1 Introduction

More than a few spectacles have been accounted that an expansion of the Universe is accelerating [1,2,3]. In the recent era, this expansion of Universe is one of the most attractive topics among the Cosmologists. Such expansion of Universe is caused due to some mysterious energy with strong negative pressure dubbed as dark energy. Dark energy can be described by using the equation of State (EoS) parameter (\(\omega\)) which involve the pressure (p) and energy density of dark energy (\(\rho\)) and characterized as \(\omega =\frac{p}{\rho }\). Afterward the unexpected innovation of dark energy [1,2,3], it has one concluded up an existence of the crucial subjects in the theoretical physics as well as the current cosmology. More about the EoS parameter is that, if the EoS parameter lies near to \(-1\) i.e. \(\omega \approx -1\): it would remain alike to standard cosmology, a slight little superior than -1 i. e. \(\omega >-1\): the quintessence dark energy or not as much of than \(-1\) i.e \(\omega <-1\): phantom dark energy and the probability \(\omega \ll -1\) is governed out by existing cosmological data. Along with K-essence, Chaplygin gas and several supplementary limits of EoS parameter are acquired from observational results that come from SNe-Ia data and a mixture of SNe-Ia data with CMB anisotropy and Galaxy clustering statistics which are \(-1.66<\omega <-0.62\) and \(-1.33<\omega <-0.79\), respectively. Also, the most recent outcome in 2009, achieved later a combination of cosmological data circles upcoming from CMB anisotropy, luminosity distances of high red-shift SNe-Ia, and galaxy clustering restrain the dark energy EoS parameter to \(-1.44<\omega <-0.92\). Also, the present-day observations support a cosmological constant \(\Lambda\) being the source of dark energy motivating the present accelerated epoch of the Universe. This cosmological model is termed as the \(\Lambda\)CDM model. Notwithstanding the fact that it is preferred by the observations, the \(\Lambda\)CDM model experiences constant cosmological problems [4,5,6,7].

To overwhelm this, including the mentioned above, various dark energy models have been proposed. As, dark energy problem might be an issue of quantum gravity. In view of that, holographic principle may play an important role in solving the dark energy issue. The holographic dark energy model has become a positive procedure as of late to contemplate the dark energy mystery. For an effective quantum field theory in a box of size L with a short distance cut-off, the total entropy should satisfy the relation \(L^{3} \Lambda ^{3} \le S_{BH} = \pi L^{2} M^{2}_{p}\) where \(M_{p}\) is the reduced Planck mass and \(S_{BH}\) is the entropy of a black hole of radius L which acts as a long distance IR cut-off: L. Therefore, this UV-IR relationship gives an upper bound on the zero point energy density as \(\rho _{\Lambda } \le L^{-2} M^{2}_{p}\) [8].

As of late, a few entropy formalism have been utilized to build and research the cosmological models. A new holographic dark energy model is developed say Renyi holographic dark energy which dependent on the absence of interactions between cosmos sectors, and this model shows more stability by itself. The energy density of Renyi holographic dark energy is characterized as follows

$$\begin{aligned} \rho _{de}=\frac{3d^{2}}{8 \pi L^{2}}(1+\pi \delta L^{2})^{-1}, \end{aligned}$$
(1)

By spreading the holographic principle over dark energy Li et al. [9] predicted, the holographic dark energy model which is a very reasonable candidate of dark energy and originate an accelerating expansion of the Universe with a great coincidence with the present cosmological observation. It is understood that the modified theories can also well explain the issue of dark energy. In the previous decade, a lot of research work has been done in the modified theories of gravitation such as f(R), f(R, T), f(G), f(R, G), and f(T, B) gravity, where R, G and T denotes the Ricci scalar, Guass-Bonnet invariant and trace of energy momentum tensor, respectively.

In view of a viable f(R) gravity models Nojiri and Odintsov [10] demonstrated the unification of early-time inflation and late-time acceleration of the Universe while deriving an exact solution with the help of a power law cosmological model Capozziello et al. [11] achieve dust matter and dark energy phase while Sharif and Yousaf [12] deliberated an impact of dark energy and dark matter models on the dynamical evolution of collapsing self-gravitating systems in this gravity. Harko et al. [13] have investigated a several aspects of f(R, T) gravity including FLRW dust universe and pointed out the matter and time dependent terms in the gravitational field equations play a vigorous role of effective cosmological constant which could lead to some major difference as compared to the standard general relativity. Sharif and Zubair [14] discussed the laws of thermodynamic at the apparent horizon through FLRW universe in this f(R, T) theory of gravity. Chaubey et al. [15] obtained a new class of Bianchi type cosmological models in the same gravity. Mishra and Sahoo [16, 17] have investigated Bianchi type-VIh cosmological model filled with perfect fluid in the framework of f(R, T) gravity. Chirde and Shekh [18] investigated non-static plane symmetric space–time filled with dark energy within the frame work of same modified gravity while the same authors Chirde and Shekh [19] investigated plane symmetric space–time with wet dark fluid which is a candidate for dark energy modeled on the equation of state \(p=\omega (\rho -\rho ^{*})\) in the same f(R, T) theory of gravity. Some recent work in the various modified theories of gravitations are mentioned in [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36].

In the reference [37, 38], the authors who provided that within the framework of general relativity the energy conditions play a significant role in cosmology, black hole thermodynamics and singularity theorems. The energy conditions are the altered ways over which one can force to develop the idea of positive-ness of the stress energy tensor in occurrence of matter, also perceive the attractive nature of the gravity. The ordinary point-wise energy conditions are the null energy condition (NEC), weak energy condition (WEC), strong energy condition (SEC) and dominant energy condition (DEC) for the case of a FLRW space–time in general relativity and the definitions specify to

  • SEC, if \(\rho +3p\ge 0\,\);

  • WEC, if \(\rho \ge 0, \rho +p\ge 0\,\);

  • NEC, if \(\rho +p \ge 0\,\);

  • DEC, if \(\rho \ge 0, |p| \le \rho \,\).

The proof of the second law of black hole thermodynamics be governed by the null energy condition (NEC) [33], whereas the well-known Hawking–Penrose singularity invokes the strong energy condition (SEC) whose violation results in the observed accelerated expansion [39]. On the other WEC is a combination of NEC \((p+\rho \ge 0)\) and \((\rho \ge 0)\). This also means that the local energy density as measured by any time-like observer be positive. The energy conditions are deliberate in several modified theories of gravity. In the references [39,40,41,42,43,44,45,46], some researchers who have recently analyzed the energy conditions in several modified theories of gravity.

Motivating with the analysis and discussion mentioned in the above references, in this article the author analyzed the dynamical investigation of homogeneous and isotropic FLRW space–time with an interacting matter and Renyi holographic dark energy model in the context of modified f(T, B) gravity by considering the UV-IR cut-off: L as a candidate of Hubble’s horizon cut-off. The paper has been sorted out as follows: In Sect. 2, the components of field equations in f(T, B) gravity relating to Renyi holographic dark energy source is formulated. The pressure and energy density of Renyi holographic dark energy in f(T, B) gravity is obtained in Sect. 3 corresponding to FLRW Universe. In Sect. 4, the explication of isotropic and homogeneous FLRW space–time is presented. Some dynamical parameters with their physical acceptability analyzed in Sect. 5 while final concluding remarks are discussed in Sect. 6.

2 The \(\varvec{f(T,B)}\) gravity and Renyi holographic dark energy

Consider the action for the combination of f(R) and f(T) gravity namely f(T, B) gravity [25] as follows

$$\begin{aligned} S=\int e\left( \frac{f(T,B)}{\kappa ^{2} } +L_{m} \right) \text{d}^{4}x , \end{aligned}$$
(2)

where \(\kappa ^{2} =8\pi G\) and f(T, B) is a function of the torsion scalar T and the boundary term \(B=\frac{2}{e} \partial _{\mu } (eT^{\mu } )\) in which \(T_{\mu } =T_{v\mu }^{\mu }\). \(L_{m}\) and \(e=\det (e_{\mu }^{i} )\) are matter action and determinant of tetrad components, respectively.

By varying the action given in Eq. (2) with respect to tetrad, the field equation is defined as

$$\begin{aligned}&2e\delta _{\nu }^{\lambda } \nabla ^{\mu } \nabla _{\mu } \partial _{B} f-2e\nabla ^{\lambda } \nabla _{\nu } \partial _{B} f+eB\partial _{B} f\delta _{\nu }^{\lambda } \nonumber \\&\quad +4e\left( \partial _{\mu } \partial _{B} f+\partial _{\mu } \partial _{T} f\right) S_{\nu }^{\mu \lambda } +4e_{\nu }^{a} \partial _{\mu } \left( eS_{a}^{\mu \lambda } \right) \partial _{T} f\nonumber \\&\quad -4e\partial _{Tf} T^{\sigma } {}_{\mu \nu } S_{a}^{\lambda \mu } -ef\delta _{\nu }^{\lambda }\nonumber \\&\quad =16\pi eT_{\nu }^{\lambda } . \end{aligned}$$
(3)

As we know, in the standard cosmological model, the universe is well thought-out by a perfect fluid. Therefore, the energy-momentum tensor for perfect fluid is written as

$$\begin{aligned} T_{\nu }^{\lambda } =(\rho +p)u_{\nu } u^{\lambda } -p\delta _{\nu }^{\lambda } , \end{aligned}$$
(4)

where \(\rho\) and p be the energy density and the pressure of the fluid inside the Universe, respectively. \(u^{\nu } =\left( 0,0,0,1\right)\) with \(u^{\nu } u_{\nu } =1\) are the comoving coordinates, where \(u^{\nu }\) is the four-velocity vector of the fluid. The nonzero element of the energy - momentum tensor is given by

$$\begin{aligned} T_{0} {}^{0} =\rho , T_{1} {}^{1} =T_{2} {}^{2} =T_{3} {}^{3} =-p. \end{aligned}$$
(5)

For the Universe where Renyi holographic dark energy and dark matter are interacting to each other the total energy density satisfies the continuity equation as following

$$\begin{aligned} {\dot{\rho }}_{m}+{\dot{\rho }}_{r}+3H(\rho _{m}+\rho _{r}+p_{r})=0 \end{aligned}$$
(6)

Once the energy densities of Renyi holographic dark energy and dark matter do not conserve separately, the equation of continuity of matter becomes

$$\begin{aligned} {\dot{\rho }}_{m}+3H(\rho _{m})=Q. \end{aligned}$$
(7)
$$\begin{aligned} {\dot{\rho }}_{r}+3H(\rho _{r}+p_{r})=-Q \end{aligned}$$
(8)

where Q implies the collaboration between dark matter and Renyi holographic dark energy. In general Q should performance a role as inverse of cosmic time. For the suitability, choose \(Q=3\sigma H\rho _{m}\) where \(\sigma\) is the coupling constant. If \(\sigma =0\), the equation of continuity condenses to the non-interacting case, such a kind of interaction is considered by the various authors some of them are mentioned in [47,48,49].

Expending the above interaction \(Q=3\sigma H\rho _{m}\), from Eq. (7) the energy density of pressureless dark matter is obtained as

$$\begin{aligned} \rho _{m}= m_{1}a^{3(\sigma -1)} \end{aligned}$$
(9)

where \(m_{1}\) be the arbitarary constant of integration.

The measurements with red-shift data from SNe-Ia supernovae has directed to the anticipation of an accelerating as well as flat Universe with \(\Omega =\Omega _{m}+\Omega _{r}=1\) where the values of \(\Omega _{m}\) and \(\Omega _{r}\) are 0.3 and 0.7, respectively. This value of the density parameters relates to a cosmological constant that is very small, nevertheless, nonzero and positive. The components of total energy density parameter \(\Omega\) such as energy densities for Renyi holographic dark energy \(\Omega _{r}\) and matter \(\Omega _{m}\) are given as [29]

$$\begin{aligned} \Omega _{r} =\frac{\rho _{r} }{\rho {}_{c\tau } } =\frac{\rho _{r} }{3H^{2} }, \end{aligned}$$
(10)
$$\begin{aligned} \Omega _{m} =\frac{\rho }{\rho _{c\tau } } =\frac{\rho }{3H^{2} }, \end{aligned}$$
(11)

where \(\rho _{c\tau }\) is the critical energy density of the universe.

3 Metric and components of field equations

Consider the spatially homogeneous and isotropic Friedman–Lemâitre–Robertson–Walker (FLRW) line element in the form

$$\begin{aligned} \text{d}s^{2} =\text{d}t^{2} -a^{2} (t)\left[ \frac{\text{d}r^{2}}{1-kr^{2} } +r^{2} \text{d}\theta ^{2} +r^{2} \sin ^{2} \theta \mathrm{\; }\text{d}\phi ^{2} \right] , \end{aligned}$$
(12)

where a be the scale factor of the universe and \((t,r,\theta ,\phi )\) are the comoving coordinates.

The angle \(\theta \mathrm{\; }\) and \(\phi\) are the the standard azimuthal and polar edges of circular directions, with \(0\le \theta \le \pi\) and \(0\le \phi \le 2\pi\). The homogeneity of the universe fixes a unique edge of reference. Additionally, k be the constant describing the curvature of the space. \(k=1\) describes a closed universe, the flat universe is acquired for \(k=0\) and \(k=-1\) relates to an open universe. In this work, the author consider the flat Universe by taking \(k=0\) with endless range.

The equation of motion (3) for the spatially homogeneous and isotropic FLRW line element (12) with the fluid of stress-energy tensor (5) can be written as

$$\begin{aligned}&-3H^{2} (3\partial {}_{B} f+2\partial _{T} f)+3H\partial _{B} {\dot{f}}-3{\dot{H}}\partial _{B} f\nonumber \\&\quad +\frac{1}{2} f=\kappa ^{2} \rho , \end{aligned}$$
(13)
$$\begin{aligned}&-(3H^{2} +{\dot{H}})(3\partial _{B} f+2\partial _{T} f)-2H\partial _{T} {\dot{f}}+\partial _{B} \ddot{f}\nonumber \\&\quad +\frac{1}{2} f=-\kappa ^{2} p. \end{aligned}$$
(14)

The overhead dot represents the differentiation with respect to cosmic time t.

The torsion scalar for the metric (12) is written as

$$\begin{aligned} T=-6H^{2} . \end{aligned}$$
(15)

The torsion scalar and Ricci scalar are related together as

$$\begin{aligned} R=-T+B, \end{aligned}$$
(16)

All the standard activity in relativity is made with Ricci scalar R, however in f(T, B) gravity it is made with torsion scalar (T) alongside boundary term (B). This issue discloses that the models are distinctive just by limit (boundary) term [25]. The limit term for the metric (12) is obtained as

$$\begin{aligned} B=-6\left( {\dot{H}}+3H^{2} \right) . \end{aligned}$$
(17)

The Ricci scalar R from (16) is found as

$$\begin{aligned} R=-6\left( {\dot{H}}+2H^{2} \right) . \end{aligned}$$

However, the standard form of Friedman equations are of the form

$$\begin{aligned} 3H^{2} =\kappa ^{2} \rho _\text{tot}, \end{aligned}$$
(18)
$$\begin{aligned} 2{\dot{H}}+3H^{2} =-\kappa ^{2} {\bar{p}}_\text{tot}. \end{aligned}$$
(19)

As the Universe overwhelms by another fluid other than an ideal fluid in f(T, B) gravity, the parameters \(\rho _\text{tot}\) and \(p_\text{tot}\) are written as follows:

$$\begin{aligned} \rho _\text{tot} =\rho _{m} +\rho _{r} , \end{aligned}$$
(20)
$$\begin{aligned} p_\text{tot} =p_{m}+p_{r} , \end{aligned}$$
(21)

Recently, considering two cases of non-interacting and interacting fluid scenario. Aditya et al. [50] investigated the dark energy phenomenon by studying the Tsallis holographic dark energy within the framework of Brans–Dicke scalar tensor theory of gravity (where Brans–Dicke scalar field is a logarithmic function of the scale factor a(t) and Hubble’s horizon as the IR cut-off).

Using Eqs. (13), (14), (18)–(19), we find \(\rho _{r}\) and \(p_{r}\) as

$$\begin{aligned} \kappa ^{2} \rho _{r}&=3H^{2} (1+3\partial _{B} f+2\partial _{T} f)-3H\partial _{B} {\dot{f}} \\ &\quad+3{\dot{H}}\partial _{B} f-\frac{1}{2} f, \end{aligned}$$
(22)
$$\begin{aligned}\kappa ^{2} p_{r} &=-3H^{2} (1+3\partial _{B} f+2\partial _{T} f) \\ &\quad-{\dot{H}}(2+3\partial _{B} f+2\partial _{T} f)-2H\partial _{T} {\dot{f}}+\partial _{B} \ddot{f}+\frac{1}{2} f, \end{aligned}$$
(23)

where the quantities \(\rho _{r}\) and \(p_{r}\) are the parts of the energy density and pressure in Renyi holographic dark energy, respectively, appears from f(T, B) gravity and these are the representative of dark energy. The expressions of \(\rho _{r}\) and \(p _{r}\) presented in Eqs. (22) and (23) are slightly differ than that of equations presented in (13) and (14) in view of the standard Fridman equations provided in (20) and (21). Here, we have considered the model of the form \(f=f(T,B)=\eta T^{m}+\beta B^{n}\).

4 Explication of isotropic homogeneous space–time

The ultimate remarkable revolution of the modern cosmology (type-Ia supernovae) is a consent on the conclusion that, the Universe has come into a state of accelerating expansion with the range of deceleration parameter \(-1\le q\le 0\). In order to isotropization of the Universe, many authors have used the power-law and exponential law cosmologies which describes only epoch-based evolution of the Universe due to the constancy of the deceleration parameter but this cosmologies do not exhibit the transition of the universe from deceleration to acceleration. In order to explain the transition of the universe from deceleration to acceleration, many authors have used the following special form of scale factor of the Universe:

$$\begin{aligned} q=\frac{\ddot{a}a}{{\dot{a}}^{2}}=-1+\frac{\alpha }{1+a^{\alpha }}, \end{aligned}$$
(24)

where a is mean scale factor of the Universe.

Keep in mind the relation of \(a=\frac{1}{1+z}\) together with the above Eq. (24), the time-redshift relation is obtained as

$$\begin{aligned} t=\frac{1}{\alpha k}\log\left( 1+\frac{1}{(1+z)^{\alpha }}\right) , \end{aligned}$$
(25)

where \(\alpha\) and k both are the positive free parameters (say constant). From Eq. (24), we observed that when \(a=0, q=\alpha -1>0\) for \(a^{\alpha }=\alpha -1\), \(q=0\) and for \(a^{\alpha }>\alpha -1\), \(q<0\). Therefore, the Universe begins with a decelerating expansion and the expansion changes from past decelerating phase to recent accelerating one. This cosmological scenario is in agreement with the remarkable revolution of the modern cosmologcal observations.

5 Dynamical parameters with physical acceptability

Using the time-redshift relation provided in Eq. (25), the expression of deceleration parameter is obtained as

$$\begin{aligned} q=-1+\frac{k(1+z)^{\alpha }}{1+(1+z)^{\alpha }} \end{aligned}$$
(26)

It is recognized that the fundamental quantity whuch enlightening the evolution of the homogeneous and isotropic universe is the deceleration parameter. A cosmological model which specified that the expressions for q(z) is of little help. The value of deceleration parameter lies in the fact that the rate at which the universe accelerates or decelerates its expansion. If the value of deceleration parameter is positive, i.e., \(q(z)>0\), the expansion of the Universe is decelerating while if it is negative, i.e., \(q(z)<0\) the expansion of the Universe is accelerating. The proposed deceleration parameter involving two free parameters \(\alpha\) and k which are valid from matter-dominated epoch \(z\gg 0\) onwards, i.e., up to \(z = -1\). The well-designed form of deceleration parameter obeys \(q(z \gg 1) = 1/2\), which is very much necessitated by cosmic structure formation. Also \(q(z = -1) = -1\) required by thermodynamic arguments.

The performance of the deceleration parameter is described in Fig. 1 where for the fix value of \(k=0.15\) the left panel (a) and right panel (b) correspondingly represents the performance of deceleration parameter (q) versus redshift (z) towards \(\alpha =0.10, 1.5, 2.0\) and \(1\le \alpha \le 2\). From the fig, one can notice that the variation of deceleration parameter is from negative to positive. For this variation the free parameters \(\alpha\) and k are most important quantities because if \(\alpha <1\) the variation does not observed while for \(\alpha >1.5\) and \(z \gg 1\), \(q(z)\ge 1\) which is not in accordance with cosmic structure formation. So as per cosmic structure formation, the Universe exhibits the transition from early deceleration to the current acceleration only for \(\alpha = 1.5\). Also, the results obtained for the transition i.e. redshift z and \(\alpha\) in the present model is consistent with spatially flat \(\Lambda\)CDM as well as are reliable with the work mentioned in literature [51,52,53,54]. Hence fix the value of \(\alpha = 1.5\) throughout the discussion of all dynamical parameters.

Fig. 1
figure 1

Deceleration parameter versus redshift for the appropriate choice of constants \(\alpha =1.0, 1.5, 2.0\) and \(k=0.15\)

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Using the same time-redshift relation (25), the Hubble’s parameter (H) in relation with the expression of expansion scalar is obtained as

$$\begin{aligned} H=\frac{\theta }{3} =k\left( 1+(1+z)^{\alpha }\right) \end{aligned}$$
(27)

In our derived model, it is found that the scalar expansion and the Hubble’s parameter both are consistent all through the expansion of the Universe, which shows that the Universe is expanding. The behavior the Hubble’s parameter is clearly seen in Fig. 2.

Fig. 2
figure 2

The Hubble’s parameter versus redshift for the appropriate choice of constants \(\alpha =1.5\) and \(k=0.15\)

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Constraints on Hubble’s parameter from observation of H ( z ) datasets

In this study, the expansion scenario of the universe be directly investigated by the Hubble parameter as a function of redshift. To measure the value of the Hubble parameter at some definite redshift generally two well-known methods such as the extraction of H(z) from line-of-sight BAO data and the differential age method are used. In order to find the best fit value of the model parameters of our obtained model, we have used the technique of 57 points of Hubble parameter values H(z) with \(\sigma _{\mu }^{2}\) errors of differential age method (31 points) and BAO and other methods (26 points).

For the theoretical, observed values of Hubble parameter and the standard error in the observed Hubble parameter see Table 1 and the best fit curve for the Hubble parameter versus redshift z is represented in Fig. 3.

Table 1 57 points of H(z) data: 31 (DA) and 26 (BAO+other)
Full size table
Fig. 3
figure 3

The plot of Hubble function H(z) versus redshift z

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From Eq. (9), the energy density of matter is obtained as

$$\begin{aligned} \rho _{m}= \frac{c_{1}}{(1+z)^{3\alpha (\sigma -1)}} \end{aligned}$$
(28)

Considering the UV-IR cut-off: L as a candidate of Hubble’s horizone cut-off i.e. \(L=\frac{1}{H}\) together with the value of the Hubble parameter obtained in Eq. (27), the energy density of Renyi holographic dark energy presented in Eq. (1) takes the form

$$\begin{aligned} \rho _{r} =\frac{3d^{2}}{8\pi }\left\{ \frac{k^{4}\left[ 1+(1+z)^{\alpha } \right] ^{4}}{\pi \delta + k^{2}\left[ 1+(1+z)^{\alpha } \right] } \right\} . \end{aligned}$$
(29)

The graphical behavior of energy density of matter and Renyi holographic dark energy versus redshift in f(T, B) gravity model for the appropriate choice of constants \(\alpha =1.5\), \(k =0.15\), \(\sigma = \delta =0.05, 0.10, 0.15\) are, respectively, well-defined in Figs. 4 and 5. From the graphical performance of both the figures, it is observed that both energy densities are consistently non-negative and increases with the the passege of redshift.

Fig. 4
figure 4

The energy density of matter versus redshift for the appropriate choice of constants \(\alpha =1.5\), \(k=0.15\) and \(\sigma = 0.05, 0.10, 0.15\)

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Fig. 5
figure 5

The energy density of matter versus redshift for the appropriate choice of constants \(\alpha =1.5\), \(k=0.15\) and \(\delta = 0.05, 0.10, 0.15\)

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Using Eq. (27), from the Eq. (23) the expression of Renyi holographic dark energy pressure is obtained as

$$\begin{aligned}p_{r} &=-(4+q)\left( H^{2}+m_{1}(2-q)H^{2n}+m_{2}H^{2m}\right)\\ &\quad+\left( \frac{4mm_{2}(n-1)(1+q)-3m_{2}}{m}H^{2m}\right)-\left( \frac{m_{1}(2-q)}{n}\right) H^{2n} \\ &\quad+ \left(\frac{m_{1}(n-1)(2-q)\left[ 2+j+3q+6(3+5q+q^{2})H\right]}{3}\right). \end{aligned}$$
(30)

where \(m_{1} =3n \beta (-6)^{n-1}\), \(m_{2} =\eta m (-6)^{m-1}\) and \(j=\dddot{a}/H^{3}a\).

The graphical behavior of Renyi holographic dark energy pressure versus redshift in f(T, B) gravity model for the appropriate choice of constants \(\alpha =1.5\), \(k =0.15\), \(m=2\) and \(n=1\) are presented in Fig. 6. From the graphical performance, the Renyi holographic dark energy pressure is initially negative and with the expansion it becomes positive (See Fig. 6) i.e. it is seen that from late Universe (\(z<-1\)) to some part of early Universe (\(z>0\)) for the interval \(0<z\le 0.3\) including present Universe (\(z=0\)) Universe, the Renyi holographic dark energy pressure is negative and for \(z>0.3\) it becomes positive.

Fig. 6
figure 6

The Renyi holographic dark energy pressure versus redshift for the appropriate choice of constants \(\alpha =1.5\), \(k=0.15\), \(m=2\) and \(n=1\)

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From Eqs. (29) and (30), the expression of the Renyi holographic dark energy equation of state parameter obtained as

$$\begin{aligned}\omega _{r} =\frac{8\pi }{(3d^{2})}\left\{\frac{\begin{array}{l}\left( \pi \delta + k^{2}\left[1+(1+z)^{\alpha } \right] \right)\{-(4+q)\left(H^{2}+m_{1}(2-q)H^{2n}+m_{2}H^{2m}\right)-\left(\frac{m_{1}(2-q)}{n}\right) H^{2n} \\ \quad+\left(\frac{m_{1}(n-1)(2-q)\left[ 2+j+3q+6(3+5q+q^{2})H\right]}{3}\right)\}+\left( \frac{4mm_{2}(n-1)(1+q)-3m_{2}}{m}\right)H^{2m}\end{array}.}{k^{4}\left[ 1+(1+z)^{\alpha } \right]^{4}}\right\}\end{aligned}$$
(31)

As, according to the definition of the equation of state, it is associated with Renyi holographic dark energy density \(\rho _{r}\) and pressure \(p_{r}\) which classifies the expansion of the Universe. If the value of the equation of state parameter is exact 1 then It represents the stiff fluid, if it is 0 then the Universe is in matter dominated phase while it is 1/3, the Universe is in the radiation dominated phase. Whereas if the equation of state lies in between \(-1\) to 0, i.e., \(-1< \omega _{r} < 0\) then the Universe shows the quintessence phase while \(\omega _{r} = -1\) shows the cosmological constant, i.e., \(\Lambda\)CDM model and the phantom era is observed when \(\omega _{r} <-1\).

In our derived Universe, the behavior of equation of state parameter of Renyi holographic dark energy versus redshift for the appropriate choice of constants is shown in Fig. 7. From Fig. 7, it is observed that for a late Universe at \(z =-1\) towards \(\delta = 0.05, 0.10\) and 0.15, the values of \(\omega _{r}\) are \(-0.5\), \(-1\) and \(-1.5\), respectively. Hence, the late Universe for \(\delta = 0.05\) represents quintessence phase, for \(\delta = 0.10\) the Universe is in good agreement with the recent observational data also consequently fit as good as \(\Lambda\)CDM while for \(\delta = 0.15\) the Universe is in phantom phase. For the present Universe at \(z=0\) towards \(\delta = 0.05, 0.10\) and 0.15, the values of \(\omega _{r}\) are \(-0.18\), \(-0.29\) and \(-0.41\), respectively, which represent the present Universe is in quintessence phase for all \(\delta\) and resumbles with the analysis of [25, 34].

Fig. 7
figure 7

The equation of state parameter of Renyi holographic dark energy versus redshift for the appropriate choice of constants \(\alpha =1.5\), \(k=0.15\), \(d=0.01,\) \(m=2\), \(n=1\), \(\eta =2.5\) \(\beta =2\) and \(\delta =0.05, 0.1, 0.15.\)

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Stability factor of of Renyi holographic dark energy model

For the stability of corresponding solutions of the derived model, the author should check that our universe is physically acceptable with the help of velocity of sound. For this, firstly it is required that the velocity of sound should be less than velocity of light and expressed as

$$\begin{aligned} \vartheta ^{2}_{s}=\frac{\partial p}{\partial \rho } \end{aligned}$$
(32)

The graphical representation of stability factor (velocity of sound) of the model versus redshift is clearly seen in the following Fig. 8, with appropriate choice of constants.

Fig. 8
figure 8

The stability factor of Renyi holographic dark energy versus redshift for the appropriate choice of constants \(\alpha =1.5\), \(k=0.15\), \(d=0.01,\) \(m=2\), \(n=1\), \(\eta =2.5\) \(\beta =2\) and \(\delta =0.05, 0.1, 0.15.\)

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From the Fig. 8, it is concluded that from late to early Universe for all \(\delta\) the velocity of sound is always \(0< \vartheta ^{2}_{s} < 1\) which is the evidance of stable universe. Hence, our derived universe is always stable.

From Eqs. (23) and (24), the components of the overall energy density parameter such as energy densities of Renyi holographic dark energy and dark matter are obtained as

$$\begin{aligned} \Omega _{r}= \frac{d^{2}}{8 \pi } \left\{ \frac{H^{2}}{\pi \delta +H^{2}} \right\} \end{aligned}$$
(33)
$$\begin{aligned} \Omega _{m}= \frac{c_{1}}{3k^{2}}\left\{ \frac{1}{(1+z)^{3\alpha (\sigma -1)}(1+(1+z))^{2}}\right\} \end{aligned}$$
(34)

The behavior of overall energy density parameter of the Universe versus redshift for the appropriate choice of constants is seen in Fig. 9. From the fig. one can see that the overall energy density parameter recommended that the model is flat.

Fig. 9
figure 9

The overall energy density parameter of the Universe versus redshift for the appropriate choice of constants \(\alpha =1.5\), \(k=0.15\), \(d=0.01,\) \(m=2\), \(n=1\), \(\eta =2.5\) \(\beta =2\) and \(\delta =0.05, 0.1, 0.15.\)

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Energy conditions

Taking Eqs. (22) and (23) into WEC, NEC and DEC constraints, we are able to prove that

  • Weak energy conditions (WEC):  \(\rho _{r}\ge 0, \rho _{r}+p_{r} \ge 0\,\);

  • Strong energy condition (NEC):  \(\rho _{r}+3p_{r} \ge 0\,\);

  • Null energy condition (NEC): \(\rho _{r}+p_{r} \ge 0\,\);

  • Dominant energy conditions (DEC): \(\rho _{r} \ge 0, |p_{r}|\le \rho \,\).

NEC:

$$\begin{aligned}p_{r} +\rho _{r}&\ge 0 \Longleftrightarrow -(4+q)\left(H^{2}+m_{1}(2-q)H^{2n}+m_{2}H^{2m}\right)\\ &\quad+\left(\frac{4mm_{2}(n-1)(1+q)-3m_{2}}{m}H^{2m}\right) -\left(\frac{m_{1}(2-q)}{n}\right) H^{2n} \\ &\quad+ \left(\frac{m_{1}(n-1)(2-q)\left[ 2+j+3q+6(3+5q+q^{2})H\right]}{3}\right)\\&\quad +\frac{3d^{2}}{8\pi }\left\{\frac{k^{4}\left[ 1+(1+z)^{\alpha } \right] ^{4}}{\pi \delta +k^{2}\left[ 1+(1+z)^{\alpha } \right]} \right\} \ge 0 .\end{aligned}$$
(35)

WEC: NEC together with

$$\begin{aligned} \rho _{r} \ge 0 \Longleftrightarrow \frac{3d^{2}}{8\pi }\left\{ \frac{k^{4}\left[ 1+(1+z)^{\alpha } \right] ^{4}}{\pi \delta + k^{2}\left[ 1+(1+z)^{\alpha } \right] } \right\} \ge 0 \end{aligned}$$
(36)

DEC:

$$\begin{aligned}&\rho _{r} \ge 0 \Longleftrightarrow \frac{3d^{2}}{8\pi }\left\{ \frac{k^{4}\left[ 1+(1+z)^{\alpha } \right] ^{4}}{\pi \delta + k^{2}\left[ 1+(1+z)^{\alpha } \right] } \right\} \nonumber \\&\quad \ge 0 {\;\;}\text{together with} \end{aligned}$$
(37)
$$\begin{aligned}\rho _{r}-|p_{r}| &\ge 0 \Longleftrightarrow\frac{3d^{2}}{8\pi}\left\{ \frac{k^{4}\left[ 1+(1+z)^{\alpha }\right] ^{4}}{\pi\delta + k^{2}\left[ 1+(1+z)^{\alpha } \right] }\right\}\\&\quad+\left|-(4+q)\left( H^{2}+m_{1}(2-q)H^{2n}+m_{2}H^{2m}\right) \right.\\&\quad-\left( \frac{m_{1}(2-q)}{n}\right) H^{2n}\left(\frac{4mm_{2}(n-1)(1+q)-3m_{2}}{m}H^{2m}\right)\\ &\left.\quad+ \left(\frac{m_{1}(n-1)(2-q)\left[ 2+j+3q+6(3+5q+q^{2})H\right]}{3}\right)\right|\ge 0 . \end{aligned}$$
(38)

In this context, the energy conditions are unbiased modest constrictions on numerous linear combinations of the energy density and pressure. Since normal matter has both positive energy density and positive pressure, hence the normal matter will automatically satisfy the NEC, WEC, DEC and SEC. Among all the four energy conditions NEC and WEC are very significant as the left over energy conditions violate due to the violation of these conditions. Also, the strong energy condition is in the limelight of conversation because according to the recent data of the accelerating universe, the SEC must be violated on cosmological scale [55, 56]. It is noted that the Renyi holographic dark energy f(T, B) gravity model satisfies the NEC, DEC and WEC. The plot of \(\rho _{r}\) and \(\rho _{r}+p_{r}\) are shown in Figs. 5, 10 and 11 for different values of \(\delta\). The negative equation of state parameter implies \(\rho _{r}+3p_{r}<0\) by which the SEC is violated on account of inconsequential dissimilar values with NEC, DEC and WECC. The behavior of SEC for the different values of \(\delta\) is clearly seen in Figs. 10 and 11.

Fig. 10
figure 10

The energy conditions of the Renyi holographic dark energy versus redshift for the appropriate choice of constants \(\alpha =1.5\), \(k=0.15\), \(d=0.01,\) \(m=2\), \(n=1\), \(\eta =2.5\) \(\beta =2\) and \(\delta =0.05.\)

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Fig. 11
figure 11

The energy conditions of the Renyi holographic dark energy versus redshift for the appropriate choice of constants \(\alpha =1.5\), \(k=0.15\), \(d=0.01,\) \(m=2\), \(n=1\), \(\eta =2.5\) \(\beta =2\) and \(\delta =0.10, 0.15\)

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6 Conclusions

In the analysis of interaction between matter and Renyi holographic dark energy model in the context of f(T, B) gravity surrounded by the framework of FLRW space–time, the conclusions that can be drawn from this work is that, the behavior of deceleration parameter is discussed for the different value of \(\alpha\) say \(\alpha = 0.05, 0.10, 0.15\) but as per cosmic structure formation, the stable Universe exhibits the transition from early deceleration to the current acceleration only for \(\alpha = 1.5\) which also consistent with spatially flat \(\Lambda\)CDM. For all the values of constants \(\sigma\) and \(\delta\) the energy density of matter and Renyi holographic dark energy both are consistently non-negative and increases with the passage of redshift. For the late (\(z<-1\)), present (\(z=0\)) and some interval of early Universe (\(0<z\le 0.3\)) the Renyi holographic dark energy pressure is negative (\(p_{r} < 0\)) while for all early Universe (\(z > 0.3\)) it is positive, i.e., (\(p_{r} > 0\)).

The behavior of equation of state parameter is discussed for fix \(\alpha = 1.5\) and different values of \(\delta\). For a late Universe and present Universe towards \(\delta = 0.05, 0.10\) and 0.15, the values of \(\omega _{r}\) are \(-0.5\), \(-1\), \(-1.5\) and \(-0.18\), \(-0.29\), \(-0.4\)1, respectively, which confirms the late Universe represents quintessence phase, the Universe is in good agreement with the recent observational data also consequently fit as good as \(\Lambda\)CDM and the Universe is in phantom phase while the present Universe is in quintessence phase for the respective values of \(\delta\). The graphical performance of overall energy density parameter of the Universe indicates that the model is flat. About the energy conditions, in the analysis of the Renyi holographic dark energy in f(T, B) gravity model the Universe fulfill the NEC, DEC and WEC while violated SEC due to the negative equation of state parameter. Hence, the violation of SEC confirm that the expansion of the Universe is accelerating. Hereafter, in the derived Universe the cosmic acceleration is through energy conditions.