained our model parameters by running 10 Markov chains.Furthermore,we have performed a Gelman-Rubin convergence test (please see the table below).The R-1 is smaller than 0.05 for all of the parameters,indicating that our chains have completely reached the convergence region.

4.1.Supernovae type Ia

We use supernovae from the Pantheon compilation made of 1048 spectroscopically confirmed Type Ia Supernovae,distributed in the redshift range 0.01 ＜z ＜2.26 [53].So far,the Pantheon compilation is the largest and it contains measurements from different supernovae surveys such as SDSS,SNLS and HST.For the purpose of estimating the best fit parameters,we compute the chi-square

where μobsis the observed distance modulus and CPantheonis the covariance matrix of the Pantheon data.

4.2.Cosmic microwave background

The χ2for CMB is expressed as follows:

where the CMB covariance matrix CCMB[54].

Fig.1.The distance modulus behavior in terms of the redshift z.

4.3.Baryon acoustic oscillation

The Baryon Acoustic Oscillation seen by galaxy surveys plays a crucial role in the determination of the evolution of the Universe.From BAO datasets we can measure the angular diameter distance DA(z) using clustering perpendicular to the line of sight.In addition,we can measure the expansion rate of the Universe H(z),through clustering along the line of sight.The angular diameter distance DA(z) and the spherical averaged scale DVare related to H(z) as follows

The peak positions of BAO are in general given in terms of DV(z)/rs(z),DA(z)/rs(z) and H(z)rs(z) measured at the drag epoch zdrag,i.e.,where baryons were released from photons.In this paper,we use correlated BAO data and uncorrelated ones [55—57].Hence,the total chi-square for BAO,is expressed as

4.4.Hubble data

For tighter constraints,we also make use of the Hubble measurements H(z).In general,the Hubble rate can be inferred either from the clustering of galaxies/quasars by measuring the BAO in the radial direction [58],or from the differential age method.Both methods lead to a compilation of 36 data points of the Hubble parameter H(z).The chisquare of the Hubble measurements is given by

Finally,the total chi-square,,is given by the sum of all the χ2previously defined:

Fig.2.The Hubble parameter behavior in terms of the redshift z.

Table 1.The three regression models and their corresponding R2.

In the previous sections,we have briefly described the f(T,B)gravity and solved the field equation with a new parametrization of the Hubble parameter.The considered form of H(z)contains two model parameters b1and b2,which have been constrained through some observational data for further analysis.We have used some external datasets,such as an observational Hubble dataset of a recent compilation of 57 data points,the Pantheon compilation of SNeIa data with 1048 data points,and also the Baryonic Acoustic Oscillation dataset with six data points,to obtain the best fit values for these model parameters (see tables 2 and 3),in order to validate our technique.

5.Some cosmographic parameters

Geometrical parameters played a significant role in analyzing the model for any gravity theory.In this section we discuss some cosmographic dimensionless parameters such as the deceleration q(z) and jerk j(z) parameter.The deceleration parameter q(z)is a quantity that shows how the expansion rate changes over time.The jerk parameter j(z) defines the rate of change of acceleration.

The deceleration parameter q(z)in H(z)form is observed as

According to astrophysical data,our universe is in the stage of cosmic acceleration.To understand the entire cosmic history of the Universe,a cosmological model must include both the decelerated and accelerated phases of the expansion.Consequently,it is necessary to investigate the behavior of the deceleration parameter q(z).The behavior of q(z) for the associated values of the model parameters constrained are shown in figure 4.It is clear that the parameter q(z) shows a transition from a decelerated to an accelerated phase.Furthermore,the range at which a transition takes place resembles that of recent observations [59].

Fig.3.The 1-σ and 2-σ confidence contours obtained from SNIa+BAO+H(z) data obtained for the statistical fitting model.

Table 2.Summary of the best fit and the mean values of the cosmological parameters.

In particular,the jerk parameter,a dimensionless third derivative of the scale factor a(t) with respect to cosmic time t,can provide us with the simplest approach to search for departures from the ΛCDM model.It is defined asj(t)=[60].In terms of redshift z and the deceleration parameter q(z),the jerk parameter j(z) can be observed as

Fig.5.Jerk parameter(j (z)) versus redshift.

Fig.6.Energy density parameter of statistical f(T,B)=aT2+cB2 gravity model(ρd)versus redshift(z).

Blandford et al [61] described how the jerk parameterization provides an alternative and a convenient method to describe cosmological models close to the ΛCDM model.A powerful feature of j(z)is that for the ΛCDM model j(z)=1(constant)always.It should be noted here that Sahni et al[62,63]drew attention to the importance of j(z)for discriminating different dark energy models,because any deviation from the value of j(z)=1 would favor a non-ΛCDM model.The behavior of j(z)for the associated values of model parameters constrained is shown in figure 5.It is clear that the parameter j(z) shows that the jerk parameter is positive throughout the evolution and finally acquires the value that tends to one,which describes the ΛCDM model.

6.Some specific models of f(T,B) gravity

One method to solve the equations is by assuming a form of f(T,B) gravity.This way seems more physically interesting because one can propose a specific form of theory f(T,B)gravity.A similar analysis can be done in f(T,B) for the different b and d provided in(18).In order to illustrate this,let us consider some specific models.From the standard considered model(18),we will study the behaviors of energy density and the equation of state parameters,as well the energy conditions for specific models.

6.1.Model-I

In this subsection,we define the f(T,B) gravity model towards b=2 and d=2.With this model the expressions of ρdand ωdfrom equations (19) and (21) are of the form

Fig.7.Equation of state parameter of statistical f(T,B)=aT2+cB2 gravity model(ωd)versus redshift(z).

The behavior of energy density of the statistical f(T,B)gravity model versus redshift is presented in figure 6.and it is observed that the energy density is consistently non-negative and increases with the passage of redshift.

The equation of state parameter is obtained as

Equation (36) represents the expression for the equation of state parameter of the statistical f(T,B)gravity model and its behavior is clearly shown in figure 7 with redshift.One can see in figure 7 that at the late Universe (z ＞0) towards the value of the equation of state parameter of statistical f(T,B)gravity model is less than -1,which represents the Universe involving phantom field dark energy era,while at z=0 and-1,the equation of state parameter of statistical f(T,B)gravity model is -1.Hence for both z=0 and -1 the Universe involves the ΛCDM era,whereas in the early Universe (z ＞0) it consists of both barotropic as well as of a quintessence field dark energy era.

Fig.8.Null,Dominant and Strong energy conditions of f(T,B)=aT2+cB2 gravity models versus redshift(z) .

Fig.9.Energy density parameter of statistical f(T,B)=aT2+cB3 gravity model(ρd)versus redshift(z).

Energy conditionsThe most famous energy conditions NEC,DEC and SEC are observed as

Equations (37),(38) and (39) represent the expression for energy conditions of the statistical f(T,B)gravity model.The behavior of the energy conditions is clearly shown in figure 8 with redshift.In the present universe,it is observed that NEC and DEC are well satisfied throughout cosmic evolution.However,the SEC is violated for our model.The violation of the SEC is due to the accelerated expansion of the Universe.

Fig.10.Equation of state parameter of statistical f(T,B)=aT2+cB3 gravity model(ρd)versus redshift(z).

6.2.Model-II

In this subsection,we define the f(T,B) gravity model towards b=2 and d=3.With this model the expressions of ρdand ωdfrom equations (19) and (21) are of the form

The behavior of the energy density of the statistical f(T,B)gravity model versus redshift is presented in figure 9.and it is observed that the energy density is consistently non-negative and increases with the passage of redshift,which is the same as that of model-I.

The equation of state parameter is obtained as

Equation(41)represents the expression for the EoS parameter of the statistical f(T,B) gravity model and its behavior is clearly shown in figure 10 with redshift.One can see in figure 10 that its behavior also is the same as that of model-I,i.e.,at the late Universe (z ＞0) toward the value of the equation of state parameter of the statistical f(T,B) gravity model is less than -1,which represents the Universe involving phantom field dark energy era,while at z=0 and -1,the equation of state parameter of the statistical f(T,B) gravity model is -1.Hence for both z=0 and -1 the Universe involves the ΛCDM era,whereas in the early Universe (z ＞0) it consists of both barotropic as well as of a quintessence field dark energy era which resembles model-I.

Energy conditions

The most famous energy conditions NEC,DEC and SEC are observed as NEC:

Fig.11.Null,Dominant and Strong energy conditions of f(T,B)=aT2+cB3 gravity models versus redshift(z) .

Fig.12.Energy density parameter of statistical f(T,B)=aT3+cB2 gravity model(ρd)versus redshift(z).

Fig.13.Equation of state parameter of the statistical f(T,B)=aT3+cB2 gravity model(ρ)d versus redshift(z) .

Fig.14.Null,Dominant and Strong energy conditions of f(T,B)=aT3+cB2 gravity models versus redshift(z) .

Equations (41),(42) and (43) represent the expression for energy conditions of the statistical f(T,B)gravity model.The behavior of the energy conditions is clearly shown in figure 11 with redshift.In the present universe,it is observed that NEC and DEC are well satisfied throughout cosmic evolution.However,the SEC is violated for our model.The violation of the SEC is due to the accelerated expansion of the Universe.

6.3.Model-III

In this subsection,we define the f(T,B) gravity model towards b=3 and d=2.With this model the expressions of ρdand ωdfrom the equations (19) and (21) are of the form

The behavior of energy density of the statistical f(T,B)gravity model versus redshift is presented in figure 12.and it is observed that the energy density is consistently nonnegative and increases with the passage of redshift,which is the same as that of model-I.

The equation of state parameter is obtained as

Equation(45)represents the expression for the EoS parameter of the statistical f(T,B) gravity model and its behavior is clearly shown in figure 10 with redshift.One can see in figure 13 that its behavior is also the same as model-I,i.e.,at the late Universe (z ＞0) toward the value of the equation of state parameter of the statistical f(T,B) gravity model is less than -1,which represents the Universe involving phantom field dark energy era,while at z=0 and -1,the equation of state parameter of statistical f(T,B)gravity model is-1.Hence for both z=0 and -1 the Universe involves the ΛCDM era,whereas in the early Universe (z ＞0) it consists of both barotropic as well as of a quintessence field dark energy era.

Energy conditions

The most famous energy conditions NEC,DEC and SEC are observed as NEC:

Equations (46),(47) and (48) represent the expression for energy conditions of the statistical f(T,B)gravity model.The behavior of the energy conditions is clearly shown in figure 14 with redshift.In the present universe,it is observed that NEC and DEC are well satisfied throughout cosmic evolution.However,the SEC is violated for our model.The violation of the SEC is due to the accelerated expansion of the Universe.

7.Conclusions

Teleparallel theories of gravity and their adaptations have received a lot of attention recently,in order to answer numerous cosmological problems.These theories are located in a torsionally supported globally flat manifold.GR has an equivalent teleparallel representation(TEGR)based on torsion(and tetrads) rather than curvature,as is well known (and metrics).Numerous modified teleparallel hypotheses have been put forth from this angle.The first,known as f(T)gravity,is a natural generalization of the TEGR action that is achieved by altering the torsion scalar T in the action.This method is comparable to the metric counterpart of f(R) gravity.The cosmological behavior of the cosmos has been remarkably well described by these two ideas.A modified teleparallel theory of gravity known as the f(T,B) theory,which,within certain bounds,can recover either f(T) or f(R) gravity,was developed with the purpose of unifying both f(R) and f(T)gravity and examining how these theories are related.In this manuscript,we have presented a cosmological analysis for a teleparallel f(T,B) theory of gravity for flat FRW space-time.

The main results for these cosmological models are:

·We have considered a well-known f(T,B)gravity model,namely f(T,B)=aTb+cBd,to examine other cosmological parameters.

·By parametrizing the Hubble parameter and estimating the best fit values of the model parameters b0,b1,and b2imposed from Supernovae type Ia,Cosmic Microwave Background,Baryon Acoustic Oscillation,and Hubble data using the MCMC method,we propose a method to determine the precise solutions to the field equations.We then observed that the model appears to be in good agreement with the observations.

·The Hubble parameter H and the redshift parameter z are related in the current study’s regression equation.We have used the best regression relation feasible to construct an appropriate functional relation between the Hubble parameter H and redshift z.We have calculated R2,where,to determine the best fit.

·We employed supernovae from the Pantheon compilation,which consists of 1048 spectroscopically verified Type Ia Supernovae spread in the redshift range 0.01 ＜z ＜2.26 [53],in our investigation for various dark energy models.

·We calculated the angular diameter distance DA(z) from BAO datasets by clustering perpendicular to the line of sight.Additionally,by seeing clustering along the line of sight,we were able to determine the Universe’s expansion rate,H(z).

·Two model parameters,b1and b2,are present in the considered form of H(z),and they have been restricted for further investigation using certain observational data.For the purpose of validating our methodology,we have used a number of external datasets,including a recently compiled observational Hubble dataset with 57 data points,the Pantheon compilation of SNeIa data with 1048 data points,and a Baryonic Acoustic Oscillation dataset with six data points,to determine the best fit values for these model parameters.

·We have also obtained three specific models for f(T,B) gravity for different values of d and b.In Model I,b=d=2;in Model II,b=2,d=3,and in Model III,b=3,d=2.In all three of these models,we have seen that the cosmos has an ΛCDM era for z=0 and-1,but the early universe for z ＞0 has both a barotropic and a quintessence field dark energy era.Thus,all three models of f(T,B)gravity are in good agreement with the ΛCDM era.Using information from the H(z),Pantheon samples,and BAO datasets,the model-independent constraints on the modified f(T,B)gravity have been investigated in the current study.This type of model-independent constraint on modified gravity from current data has been recently investigated [46—50] and is already mentioned in the introduction.

Also of note is that in the current epoch,all models under study anticipate a violation of the strong energy condition.It is a well-known fact that applying the energy conditions to modified theories of gravity is an open investigation that ultimately leads to a conflict between theory and observation.

·In recent years,f(T,B) gravity has been used to study a number of cosmic aspects.[39] demonstrates how f(T,B) gravity can mimic the de Sitter universe,power law,and CDM models using cosmological reconstruction methods.[64] used this method to present the bouncing solution in this extended teleparallel gravity and to explore the singularity and little rip cosmology,and[65]carried out a stability analysis on the cosmological models that were modelled in conflict with the observational data for the accelerating universe.Four cosmological models that have shown promise in meeting late-time cosmic acceleration measurements that can generate quintessence behavior and experience transition along the phantom-divide line are presented in [66] as the observational side of this class of models.

·In 2023,Kadam et al [67] investigated a different type of cosmological model in f(T,B) gravity,where the Witzenbo¨ ck connection has been used instead of the usual levi-Civita connection by using f(T,B)=αT+βBn.A number of writers have looked into two different f(T,B)gravity model types so far:the first using the Levi-Civita connection and the second using the Witzenbo¨ ck connection.Kadam et al[67]and other authors obtained the cosmological models in f(T,B)gravity by using f(T,B)=αT+βBn.But in the present paper,we have investigated the cosmological models in f(T,B) gravity by using f(T,B)=aTb+cBd.It follows that our model is superior and more general than the prior one.

All of the aforementioned findings give us reason to assume that an extension in the form of f(T,B) gravity could result in intriguing situations where we can think about how torsion and boundary terms combine with astrophysical evidence to shed insight on the study of the late-time accelerating universe.

Acknowledgments

The authors (S H Shekh and A Pradhan) are appreciative of the help and resources given by the IUCAA in Pune,India.Additionally,the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan provided funding for this study (Grant No.AP09058240).The authors are thankful to the anonymous reviewers and editor for their constructive comments,which helped to improve the paper’s quality in its present form.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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