# Noether symmetry approach in *f*(*T*, *B*) teleparallel cosmology

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## Abstract

We consider the cosmology derived from *f*(*T*, *B*) gravity where *T* is the torsion scalar and \(B=\frac{2}{e}\partial _{\mu }(e T^{\mu })\) a boundary term. In particular we discuss how it is possible to recover, under the same standard, the teleparallel *f*(*T*) gravity, the curvature *f*(*R*) gravity, and the teleparallel–curvature *f*(*R*, *T*) gravity, which are particular cases of *f*(*T*, *B*). We adopt the Noether Symmetry Approach to study the related dynamical systems and to find cosmological solutions.

## 1 Introduction

Nowadays, one of the most important problems in Physics is to understand the late-time accelerated expansion of the universe. Besides, large scale structure, ranging from galaxies to superclusters, presents the problem of missing matter, i.e. the luminous matter is not sufficient in order to guarantee the stability and the evolution of self-gravitating astrophysical systems. There are several candidates to explain these phenomena and the most popular ones are the dark energy and the dark matter, i.e. cosmic fluids gravitationally interacting and leading the evolution of the Hubble flow but without any electromagnetic counterpart.

In particular, the physics underlying the dark energy is still not understood since it behaves as a repulsive gravitational force in contrast to the usual gravitational field. In general, there exist two ways to study the dark energy problem: (1) one can retain the General Relativity (GR) and introduce a new kind of fluid which possess a negative pressure (e.g. by introducing a scalar field), or (2) one can think that GR needs to be modified at high energy levels and hence the dark energy comes out from these modifications. One can change the left hand side of the Einstein field equations related with the theory of gravity (e.g. extending GR or considering alternatives [1]) or one can modify the right hand side of it by changing the matter content of the universe (i.e., the energy-momentum tensor).

Beside the issue to explain the energy-matter content of the universe, and then the source of accelerated expansion and structure aggregation, competing theories of gravity are posing several fundamental questions on the nature of gravitational field. In particular, if torsion has to be involved in the dynamics, if the equivalence principle is valid in any case, if geodesic structure and metric structure are related or not, if theories of gravity have to be formulated in metric, metric-affine or purely affine approaches [2, 3]. In particular, the teleparallel formulation of gravity has recently acquired a lot of interest due to its applications at cosmological and fundamental level.

In this paper, we are interested in studying cosmology in teleparallel modified theories of gravity, which in contrast to GR, consider a curvatureless spacetime with a non-zero torsion. In this perspective, one needs to introduce the so-called Weitzenböck connection instead of the standard Levi-Civita connection [4]. By doing so, a spacetime endorsed with non-zero torsion and a vanishing curvature is achieved.

From the geometrical point of view, these spacetimes are different from the ones considered in GR. From the point of view of the field equations, the teleparallel equivalent of General Relativity (TEGR) is equivalent to GR (see [6, 7] for further notions of TEGR). A natural extension of TEGR is, instead of considering only the trace of the torsion tensor *T* in the action, to introduce a function *f*(*T*) in it (see the review paper of Ref. [5] for a discussion and references therein). *f*(*T*) gravity remains a second order theory whereas the straightforward GR extension, *f*(*R*) gravity is a fourth-order one in metric formalism. However, the price to pay in this approach is that *f*(*T*) is not invariant under local Lorentz transformations, and hence different vierbeins could arise by different field equations [8, 9]. This theory has been used in cosmology to understand the accelerating cosmic expansion of the universe [10, 11, 12, 15], reconstruct cosmological models based on observational data [13, 14, 16, 17], among other models studied.

Recently, a new generalization of the standard *f*(*T*) gravity was proposed in [18]. In this theory, the function *f*(*T*) is extended to *f*(*T*, *B*), where *B* is a boundary term related with the Ricci scalar via \(R=-T+B\). By adding this dependency, one can recover *f*(*T*) and *f*(*R*) under suitable limits. Some cosmological features as reconstruction techniques and thermodynamics have been studied in [19] under the standard of this new theory. In [20], a non-minimally coupled scalar field with both the boundary term and the torsion scalar was presented in view to study cosmology by dynamical system techniques. There, it was shown that a dynamical crossing of the phantom barrier is possible and also without fine tuning, the system evolves to a late-time acceleration attractor solution. Some exact solutions and its thermodynamics properties were also discussed in [21]. In summary *f*(*T*, *B*) gravity presents several interesting features by which it is possible to unify, under the same standard, issues coming from *f*(*T*) and *f*(*R*) gravity.

Here, we will explore cosmological solutions coming from *f*(*T*, *B*) by the so-called *Noether symmetry approach*. This technique proved to be very useful for several reasons: (1) it allows one to fix physically interesting cosmological models related to the conserved quantities (i.e. in particular couplings and potentials) [22]; (2) the existence of Noether symmetries allows one to reduce dynamics and then to achieve exact solutions [23]; (3) symmetries act as a sort of selection rules to obtain viable models in quantum cosmology [24].

The plan of this paper is as follows: In Sect. 2, we briefly introduce TEGR and its extensions like *f*(*T*) gravity and *f*(*T*, *B*) gravity. *f*(*T*, *B*) cosmology and particular cases that can be derived from it are introduced in Sect. 3. Section 4 is devoted to the study of Noether’s symmetries for *f*(*T*, *B*) gravity. In particular, we derive the Noether vector field and derive the Noether conditions for the function *f*(*T*, *B*). In the related subsections, we study particular cases of *f*(*T*, *B*) function discussing, in particular, how it reduces to *f*(*T*), *f*(*R*) and *f*(*R*, *T*) gravities. The main point of this section is to demonstrate how several classes of modified gravity theories can be reduced to the *f*(*T*, *B*) paradigm. Discussion and conclusions are drawn in Sect. 5. In our notation, Greek and Latin indices denote spacetime and tangent coordinates, respectively, and the signature \((+,-,-,-)\) is adopted for the metric.

## 2 Teleparallel equivalent of general relativity and its modifications

*f*(

*T*) gravity. In this theory, the vierbeins or tetrad fields \(e_{\mu }^{a}\) are the dynamical variables which form an orthonormal basis for the tangent space at each point \(x^{\mu }\) of the spacetime manifold. Hence, the tetrads \(e_{\mu }^{n}\) and their inverses \(E_{m}^{\mu }\) obey the following orthogonality relations:

*T*and it is usually called the torsion scalar. This quantity is a topological object and the TEGR is constructed by defining the action

*R*and the torsion scalar

*T*differs by a boundary term via

*f*(

*T*) is a smooth function of the torsion scalar. It is easy to see that, by setting \(f(T)=T\), the TEGR action is recovered. In this theory is not possible to find the teleparallel equivalent of

*f*(

*R*) gravity since now the boundary term in (7) contributes to the field equations. Since

*T*itself is not invariant under local Lorentz transformations, this theory is also not invariant under Lorentz transformations. An important fact is that this theory is a second order one and hence, mathematically, it is easier than

*f*(

*R*) gravity. The above action (8) can be immediately generalized by assuming that the function

*f*(

*T*) depends also on the boundary term

*B*. The action reads as follows [18]:

*f*is a smooth function of two scalar fields, i.e. both the scalar torsion

*T*and the boundary term

*B*. The motivation of this action comes out from the fact that from

*f*(

*T*) gravity, it is not possible to find an equivalent theory of its metric counterpart, the

*f*(

*R*) gravity. From the above action, we can easily see that the

*f*(

*R*) and

*f*(

*T*) can be recovered by assuming \(f(T,B)=f(-T+B)=f(R)\) and \(f(T,B)=f(T)\), respectively.

*f*(

*T*),

*f*(

*R*), and

*f*(

*R*,

*T*) gravities.

## 3 *f*(*T*, *B*) cosmology

*f*(

*T*,

*B*) gravity. In particular to find exact cosmological solutions by the Noether symmetry approach. Let us consider

*f*(

*T*,

*B*) cosmology in a flat Friedmann–Lemaître–Robertson–Walker (FLRW) universe. The spatially flat FLRW metric in Cartesian coordinates reads as follows:

*a*(

*t*) is the scale factor of the universe. This metric can be constructed by the following tetrad field:

*f*(

*T*,

*B*) is not invariant under Lorentz transformations, one needs to be very careful with the choice of the tetrad. For instance, the unwanted condition \(f_{TT}=0\) appears when one considers a flat diagonal FLRW tetrad in spherical coordinates. The above vierbein is a “good tetrad” in the sense of Ref. [26] since it will not constrain our system.

*p*(

*t*) are the energy density and pressure of the cosmological fluid respectively. It is clear that by setting \(f(T,B)=-f(T-B)=-f(-R)\) we recover the FLRW equations in

*f*(

*R*) gravity with the standard notation (see for example [3, 27]). Immediately we have

*f*(

*T*) gravity given by

*f*(

*R*,

*T*) can be viewed as a special case of

*f*(

*T*,

*B*) gravity since we can choose \(f(T,B)=f(-T+B,T)=f(R,T)\). In this sense, it can be argued that

*f*(

*T*,

*B*) should be a more natural theory to consider than

*f*(

*R*,

*T*) as we will discuss below.

*t*. Here, \({\mathbb Q}\equiv \{a,T,B\}\) is the configuration space from which it is possible to derive \({\mathbb {TQ}}\equiv \{a,\dot{a}, T, \dot{T}, B,{\dot{B}}\}\), the corresponding tangent space on which \(\mathcal{L}\) is defined as an application. The variables

*a*(

*t*),

*T*(

*t*), and

*B*(

*t*) are, respectively, the scale factor, the torsion scalar and the boundary term defined in the FLRW metric. The Euler–Lagrange equations are given by

*a*,

*T*,

*B*in order to derive the

*f*(

*T*,

*B*) action as follows:

*T*and

*B*, we find

*a*,

*T*, and

*B*. If we choose \(f(T,B)=f(T)\), we recover the teleparallel

*f*(

*T*) cosmology with the Lagrangian [30]

*f*(

*R*) gravity [31]

*T*and

*B*are given by

*f*(

*R*,

*T*) point-like Lagrangian is given by

## 4 Noether symmetry approach for *f*(*T*, *B*) cosmology

The Noether symmetry approach has been widely used in the literature to find cosmological solutions in modified gravity (see [22] for a comprehensive review). The main idea is to find symmetries in a given model and then to use them to reduce related dynamical systems and find exact solutions. As a byproduct, the existence of the symmetries selects the functions inside the models (e.g. couplings and self-interaction potentials) that, in most cases, have a physical meaning. In this sense, the existence of a Noether symmetry is a sort of selection rule. Essentially, the technique consists in deriving constants of motions. Any constant of motion is related to a conserved quantity that allows one to reduce the dynamical system and then to obtain exact solutions. If the number of constants is equal to the number of degrees of freedom, the system is completely integrable.

*X*is the Noether vector field and \(L_{X}\) is the Lie derivative. For generalized coordinates \(q_{i}\), we can construct the Noether vector field

*X*. We have

*X*in the space \({\mathbb Q}\equiv \{a,T,B\}\) is

*a*,

*T*, and

*B*. Therefore we have

*n*dimensional configuration space, we have \(1+n(n+1)/2\) equations derived from Eq. (37). In our case, the configuration space is three dimensional, so we have seven partial differential equations. Explicitly, from (37), we find the following system of partial differential equations:

*f*(

*T*,

*B*). There are two different strategies to solve it and to find symmetries: (1) one can directly solve the system (43)–(49) and then find the unknown functions; (2) one can impose specific forms of

*f*(

*T*,

*B*) and search for the related symmetries [25]. From a physical viewpoint, the second approach is better because it allows one to study reliable models. By the first strategy, solutions can be achieved but, in most cases, they are implicit functions that do not allow a physical analysis [22]. We will adopt the second one to discuss the

*f*(

*T*,

*B*) cosmology.

### 4.1 Case 1: \(f(T,B)=b_{0}B^{k}+t_{0}T^{m}\)

For a power-law like function given by \(f(T,B)=b_{0}B^{k}+t_{0}T^{m}\), where \(b_{0}\), \(t_{0}\), *k* and *m* are constants, we find that the unique solution of (43)–(49) is for \(k=1\). This is trivial because it gives \(f(T,B)=b_{0}B+t_{0}T^{m}\), which is the same as a power-law *f*(*T*) function. This comes from the fact that *B* is a boundary term so that a linear form of the function in *B* does not introduce any change in the field equations. Hence, this kind of function gives the same results as reported in [30].

### 4.2 Case 2: \(f(T,B)=f_{0}B^{k}T^{m}\)

*k*and

*m*are constants. From (46)–(48), it is \(\alpha =\alpha (a)\). If we replace the function (50) into (43)–(49), we find the following Noether vector:

*a*(

*t*) gives, for \(k\ne 1\) and \(f_{0}\ne 0\),

*T*and

*B*given by (21) and (22) we find that Eqs. (53) and (54) become

### 4.3 Case 3: \(f(T,B)=-T+F(B)\)

*g*(

*a*) is an arbitrary function of the scale factor. Therefore, from (67) one finds that

*u*(

*t*) is

*E*being the Hamiltonian (the energy) of the system and \(V(t)=2C_{2} u_{3} e^{t u_{1}}\) can be understood as an energy potential, we find the following constraint:

In the next subsections, remarkable theories that can be recovered from *f*(*T*, *B*) gravity are discussed. We will see that all symmetries found in earlier studies for *f*(*T*), *f*(*R*), and *f*(*R*, *T*) can be achieved starting from the Noether symmetry equations of Eqs. (43)–(49) derived for *f*(*T*, *B*) cosmology.

### 4.4 Case 4: \(f(T,B)=f(T)\)

*f*(

*T*) gravity. The cases studied in [28, 29] are straightforwardly obtained. Equations (46)–(48) are identically satisfied since \(f_{TB}=f_{BB}=0\). The other equations become

*a*and the r.h.s. only on

*T*, so that

*C*is a constant. Thus, it is easy to solve the above equation yielding

*f*(

*T*) gravity admits power-law cosmological solutions of the form of \(a(t)\propto t^{-2C/C_{3}}\). A more general study of power-law

*f*(

*T*) cosmology is in [30].

### 4.5 Case 5: \(f(T,B)=f(-T+B)=f(R)\)

*f*(

*R*) gravity by assuming \(f(T,B)=f(-T+B)=f(R)\). Hence, \(f_{R}(R)=f'(-T+B)=-f_{T}=f_{B}\) and the system of differential equations (43)–(49) related to the Noether symmetry in

*f*(

*R*) gravity becomes

*f*(

*R*) gravity. In doing this, Eqs. (96) and (97) are identical and hence the Noether equations become

*T*and

*B*are summed twice. Therefore, by changing \(\gamma =\frac{1}{2}\tilde{\beta }\) we find the same equations as in [31], that is,

*g*(

*a*) is an arbitrary function depending on

*a*. Note that the latter solution is very similar to the one found in (71) for the case \(f(T,B)=-T+F(B)\). Now if we replace this solution into (106), we obtain

*f*(

*R*) gravity admits power-law solution of the form

### 4.6 Case 6: \(f(T,B)=f(-T+B,T)=f(R,T)\)

*f*(

*R*,

*T*), we require to change the function \(\gamma (T,B,a)\rightarrow \gamma (T,B,a)+\beta (T,B,a)\). Additionally, for the derivative terms, we need to use Eqs. (31)–(35) and hence the transformation \(\partial /\partial T \rightarrow \partial /\partial T-\partial /\partial R\). After these replacements, the Noether conditions become

*f*(

*R*,

*T*) result the same as those studied in [33] with the same physical implications.

## 5 Discussion

In this paper, we discussed an extension of modified teleparallel gravity including functions of the torsion scalar *T* and its related boundary term \(B=\frac{2}{e}\partial _{\mu }(e T^{\mu })\). In such a way, a gravitational theory with two fields, i.e. *T* and *B*, can be taken into account. If not assumed in a trivial way, that is linear in *B*, interesting features come out from the combinations of *T* and *B*, in particular, the possibility to relate *f*(*T*) and *f*(*R*) gravity under the same standard. This means that not only GR and TEGR (respectively, theories linear in the Ricci scalar *R* and the torsion scalar *T* in their actions) result in the “same” effective theory but also their extensions, also if conceptually very different, can show analogies and similitudes.

Here we consider the Noether symmetry approach in order to investigate the related cosmologies. The main result is that the Noether vector fields emerging from *f*(*T*, *B*) gravity are a general standard to find solutions both in the starting theory and in the particular cases like *f*(*T*), *f*(*R*), and *f*(*R*, *T*). In this last case, the Noether technique allows one to deal with curvature *R* and torsion *T* scalars as two scalar fields.

The related cosmological solutions are of physical interest and, essentially, all the main cosmological behaviors can be recovered. However, this is only a preliminary study where no effective comparison with observations has been made and only toy models have been analyzed in order to test the technique.

In forthcoming studies, we will adopt an approach for *f*(*T*, *B*) gravity as in [30], where the condition (37) is extended to the possibility of discussing singular Lagrangians. Furthermore, a similar approach can be used for teleparallel modified Gauss–Bonnet gravity \(f(T,B,T_{G},B_{G})\) as studied in [34]. Under this standard, other interesting models can naturally arise by taking into account some specific functions of *T* and *B* as \(f=f(-T+B,-T_{G}+B_{G})=f(R,G)\) (modified Gauss–Bonnet) or \(f=f(T,T_{G})\) gravity (modified teleparallel Gauss–Bonnet). The final issue is to define a *mother theory* by which all extensions and modifications of GR can be generated.

## Notes

### Acknowledgements

S.B. is supported by the Comisión Nacional de Investigación Científica y Tecnológica (Becas Chile Grant No. 72150066). S.C. acknowledges the financial support of INFN (*iniziative specifiche* TEONGRAV and QGSKY). This article is based upon work from COST Action CA15117 (CANTATA), supported by COST (European Cooperation in Science and Technology).

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