Cosmological Models within f(T, B) Gravity in a Holographic Framework

Abstract

We investigate the cosmological evolution of the universe for a spatially flat FLRW background space within the context of $f(T,B)$ gravity, which is a recently formulated teleparallel theory that connects both $f\left(T\right)$ and $f\left(R\right)$ gravity under suitable limits. The analysis focuses on four different $f(T,B)$ cosmological models corresponding to various choices of scale factor, namely, emergent, logamediate, and intermediate. In addition to this, we assume a power law-like function of $f(T,B)$ gravity. The reconstruction of $f(T,B)$ gravity considers the Holographic Ricci Dark Energy (HRDE) as the background fluid. We analyze the equation of state parameters and the squared speed of sound for the reconstructed models. Finally, we conduct a thermodynamical analysis for each reconstructed model. The generalized second law of thermodynamics (GSLT) is valid for the four different $f(T,B)$ cosmological models.

1. Introduction

The current cosmology scenario is teeming with a plethora of models that aim to explain what is perhaps simultaneously the most exciting and confounding phenomenon observed in the past few decades, the late-time acceleration [1,2,3,4]. It was first discovered in 1998 when observations of SNeIa collated by the high-redshift SN team [5] and SN cosmology project [6] appeared as illuminating candles, indicating that the universe’s expansion is accelerating. Ever since, increasing observational evidence [7,8] has only affirmed the accelerated expanding paradigm of the Universe. These measurements and observations have resulted in the introduction of a new mysterious energy component known as dark energy [9,10,11,12,13,14], which is attributed to a negative pressure. Over the years, robust efforts have been made to understand the acceleration phenomenon. For this purpose, researchers have proposed various dark energy models such as the cosmological constant [15,16], which is considered to be the simplest model phenomenologically and is known as the ΛCDM [17] model when cold dark matter constitutes the standard cosmological model. Models without the cosmological constant include scalar fields [18], tachyon fields [19], Chaplygin gas [20,21], bouncing models [22,23], braneworld models [24,25,26], and so on. These models have been studied in the framework of General Relativity (GR) [27], where the space–time is mediated by by curvature. However, the interest in modified [28,29,30,31] and extended [32,33,34] theories of gravity has been increasingly attracting the attention of cosmologists in recent years, as it seems to be a promising alternative to GR in providing a systematic and geometric explanation of numerous cosmological phenomena. In [35], readers can find an extensive review of modified gravities that have been looked into as gravitational alternatives for dark energy. The authors have shown that the rich cosmological structure within the realms of modified gravity could naturally lead to an effective cosmological constant, quintessence, or phantom era in the late universe, with the possibility of a transition from deceleration to acceleration or crossing of the phantom divide if necessary, due to gravitational terms which increase with decreasing scalar curvature. In addition, they demonstrated that some of the models discussed in their work could pass the solar system tests.

In modified theories, also known as f theories, the Einstein–Hilbert action [36] of GR provided by $S=\int \left({\displaystyle \frac{R}{2{\kappa }^2}}+L_m\right)\sqrt{-g}{d}^{4}x$ (where R is the Ricci scalar, $\kappa$ is the gravitational coupling constant, and $\sqrt{-g}$ is the determinant of the metric tensor) is replaced by a more general action. The $f\left(R\right)$ gravity theory [37,38,39] is considered to be the simplest f theory. This approach introduces an arbitrary function of the Ricci scalar; the Einstein–Hilbert action is recovered when $f\left(R\right)$ is a linear function. Comparisons with observational data in the case of $f\left(R\right)$ theory have been studied in [40,41,42,43,44]. An important work in the context of a unified description of the inflationary era with dark energy within a modified gravity framework was carried out by [45]. In this work, the authors have provided the latest developments in modified gravity, aiming to provide a virtual “toolbox” containing all the necessary information on inflation, dark energy, and bouncing solutions in the context of various forms of modified gravity. Other proposed modified gravity models include $f(R,\mathcal{T})$ gravity ($\mathcal{T}$ is the trace of the energy–momentum tensor ${\mathcal{T}}_{\alpha \beta }$) [46,47], $f(R,\mathcal{T},\mathcal{Q})$ gravity ($\mathcal{Q}=R_{\alpha \beta }{\mathcal{T}}_{\alpha \beta }$) [48], scalar–tensor theories [49,50,51], and more. In [52], the authors have discussed the structure and cosmological properties of various modified theories, including $f\left(R\right)$ theories, scalar–tensor theory, Gauss–Bonnet theory, nonlocal gravity, non-minimally coupled models, Horava–Lifshitz $f\left(R\right)$ gravity, etc. The paper is focused on the possible unification of early-time inflation with late-time acceleration within such theories when assuming a spatially flat FRW cosmology. They demonstrated that the qualitative possibility of such a unification is a very natural property for the discussed alternative gravities.

In addition, another critical alternative theory of gravitation in terms of torsion has been introduced, known as the Teleparallel Equivalent of General Telativity (TEGR) [53,54]. It was first proposed by Einstein; in this theory, the Levi-Civita Connection is substituted by a so-called Weitzenbock connection [55]. Thus, while GR is based on Riemannian geometric foundations, the teleparallel theory of gravity is based on the work by Weitzenbock and others, who laid the foundations for a torsional rather than curvature-based formulation of gravity. Extensive research on torsional gravity in recent years, specifically $f\left(T\right)$ gravity, can be found in [56,57,58,59,60]. We mention here that the local Lorentz invariance breaks down in the case of $f\left(T\right)$ gravity formulation, which is its major problem. Thus, extended and modified forms of teleparallel gravity have been introduced to construct a covariant formulation of $f\left(T\right)$ gravity. For example, the new approach in [61] includes choosing a nonzero spin connection and a pure gauge. Readers can also refer to [62], where a more general approach is considered containing the squares of the irreducible parts of the torsion $f(T_{ax},T_{vec},T_{ten})$. Despite the loss of Lorentz invariance, the standard teleparallel approach is still very prevalent among research topics in the literature. This arises from the fact that the covariant issue can somehow be “abated” (though only at the level of the field equations) by choosing the correct tetrads [63]. We can also look into the study of the existence and stability conditions for certain notable exact relativistic solutions within the context of a higher-order modified teleparallel gravity theory, as carried out by A. Paliathanasis [64]. Utilizing a Lagrange multiplier, their theory is equivalent to General Relativity with a minimally coupled noncanonical field. This work explores de Sitter and ideal gas solutions in vacuum as well as the behavior of scaling solutions in the presence of matter, and demonstrates how the non-canonical scalar field reproduces the Hubble function of $\Lambda$-cosmology. In the case of FLRW cosmology, it is always possible to obtain “good tetrads” for the non-trivial cosmological solutions. An additional alluring property correlating GR and TEGR is that the Ricci scalar is equivalent to the sum of the torsion and total divergence term B (the boundary term). In this context, we may refer to [65], in which the authors have proposed an interesting model termed the $f(T,B)$ model, wherein the torsion and boundary scalars contribute independently of each other through the arbitrary function f. Notably, this theory becomes equivalent to $f\left(R\right)$ theory for choosing a special form of $f(-T+B)$. It has been shown [66] that ΛCDM models can be reconstructed for teleparallel gravity and that holographic dark energy models can be described. In this context, the work by A. Paliathanasis and G. Leon should also be mentioned [67]. In their analysis of $f(T,B)$ gravity in a non-flat universe, assuming $f(T,B)$ to be a linear function and introducing a scalar field, the results show that the model admits a mini-superspace description and generalizes certain results pertaining to $f(T,B)$ gravity for a spatially flat FLRW geometry. The findings of this study are crucial for gaining insights into spatial curvature within the framework of teleparallelism. Moreover, they spark interest in further exploration of the $f(T,B)$ theory due to its particular asymptotic behavior.

While investigating the various cosmological scenarios, we may also consider various revolutionary theories emerging from string theory and black hole thermodynamics. These startling theories have illuminated some unexpected corners of the nature of space–time and its relation to energy, matter, and entropy, which in turn have had grave implications in cosmology. The holographic principle [68,69,70] is an example of a radical change in modern concepts. This principle requires that the degrees of freedom of a spatial region reside on the surface of the region rather than in the interior. Additionally, it states that the number of degrees of freedom per unit area should not be greater than one per Planck area; thus, the area of a region in Planck units must not be exceeded by its entropy. Fischler and Susskind [71] first proposed a cosmological version of the holographic principle. The holographic Nojiri–Odintsov model [72] is the most general holographic dark energy (HDE) model, and all other known HDE models [73,74,75,76,77,78] are particular examples of this model. A holographic approach to describing the early acceleration and the late-time acceleration eras of our universe can be found in [79]. So-called “holographic unification” has been demonstrated in the context of $f\left(R\right)$ and $f\left(G\right)$ gravity theories, wherein the IR cutoffs are taken in terms of the particle horizon or future horizon and their derivatives. This work proves how the holographic principle can be instrumental in unifying the cosmological eras of the universe. Another work that deserves mention in the context of such a unification scenario is the study carried out by [80]. Their work proposes a modified holographic cutoff, which provides a smooth and unified cosmological scenario from a constant-roll inflation era to the dark energy era at the late-time of the universe. Inspired by the prevailing ideas on holographic dark energy, Gao et al. [81] proposed the HRDE model, in which the IR cutoff in the holographic model is taken to be the average radius of the Ricci scalar curvature, i.e., ${\left|\mathcal{R}\right|}^{-{\displaystyle \frac{1}{2}}}$. In this case, the holographic dark energy density is ${\rho }_{\Lambda }\propto \mathcal{R}$. Highly generalized versions of HDE were presented in [82], while [83] provides a more detailed description. These studies have concluded that the HRDE model works reasonably well in explaining observations such as cosmic acceleration, possibly leading to understanding the problem of cosmic coincidence. Section 3 of our work is dedicated to reconstructing $f(T,B)$ gravity with the HRDE taken as the background fluid. Therefore, in our work we have aimed to apply cosmological reconstruction methods to this theory, assuming the Holographic Ricci dark energy (HRDE) as the background fluid in three different scenarios corresponding to three different forms of scale factor (emergent, intermediate, and logamediate), then proceeding to study various cosmological properties of this model such as its thermodynamics and the EoS parameter in order to investigate the late-time acceleration within the context of our model.

There is an established connection between gravitation and thermodynamics; thus, it might be inferred that a connection can be created between the horizon entropy and the area of a black hole. Investigations into the second law of thermodynamics in the context of horizon cosmology can be found in [84]. In particular, the authors consider different forms of horizon entropy; for each of these, they focus on different cosmological epochs of the universe. In [85], the authors show such a connection between the FLRW equations and the first law of thermodynamics (FLT) at the apparent horizon for $T_h={\displaystyle \frac{1}{2\pi r_A}}$, $S={\displaystyle \frac{\pi r_A^2}{G}}$, where $T_h$ is the temperature and $r_A$ is the radius of the apparent horizon. It was shown in [86] that the Friedmann equations in GR can be written as $dE=T_{h}dS+WdV$, where the work term W is defined as $W={\displaystyle \frac{1}{2}}(\rho -p)$. In addition, their work has been extended to braneworld gravity [87,88], scalar–tensor gravity [89], $f\left(R\right)$ gravity [90], and, Lovelock gravity [91]. In [92], the authors determined a general form of entropy that connects Friedmann equations for any gravity theory with the apparent horizon thermodynamics, then proceeded to find the respective entropies for several modified theories of gravity. In the same work, they also proposed a modified thermodynamic law for the apparent horizon that is free from specific difficulties and proves valid for all EOSs of the matter field. In the context of $f(T,B)$ gravity, the generalized first and second laws of thermodynamics have been studied for different forms of the function in [93,94,95]. Here, we are interested in studying the generalized second law of thermodynamics for the different reconstructed $f(T,B)$ models corresponding to different scale factors, which we do both with and without using the first law.

The rest of our work is organized as follows. In Section 2, we briefly introduce the $f(T,B)$ cosmology with all the basic formalisms required for the reconstruction of $f(T,B)$ gravity. Next, $f(T,B)$ gravity is incorporated in Section 3, where each subsection deals with a different scale factor. Finally, the thermodynamical analysis for each obtained reconstructed model is presented in Section 4.

2. f(T, B) Cosmology

Before delving into the cosmological reconstruction of $f(T,B)$ models, we first explore the cosmology that arises from $f(T,B)$ gravity, a fourth-order generalized teleparallel theory of gravity considering a flat homogeneous and isotropic metric. The FLRW metric, which describes the space–time in Cartesian coordinates, is provided by [96]

$$d{s}^2=-d{t}^2+a{\left(t\right)}^2(d{x}^2+d{y}^2+d{z}^2),$$

(1)

where $a\left(t\right)$ is the scale factor. The choice of tetrad is [97]

$$e_{\mu }^{\alpha }=diag(1,a\left(t\right),a\left(t\right),a\left(t\right)),$$

(2)

in which the spin connections are allowed to be zero, i.e., ${\omega }_{b\mu }^a=0$. In $f(T,B)$ gravity, the integral of the gravitational action is a function f of the scalar T and of the boundary term B, i.e., [98]

$$S={\displaystyle \frac{1}{16\pi G}}\int d^{4}xef(T,B),$$

(3)

where $e=det\left(e_{\mu }^i\right)=\sqrt{-g}$. We note here that infinite choices for the tetrad satisfy Equation (2), yet only a small subset are considered good tetrads, meaning that they have a vanishing spin connection. It can be proven that this choice of tetrad shows the second- and fourth-order contributions of the torsion scalar T and the boundary term [99]:

$$T=6H^2$$

(4)

and

$$B=6(\dot{H}+3H^2).$$

(5)

Hence, $f\left(R\right)$ gravity exists as a subset of $f(T,B)$ gravity in which

$$f(T,B):=f(-T+B)=f\left(R\right).$$

(6)

This choice of tetrad shows the second- and fourth-order contributions of the torsion scalar T and the boundary term B.

Now, if the universe is considered to be filled with a perfect fluid and the FLRW tetrad in (2) is taken, then the field equations for $f(T,B)$ gravity become

$$-3H^2(3f_B+2f_T)+3H\dot{f_B}-3\dot{H}{f}_B+{\displaystyle \frac{1}{2}}f(T,B)={\kappa }^2{\rho }_m,$$

(7)

$$-3H^2(3f_B+2f_T)-\dot{H}(3f_B+2f_T)-2H\dot{f_T}+\ddot{f_B}+{\displaystyle \frac{1}{2}}f(T,B)=-{\kappa }^2{p}_m,$$

(8)

where $H={\displaystyle \frac{\dot{a}}{a}}$ is the Hubble parameter, the dots represent the differentiation with respect to t, and $f_T$ represents the derivative of $f(T,B)$ with respect to T. Similarly, $f_B$ denotes the derivative of $f(T,B)$ with respect to B, while ${\rho }_m$ and $p_m$ represent the energy density and pressure of the matter content, respectively. Equations (7) and (8) can be written in fluid form as

$$3H^2={\kappa }_{eff}^2({\rho }_m+{\rho }_{TB}),$$

(9)

$$2\dot{H}=-{\kappa }_{eff}^2({\rho }_m+p_m+{\rho }_{TB}+p_{TB}).$$

(10)

The above equations are akin to standard FLRW equations in GR. Taking ${\kappa }_{eff}^2=-{\displaystyle \frac{{\kappa }^2}{2f_T}}$, the quantities appearing in the above equations can be written in terms of $f(T,B)$ gravity as follows:

$${\rho }_{TB}={\displaystyle \frac{1}{{\kappa }^2}}\left[-3H\dot{f_B}+(3\dot{H}+9H^2)f_B-{\displaystyle \frac{1}{2}}f(T,B)\right],$$

(11)

$$p_{TB}={\displaystyle \frac{1}{{\kappa }^2}}\left[{\displaystyle \frac{1}{2}}f(T,B)+\dot{H}(2f_T-3f_B)-2H\dot{f_T}-9H^2{f}_B+\ddot{f_B}\right].$$

(12)

The basic equations in this section help in proceeding with the reconstruction and thermodynamical analysis of our model.

3. Reconstruction of f(T, B) Gravity

The infrared cutoff of a quantum field theory, which is connected to the vacuum energy, as well as the theory’s maximum distance, are established by the holographic principle, which has its roots in black hole thermodynamics and string theory. In this section, we have endeavored to reconstruct the $f(T,B)$ gravity associated with the flat FLRW cosmology in the background of Holographic Ricci Dark Energy (HRDE). Different scale factors, viz. emergent, intermediate, and logamediate scale factors, have been employed to construct different cosmological models and study the corresponding EoS parameter. The universe is considered to be filled by a holographic fluid, the energy density of which is given as the holographic Ricci dark energy (HRDE) [100]:

$${\rho }_{\Lambda }=3c^2(\dot{H}+2H^2).$$

(13)

In the subsections below, we reconstruct three different $f(T,B)$ cosmological models corresponding to three different types of scale factors within the background of HRDE and study their properties, such as thermodynamics, in $f(T,B)$ cosmology.

3.1. Emergent Cosmological Model

Extensive investigations [101,102,103,104] have been carried out into the possibilities of an emergent universe that is ever-existing and large enough that space–time may be treated classically. We may mention here that as there is no time-like singularity in these models, the universe is in an almost static state in the infinite past. Nonetheless, it eventually evolves into an inflationary stage. Thus, a model of a perpetually existing universe which eventually enters into the Big Bang epoch is of considerable interest to us. A general framework for an emergent universe has been shown in [105]. This section aims to study the EoS for such a universe considering the HRDE as the exotic fluid within the context of $f(T,B)$ gravity.

The emergent scale factor is written as follows [106]:

$$a\left[t\right]=a_0{(e^{t\nu }+\lambda )}^n$$

(14)

where $a_0>0$, $\lambda >1$, $\nu >0$ and $n>1$. Because $H={\displaystyle \frac{\dot{a}}{a}}$, we obtain the Hubble parameter as a function of t:

$$H={\displaystyle \frac{\nu nlog\left(e\right)e^{\nu t}}{e^{\nu t}+\lambda }}.$$

(15)

Thus, the energy density of the HRDE is obtained by substituting the above equation in Equation (13) as follows:

$${\displaystyle \frac{3c^2}{e^{\nu t}+\lambda }}\left[n{\nu }^{2}lo{g}^2\left(e\right)\left({\displaystyle \frac{2e^{2\nu t}}{e^{\nu t}+\lambda }}-{\displaystyle \frac{e^{2\nu t}}{e^{\nu t}+\lambda }}+e^{\nu t}\right)\right].$$

(16)

In addition, T and B can be obtained from Equations (4) and (5), respectively, as

$$T={\displaystyle \frac{6{\nu }^2{n}^2{e}^{2\nu t}{log}^2\left(e\right)}{{(e^{\nu t}+\lambda )}^2}}$$

(17)

and

$$B={\displaystyle \frac{6n{\nu }^2{e}^{\nu t}{log}^2\left(e\right)(3n{e}^{\nu t}+\lambda )}{{(e^{\nu t}+\lambda )}^2}}.$$

(18)

Thus, the derivative of B with respect to t is

$$\dot{B}={\displaystyle \frac{6n{\nu }^3{e}^{t\nu }\lambda }{{(e^{t\nu }+\lambda )}^3}}[e^{t\nu }(6n-1)+\lambda ]].$$

(19)

Assuming that the function can be written in the form

$$f(T,B)=f_1\left(T\right)+f_2\left(B\right),$$

(20)

using Equations (17) and (18), Equation (7) becomes

$${\displaystyle \frac{1}{2}}{f}_1\left(T\right)-T{f}_{1,T}-{\kappa }^2{\rho }_m=K,$$

(21)

$$B{f}_B-B^2{(3e^{t\nu }+\lambda )}^2\left[1-{\displaystyle \frac{e^{t\nu }(9n+\lambda )}{(3e^{t\nu }+\lambda )}}{)}^2\right]f_{2,BB}-f_{2,B}=2K,$$

(22)

where K is a constant for the method of separation, $f_{1,T}={\displaystyle \frac{d{f}_1}{dT}}$, and $f_{2,B}={\displaystyle \frac{d{f}_2}{dB}}$. For the reconstruction of this model, we equate ${\rho }_{TB}$ with ${\rho }_{\Lambda }$, which is the energy density of HRDE, in Equation (9), which then changes to the following form:

$$3H^2=-{\displaystyle \frac{1}{2f_T}}({\rho }_{m0}{a}^{-3}+3c^2(\dot{H}+2H^2)).$$

(23)

Here, we have taken $\kappa =1$ and ${\rho }_m={\rho }_{m0}{a}^{-3}$ by using the matter conservation equation. Thus, by making the essential substitutions we have obtained the reconstructed $f(T,B)$ and plotted its behavior against the redshift z, as shown in Figure 1.

Next, we use Equation (11) to derive the reconstructed ${\rho }_{TB}$ as follows:

$$\begin{array}{cc}\hfill {\rho }_{TB}=& {\displaystyle \frac{1}{2}}\left[-C_1-Z{\displaystyle \frac{n\nu Y^{-1-3n}log\left(e\right)}{a_0^3(2+9n(1+n))X}}(e^{\nu t}n{\rho }_m+3a_0^3{Y}^{3n}(2n{e}^{t\nu }+\lambda ))\right]\hfill \\ \hfill & +{\displaystyle \frac{J\nu log\left[e\right]}{X^2}}[-3a_0^3(1+3n)(2+3n)\lambda Y^{3n}(I_1+I_2)]\hfill \\ \hfill & +{\displaystyle \frac{J\nu log\left[e\right]}{X^2}}[-3a_0^3(1+3n)(2+3n){\lambda }^3{Y}^{3n}{e}^{t\nu }(1+2n)(12n{e}^{t\nu }+1)]\hfill \\ \hfill & +{\displaystyle \frac{2J{n}^2\nu log\left[e\right]{\rho }_m}{X^2}}[-3nL-\lambda M-3N-15n{e}^{t\nu }{\lambda }^3-{\lambda }^4]\hfill \end{array}$$

(24)

where

$$\begin{array}{cc}\hfill X& =e^{t\nu }(6n-1)+\lambda \hfill \\ \hfill Y& {=}^{t\nu }+\lambda \hfill \\ \hfill Z& =-6e^{t\nu }(2+9n(1+n))\hfill \\ \hfill I_1& =2e^{3t\nu }(4n-1(3n+1(6n-1)))\hfill \\ \hfill I_2& =3e^{t\nu }{(2n+1)}^2(6n-1)\lambda \hfill \\ \hfill J& ={\displaystyle \frac{n\nu log\left[e\right]Y^{-2-3n}}{2a_0(2+9n(1+n))}}\hfill \\ \hfill L& =e^{4t\nu }(1+3n)(2+3n)(6n-1)\hfill \\ \hfill M& =e^{3t\nu }(2+3n)(9n-1(7n-1))\lambda \hfill \\ \hfill N& =e^{2t\nu }(4n+1)(6n-1){\lambda }^2\hfill \end{array}$$

and the pressure $p_{\left(TB\right)}$ has been obtained using the following conservation equation with the additional geometric component:

$$\dot{{\rho }_{TB}}+3H({\rho }_{TB}+p_{TB})={\displaystyle \frac{T}{2{\kappa }^2}}\left(2\dot{f_T}\right).$$

(25)

Here, $f_T={\displaystyle \frac{\dot{f}}{\dot{T}}}$, i.e., the derivative of $f_{TB}$ with respect to T; thus, $\dot{f_T}$ represents the derivative of $f_T$ with respect to t. These substitutions help us to derive the reconstructed pressure $p_{TB}$. Finally, using the reconstructed pressure and density, we obtain the EoS parameter ($\omega$) by substituting the equations in $\omega ={\displaystyle \frac{p_{TB}}{{\rho }_{TB}}}$. The graph for the evolution of the EoS parameter against the redshift is shown in Figure 2. Figure 2 shows that for varying choices of $\lambda$, the EoS parameter shows a phantom behavior and asymptotically tends to $-1$ at a later stage.

The nature of dark energy can be investigated through its EoS parameter and the sound speed of perturbations [107] to the dark energy density and pressure. These perturbations can be described mathematically through the sound speed [108]:

$$c_S^2={\displaystyle \frac{\dot{p_{TB}}}{\dot{{\rho }_{TB}}}}.$$

(26)

For our case, the derivative of the reconstructed density obtained in Equation (24) and the pressure $p_{TB}$ can be obtained and substituted in Equation (26) to obtain the sound speed for our particular emergent model in $f(T,B)$ gravity. The behavior of $c_S^2$ is plotted against the redshift z, with the results shown in Figure 3.

3.2. Intermediate Cosmological Model

In the intermediate inflationary model, first described by Barrow [109], the scale factor increases at an intermediate rate between that of power law models [110] and conventional de Sitter models [111]. The main assumption in his paper is that the pressure and density are related by the following equation of state:

$$p+\rho =\gamma {\rho }^{\lambda },\gamma \ne 0,\lambda =\mathrm{constants}$$

where, the standard equation of state $p=(\gamma -1)\rho$ for a perfect fluid is obtained when $\lambda =1$. When $\gamma >0$ and $\lambda >1$, the intermediate inflationary scenario is created, in which the scale factor expands with [112]

$$a=e^{A{t}^{\beta }}$$

(27)

and is known as the intermediate scale factor. Here, $A>0$ and $0<\beta <1$ are constants. The universe exhibits slower expansion for standard de Sitter inflation, which occurs when $\beta =1$, yet faster expansion than the power-law inflation, with $a=t^p$ and $p>1$ as a constant. These models have many interesting properties, specifically regarding the perturbation spectra that they generate, which have been studied in [113]. Our work uses the scale factor provided above to study the intermediate scenario for $f(T,B)$ gravity. The methodology adopted in this subsection is similar to that applied for the emergent model. Thus, the Hubble parameter is obtained from Equation (27) as follows:

$$H=A\beta t^{\beta -1}.$$

(28)

In this case, the energy density of the HRDE is obtained as follows:

$${\rho }_{\Lambda }=3c^2(2A^2{\beta }^2{t}^{2\beta -2}+A(\beta -1)\beta t^{\beta -2}).$$

(29)

Again, T and B are obtained by employing Equations (4) and (5). Thus,

$$T=6A^2{\beta }^2{t}^{2(\beta -1)}$$

(30)

and

$$B=6(3A^2{\beta }^2{t}^{2\beta -2}+A(\beta -1)\beta t^{\beta -2}).$$

(31)

Thus, the derivative of B with respect to t is

$$\dot{B}=6A\beta (\beta -1)t^{\beta -3}[(\beta -2)+6A\beta t^{\beta }].$$

(32)

To find the specific form of the function $f(T,B)$, we assume the functional form as provided in Equation (45). Thus, using Equations (30) and (31), Equation (7) becomes

$${\displaystyle \frac{1}{2}}{f}_1\left(T\right)-T{f}_{1,T}-{\kappa }^2{\rho }_m=K,$$

(33)

$$B{f}_B+{\displaystyle \frac{B^2}{{(3A\beta -1)}^2}}(\beta -2+6A\beta )f_{2,BB}-f_{2,B}=2K.$$

(34)

Similar to the previous model, the reconstructed $f(T,B)$ is obtained using Equation (23). Figure 4 shows its behavior plotted against time.

Using the derivative of $f(T,B)$ with respect to t and making appropriate substitutions in Equation (11), we can reconstruct the energy density for the intermediate scale factor in $f(T,B)$ cosmology. Thus,

$$\begin{array}{cc}\hfill {\rho }_{TB}=& {\displaystyle \frac{1}{2}}\left[-C_1-{\displaystyle \frac{2\beta A{e}^{-3A{t}^{\beta }}{t}^{\beta -2}{x}_1(-A{\rho }_m{t}^{\beta }\beta x_2)}{{(\beta +6\beta A{t}^{\beta }-2)}^2}}\right]\hfill \\ \hfill & +{\displaystyle \frac{A\beta e^{-3A{t}^{\beta }}{t}^{\beta -2}(y_1+y_2)}{\beta +6\beta A{t}^{\beta }-2}}+{\displaystyle \frac{\beta e^{-3A{t}^{\beta }}(z_1+z_2)}{9t^2(\beta -2)\beta }}\hfill \\ \hfill & +{\displaystyle \frac{2(\beta -1){\rho }_m{(\beta -2)}^{2}ExpIntegralE\left[\frac{2+\beta }{\beta },3A{t}^{\beta }\right]}{9t^2(\beta -2)+\beta }}\hfill \end{array}$$

(35)

where

$$\begin{array}{cc}\hfill x_1=& A\beta {\rho }_m{t}^{\beta }+3e^{3A{t}^{\beta }}(2A\beta t^{\beta }+\beta -1)\hfill \\ \hfill x_2=& 6A\beta (3\beta +3t-4)t^{\beta }+(\beta -2)(2\beta +3t-4)\hfill \\ \hfill y_1=& -A\beta {\rho }_m{t}^{\beta }(\beta (3A{t}^{\beta }-2)+3)\hfill \\ \hfill y_2=& 3e^{3A{t}^{\beta }}(\beta \left(2A(2\beta -3)t^{\beta }+\beta -4\right)+3)\hfill \\ \hfill z_1=& (\beta -2)(\beta -1)(-{\rho }_m)(3A\beta t^{\beta }+\beta -2)\hfill \\ \hfill z_2=& 27A\beta t^{\beta }{e}^{3A{t}^{\beta }}((\beta -2)\beta (A{t}^{\beta }+1)+1).\hfill \end{array}$$

We note that ExpIntegralE(n,z) provides the exponential integral function $E_n\left(z\right)$. Additionally, $E_n\left(z\right)={\int }_1^{\infty }{\displaystyle \frac{e^{t(-z)}}{t^n}}dt$ and $E_n\left(z\right)$ has a branch cut discontinuity in the complex z-plane running from 0 to ∞. However, for certain special arguments, ExpIntegralE can be evaluated to exact values. In the present case and for this set of arguments, we obtain specific values of the function and plots for the equations involving $ExpIntegralE$. We then use the reconstructed ${\rho }_{TB}$ to obtain the reconstructed pressure ($p_{TB}$) by substituting the energy density in the conservation equation in Equation (25). In this way, the obtained reconstructed EoS parameter is plotted against time, as presented in Figure 5, where it can be seen that the EoS parameter exceeds the phantom boundary ($\omega =-1$) and shows quintessence behavior ($\omega >-1$).

Similar to the previous section, the derivative of $p_{TB}$ and ${\rho }_{TB}$ can be obtained, and the sound speed is derived from Equation (26). The graph in Figure 6 shows $c_S^2$ plotted against time.

3.3. Logamediate Cosmological Model

When weak general conditions [114] are applied to cosmological models, a class of potential indefinite cosmological solutions known as logamediate inflation arises. The existence of eight possible asymptotic solutions for cosmology dynamics was put forth by Barrow [115]. Of these possible solutions, three lead to non-inflationary expansions and three others respectively give rise to power law, de Sitter, and intermediate expansion. The remaining two solutions show asymptotic expansions of the logamediate form. Additionally, logamediate inflation naturally arises in a few scalar–tensor theories [116]. In the case of logamediate cosmology, the scale factor evolves as [117]

$$a\left[t\right]=e^{A{\left(log\left[t\right]\right)}^{\lambda }}$$

(36)

where $A>0$ and $\lambda >1$. In this case, the Hubble parameter is

$$H={\displaystyle \frac{A\lambda lo{g}^{\lambda -1}\left(t\right)}{t}}.$$

(37)

The energy density of the HRDE for this logamediate scale factor case is provided by

$${\rho }_{\Lambda }={\displaystyle \frac{3A\lambda c^2}{t^2}}\left((\lambda -1)Lo{g}^{\lambda -2}\left[t\right]-Lo{g}^{\lambda -1}\left[t\right]+2A\lambda lo{g}^{2\lambda -2}\left[t\right]\right).$$

(38)

Additionally, the torsion scalar and boundary term are respectively derived as follows:

$$T={\displaystyle \frac{6A^2{\lambda }^{2}Lo{g}^{2\lambda -2}\left(t\right)}{t^2}}$$

(39)

and

$$B={\displaystyle \frac{6A\lambda }{t^2}}\left((\lambda -1)Lo{g}^{\lambda -2}\left[t\right]-Lo{g}^{\lambda -1}\left[t\right]+3A\lambda lo{g}^{2\lambda -2}\left[t\right]\right).$$

(40)

The derivative of B with respect to t is

$$\begin{array}{cc}\hfill \dot{B}=& {\displaystyle \frac{6A\lambda {\left[ln\left(t\right)\right]}^{\lambda -3}}{t^3}}\hfill \\ \hfill & \left[2-3\lambda +{\lambda }^2-ln\left(t\right)(3\lambda -3+2ln\left(t\right))+6A\lambda ln{\left(t\right)}^{\lambda }(\lambda -1-ln\left(t\right))\right].\hfill \end{array}$$

(41)

To find the specific form of the function $f(T,B)$, we assume the functional form as provided in Equation (45). Thus, using Equations (39) and (40), Equation (7) becomes:

$${\displaystyle \frac{1}{2}}{f}_1\left(T\right)-T{f}_{1,T}-{\kappa }^2{\rho }_m=K,$$

(42)

$$6B{f}_B-{\displaystyle \frac{6A\lambda \left[ln{\left(t\right)}^{\lambda -1}\right]}{t}}\left[{\displaystyle \frac{ln{\left(t\right)}^{-1}}{t^2}}Y\right]f_{2,BB}-f_{2,B}=2K,$$

(43)

where

$$Y=2-3\lambda +{\lambda }^2-ln\left(t\right)[3(\lambda -1)+2ln\left(t\right)]+6A\lambda ln{\left(t\right)}^{\lambda }[(\lambda -1)-ln\left(t\right)].$$

(44)

Just for the intermediate and emergent cosmological models, $f(T,B),$ can be determined from Equation (23); the graph obtained in this way is shown in Figure 7.

The plot for the EoS parameter is plotted and shown in Figure 8. From the figure, it is possible to observe the crossing of the phantom boundary in the later stages of the universe, resulting in a transition from quintessence to phantom, hinting at a possible Big Rip singularity.

3.4. Power Law Model

In this case, we consider a Lagrangian of separated power law-style model for the boundary term and the torsion scalar as provided in [118]:

$$f(T,B)=l{B}^k+n{T}^m$$

(45)

where k, l, m, and n are all constant terms. Thus, the derivative corresponding to the $f_T$ and $f_B$ functions for this model are as follows:

$$\begin{array}{cc}\hfill f_T=& nm{T}^{m-1}\hfill \\ \hfill \dot{f_T}=& 12nm(m-1)H\dot{H}{\left(6H^2\right)}^{m-2}\hfill \\ \hfill f_B=& lk{B}^{k-1}\hfill \\ \hfill \dot{f_B}=& 6lk(k-1)B^{k-2}(6H\dot{H}+\ddot{H})\hfill \\ \hfill \ddot{f_B}=& 6lk(k-1)B^{k-2}\left[{\displaystyle \frac{k-1}{3H^2+\dot{H}}}(6H\dot{H}+\ddot{H})+6{\dot{H}}^2+6H\ddot{H}+\stackrel{⃛}{H}\right].\hfill \end{array}$$

(46)

Additionally, we have chosen the intermediate scale factor as provided in Equation (27) as our choice of scale factor in this subsection. The effective density is obtained by substituting Equations (28), (45), and (46) in Equation (11). In this way, we obtain

$${\rho }_{TB}={\displaystyle \frac{l(k-1){\left(6A\beta t^{\beta -2}{\xi }_1\right)}^k({\xi }_1^2-k(\beta -1){\xi }_2)}{2{\xi }_1^2}}-{\displaystyle \frac{6^m{\left(A{t}^{\beta -1}\beta \right)}^{2m}n}{2}}$$

(47)

where

$$\begin{array}{cc}\hfill {\xi }_1=& \beta -1+3A\beta t^{\beta },\hfill \\ \hfill {\xi }_2=& \beta -2+6A\beta t^{\beta }.\hfill \end{array}$$

Similar substitutions can be made in Equation (12) to obtain the pressure:

$$\begin{array}{cc}\hfill p_{TB}=& {\displaystyle \frac{t^{-\beta }}{6A\beta }}\left[6^m{\left(A{t}^{\beta -1}\beta \right)}^{2m}n((4m^2-6m)(\beta -1)-3A\beta t^{\beta })\right]\hfill \\ \hfill & -{\displaystyle \frac{t^{-\beta }}{6A\beta }}\left[2^k{3}^{(k+1)}{w}_1{w}_2^k+36k{w}_1{\left(6w_2\right)}^{(k-2)}{w}_3\right]\hfill \end{array}$$

(48)

where

$$\begin{array}{cc}\hfill w_1=& A(k-1)l\beta t^{\beta },\hfill \\ \hfill w_2=& A\beta t^{(\beta -2)}(\beta -1+3A{t}^{\beta }\beta ),\hfill \\ \hfill w_3=& {\displaystyle \frac{(\beta -1)\left[A{t}^{\beta }(\beta -2)(\beta -3)\beta +6{\left(A\beta t^{\beta }\right)}^2(2\beta -3)+\frac{(k-1)t^3(\beta -2+6A{t}^{\beta }\beta )}{\beta -1+3A{t}^{\beta }\beta }\right]}{t^4}}.\hfill \end{array}$$

We can also obtain the EoS parameter for this model using Equations (11) and (12):

$$\begin{array}{cc}\hfill \omega =& {\displaystyle \frac{p_{TB}}{{\rho }_{TB}}}\hfill \\ \hfill =& -1+{\displaystyle \frac{\ddot{f_B}-3H\dot{f_B}-2\dot{H}{f}_T-2H\dot{f_T}}{3H^2(3f_B+2f_T)-3H\dot{f_B}+3\dot{H}{f}_B-\frac{1}{2}f}}.\hfill \end{array}$$

(49)

Thus, by making essential substitutions, we can obtain the EoS parameter of this model as analyzed by considering three cases (see Figure 9):

• Case 1: Varying k and m with $k<m$.
• Case 2: Varying k and m with $m<k$.
• Case 3: Varying m and n as negative values.

From the figures, it can be seen that with Case 1 (cf. Figure 9a) there is a transition from quintessence in the early universe to phantom at a later stage, while for Case 2 (cf. Figure 9b) we see an accelerated expansion phase ($\omega <-{\displaystyle \frac{1}{3}}$) for $t\le 0.2$, after which the acceleration is not preserved until $t0.7$. In the later stages of the universe, the EoS evolves into a phantom phase and the acceleration is preserved. In Case 3 (cf. Figure 9c), the crossing of the phantom boundary can be observed.

4. Thermodynamics of f(T, B) Gravity

At this point in our study, we intend to explore the validity of the generalized second law (GSL) of thermodynamics [119] using the previously obtained reconstructed energy density (${\rho }_{TB}$) and pressure ($p_{TB}$) in the emergent, intermediate, and logamediate scenarios bounded by the apparent horizon ($r_A$), which we do both with and without using the first law of thermodynamics. It should be mentioned here that we are considering a flat ($k=1$) FRW cosmology with the line element provided in Equation (1) and an equilibrium description of thermodynamics, i.e., the internal temperature is the same as that of the apparent horizon. To begin, we may define the GSL as an implication that the sum of the entropy inside the horizon ($S_{ih}$) and the entropy of the boundary of the horizon always increases against the evolution of time. For our study, we consider the apparent horizon $r_A=a\left(t\right)r$, which in terms of the Hubble parameter is written as

$$r_A={\displaystyle \frac{1}{H}},$$

(50)

and the derivative with respect to time, obtained as follows:

$$\dot{r_A}=-{\displaystyle \frac{\dot{H}}{H^2}}.$$

(51)

We take the Hawking temperature associated with the apparent horizon and define it through the surface gravity ${\kappa }_{sg}$ as

$$T_h={\displaystyle \frac{{\kappa }_{sg}}{2\pi }},$$

(52)

where ${\kappa }_{sg}={\displaystyle \frac{r_A}{2}}(2H^2+\dot{H})$.

An important equation while studying the validity of GSL is the Gibbs equation [120], which is written for the entropy within the apparent horizon as

$$T_{h}d{S}_{ih}=d\left({\rho }_{tot}V\right)+p_{tot}dV=Vd{\rho }_{tot}+(p_{tot}+{\rho }_{tot})dV,$$

(53)

where ${\rho }_{tot}$ and $p_{tot}$ are respectively the total energy density and pressure contributed by the matter and dark energy. Because we are considering a non-interacting model consisting of a perfect fluid and pressureless matter within the realm of modified $f(T,B)$ gravity, the continuity equation is written as follows:

$$\begin{array}{cc}\hfill & \dot{\rho }+3H({\rho }_m+p_m)=0\hfill \\ \hfill & \dot{{\rho }_{TB}}+3H({\rho }_{TB}+p_{TB})=0\hfill \end{array}$$

(54)

where ${\rho }_m={\rho }_{m0}{a}^{-3}$ and $p_m=0$. Using Gibbs’ equation as provided in Equation (52) together with the continuity equation provided in Equation (53), we obtain

$$T_h\dot{S_{ih}}=4\pi r_A^2(\dot{r_A}-1)({\rho }_{tot}+p_{tot}),$$

(55)

in which $\dot{S_{ih}}$ is the rate of internal energy change with respect to time, $p_{tot}=p_{TB}$, and ${\rho }_{tot}={\rho }_{m0}+{\rho }_{TB}$. In addition, the quantities $p_{TB}$ and ${\rho }_{TB}$ are the reconstructed pressure and density obtained in the previous section for the different scale factors. We now find the expression for the total entropy change with and without using the first law of thermodynamics.

4.1. GSL Using the First Law

In the representation of the field equations of $f(T,B)$ gravity provided in Equations (2) and (3), the derivative of the apparent horizon ($r_A$) with respect to time is

$$2\dot{r_A}={\kappa }^2{r}_A^3({\rho }_{tot}+p_{tot})H.$$

(56)

The Berkenstein–Hawking horizon entropy in the context of general relativity is provided by [121]

$$S_{oh}={\displaystyle \frac{A}{4G}},$$

(57)

where $A=4\pi r_A^2$ and G is Newton’s constant. Bahamonde et al. [122] used the above relation together with the Misner–Sharp energy provided by

$$E=V{\rho }_{tot}$$

(58)

to obtain the following:

$$T_{h}d{S}_{oh}=dE-WdV$$

(59)

where $W={\displaystyle \frac{1}{2}}(rh{o}_{tot}+p_{tot})$ and $V={\displaystyle \frac{4}{3}}\pi r_A^3$. Thus, they proved that it is possible for the traditional first law of thermodynamics $T_{h}d{S}_{oh}=dE-WdV$ to be met while considering the equilibrium thermodynamic description of $f(T,B)$ gravity. We may note tha the heat flow $\delta Q$ through the horizon is simply the amount of energy crossing it during the time interval $dt$. Thus, the first law of thermodynamics (Clausus relation) on the horizon can be written as $T_{h}d{S}_{oh}=\delta Q=-dE$. We can then use the unified first law to obtain

$$T_{h}d{S}_{oh}=4\pi r_A^{3}H({\rho }_{tot}+p_{tot})dt.$$

(60)

From the above equation, we obtain the derivative of $S_{oh}$ as follows:

$$\dot{S_{oh}}={\displaystyle \frac{4\pi r_A^{3}H}{T_h}}(\rho +p).$$

(61)

Finally, adding Equations (52) and (58), we obtain the following equation for the time derivative of the total entropy:

$$\dot{S_{tot}}=\dot{S_{oh}}+\dot{S_{ih}}={\displaystyle \frac{4\pi r_A^2}{T_h}}({\rho }_{tot}+p_{tot})\dot{r_A}.$$

(62)

From the definition of GSL, we can infer that the validity of GSL requires the following condition:

$$\dot{S_{tot}}=\dot{S_{ih}}+\dot{S_{oh}}\ge 0.$$

(63)

4.2. GSL without Using the First Law

We can also investigate GSL without using the first law of thermodynamics. If we consider the Berkenstein–Hawking entropy in Equation (54) and take its derivative with respect to time, for $f(T,B)$ gravity we can find that

$$T_h\dot{S_{oh}}={\displaystyle \frac{\dot{r_A}}{G}}\left(1-{\displaystyle \frac{\dot{r_A}}{2}}\right).$$

(64)

Therefore, in this case the time derivative of the total entropy is the sum of Equations (52) and (61).

Thus, these general equations may be used to investigate the validity of GSL when considering $f(T,B)$ cosmologies. In our work, we have considered the three different scale factors used previously, viz., emergent, intermediate, and logamediate, each of which yields different formulations of $r_A$ in terms of t. In addition, the various reconstructed forms of density (${\rho }_{TB}$) and pressure ($p_{TB}$) that we have formulated in the previous section correspond to different scale factors ($a\left[t\right]$). In addition, the power-law model was substituted in the appropriate equations, helping us to obtain three different mathematical forms of $\dot{S_{tot}}$, each corresponding to the different models reconstructed in the previous section. The graphs obtained by plotting the total entropy change against time are shown in Figure 9 and Figure 10. In all the plots, it can be observed that the $\dot{S_{tot}}$ remains at a positive level, which indicates the validity of GSL for all of our $f(T,B)$ cosmological models.

5. Conclusions

This paper presents an analysis of $f(T,B)$ gravity theory for a homogeneous and isotropic metric. We have focused mainly on investigating the reconstruction of four different $f(Q,T)$ models using different forms of scale factor and a model based on the power law-like form of f. The three chosen scale factors are emergent, intermediate, and logamediate, and the background fluid for reconstruction is considered to be the Holographic Ricci Dark Energy (HRDE), a particular case of a highly generalized holographic dark energy, namely, Nojiri–Odintsov HDE [123]. The first phase of our study examines four cosmological scenarios, where we obtain the reconstructed densities and pressure, EoS parameters, and squared speed of sound for the different models. Utilizing Friedmann’s equations and the HRDE, we reconstruct the function f for the three different choices of the scale factor. The following are our main results:

• In the case of the emergent scale factor, we find (cf. Figure 1) that the function is a decreasing function with respect to z. The EoS parameter (see Figure 2) shows phantom behavior and tends to $-1$ when plotted against the redshift z. The squared speed of sound shows a decrease in value with respect to the redshift but stays positive, indicating the stability of the density perturbations (and possibly the model).
• For the model with the intermediate scale factor, we observe that the function f increases with time (cf. Figure 4). From Figure 5, it can be seen that the EoS parameter shows quintessence behavior in a later stage and acceleration ($\omega <-{\displaystyle \frac{1}{3}}$) in the early stage. The squared speed of sound is greater than 0 when plotted against time (cf. Figure 6).
• In our third model, using the logamediate scale factor, we proceeded with the reconstruction of $f(T,B)$ gravity as in the previous two models. The reconstructed function when plotted against time (cf. Figure 7) shows a monotonic decrease with time, and asymptotically tends to 0 at $t1$. A transition from quintessence to phantom behavior is exhibited by the EoS parameter in this case(see Figure 8).
• For our fourth cosmological $f(T,B)$ model, we take a power law-like function for the torsion and boundary scalar. Choosing the intermediate scale factor, we reconstruct the EoS parameter, the methodology for which is discussed in Section 3.4. We obtain the EoS parameters for three different cases based on the constants that we assume in the functional form of $f(T,B)$. Figure 9 shows that the behavior of the EoS parameter for these different cases is mainly phantom-like, a result that is similar to the findings in [124].

The outcomes of this study lead to the interpretation that the models are stable under holographic background fluid in the cosmological settings of $f(T,B)$ gravity, irrespective of the choice of scale factors under consideration. The squared speed of sound appears to be positive in all cases, proving the stability of the models under the given cosmological framework. The motivation behind our consideration of this holographic fluid is discussed in the introduction; the obtained results prove that such a choice of reconstruction generates viable results. We know that the modified theories of gravity are not dependent upon any specific choice of dark energy; however, as the holographic dark energy is usually considered to be a phenomenological model used to explain the late-time acceleration, we thought to incorporate this fluid into the $f(T,B)$ framework so as to provide some insight into the cosmological outcomes.

The last part of our study focuses on thermodynamical analysis of the four models presented in the previous section, which is carried out considering the apparent horizon and using the different reconstructed densities and pressures from each of the four models to validate the generalized second law of thermodynamics (GSLT) both with and without using the first law. Figure 10 and Figure 11 show the plot of $\dot{S_{tot}}$ against time; it can be seen that the GSLT is satisfied for all of the models. As part of our future work, we intend to expand our study to include the event horizon and investigate our model’s thermodynamical consistency by examining the GSLT’s validity. In this context, it is necessary to mention an essential study [125] in which the authors examined the validity of the GSLT in the context of scalar–tensor gravity theory at the event horizon with modified Hawking temperature. We also draw attention to another vital work by [126], wherein the authors reviewed finite-time cosmological singularities in various cosmological contexts and showed the nature of these singularities. In future work, we would like to explore the possibility of our model giving rise to such singularities.