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Erratum: Teleparallel quintessence with a nonminimal coupling to a boundary term [Phys. Rev. D 92 , 084034 (2015)]
Abstract
DOI:http://dx.doi.org/10.1103/PhysRevD.93.109901
... This can be also seen in the case where one includes a scalar field in the action. For example, in [45,46], the authors found a Teleparallel theory containing non-minimal couplings between a scalar field and both the scalar torsion T and the boundary term B, finding that this theory contains the standard non-minimally coupled theories based on the curvature [47,48]. This was further generalised to quintom models [49], non-local models [50,51] and specific scalar tensor scenarios [52][53][54], finding again that Teleparallel theories are broader than standard modified theories. ...
... The theory F 3 = F 4 = 0 was studied in [45] with F 1 = ξφ 2 and F 2 = χφ 2 and then generalised for any F 1 and F 2 in [46]. The latter theory is also connected to the Teleparallel dark energy [64,65]. ...
... For a comprehensive review about dynamical systems in cosmology, see [69]. Similarly as in [45], let us introduce the following dimensionless variables ...
The present paper represents an attempt for a very generic string inspired theory of gravitation, based on a stringy action in the teleparallel gravity which includes a specific functional which depends on the scalar field and its kinetic energy, as well as the torsion and boundary terms, embedding also possible effects from the teleparallel Gauss–Bonnet invariants. We focus our study in FLRW cosmology. After we deduce the cosmological equations for the associated generic theory of gravitation, we focus on string inspired couplings which are studied by considering different analytical techniques. The first analytical technique is based on the linear stability theory, by introducing proper dimensionless variables which enables us to study the structure of the phase space and the associated physical effects. In this case we have obtained different cosmological solutions which corresponds to matter and dark energy dominated solutions, achieving a possible transition between matter and dark energy dominated epochs. For each type of cosmological solutions we have discussed the corresponding physical features, attaining viable constraints for the coupling constants due to dynamical effects. The dynamical study of the physical features included also a numerical analysis by fine–tuning the initial conditions deep into the matter era, obtaining possible trajectories for the effective equation of state for specific coupling functions.
... In Ref. [23], a review on fðTÞ theories is presented. Scalar fields have also been considered in numerous ways with some indicative being [24][25][26][27][28][29][30][31][32][33][34][35]. In the recent series of papers [36][37][38], a serious attempt to construct a general scalar-torsion theory has been made. ...
... This can be eliminated by redefining the new Horndeski Lagrangian term asG tele ¼ G tele þ TG 4 ðϕ; XÞ, so that scalar field couplings with the boundary term alone would then be possible in Eq. (40). In fact, studies of the coupling of the boundary term with the scalar field already exist in the literature [31,34,35]. ...
... Moreover, since fðTÞ is a subclass of teleparallel Horndeski, one can also conclude that teleparallel Horndeski can explain both dark energy and inflation [23,39,[73][74][75], can have bounces cosmological solutions [76,77] and also can alleviate the H 0 tension [70]. Since this theory also contains the teleparallel scalar-tensor theories studied in [30,31], it can describe a crossing of the phantom barrier, quintessencelike or phantomlike behavior, and also a late time accelerating attractor solution without requiring any fine-tuning of the parameters. Then, teleparallel Horndeski has an enlarged number of theories as compared with standard Horndeski that can explain the cosmological observations without introducing a cosmological constant. ...
Horndeski gravity is the most general scalar tensor theory, with a single scalar field, leading to second-order field equations and after the GW170817 it has been severely constrained. Since this theory is very important in modified gravity, it is then worth studying possible similar theories starting from other frameworks. In this paper, we study the analog of Horndeski’s theory in the teleparallel gravity framework where gravity is mediated through torsion instead of curvature. We show that, even though many terms are the same as in the curvature case, we have much richer phenomenology in the teleparallel setting because of the nature of the torsion tensor. Moreover, teleparallel Horndeski contains the standard Horndeski gravity as a subcase and also contains many modified teleparallel theories considered in the past, such as f(T) gravity or teleparallel dark energy. Thus, due to the appearance of a new term in the Lagrangian, this theory can explain dark energy without a cosmological constant, may describe a crossing of the phantom barrier, explain inflation and also solve the tension for H0, making it a good candidate for a correct modified theory of gravity.
... where the torsion scalar is replaced by a nonlinear function f (T ) [16][17][18][19][20][21], represent a viable framework naively inspired by f (R)gravity. This prescription can be even extended by a more general form based on f (T, B) functions in order to include both torsion and the boundary term [22,23]. This leads to a generalization of both f (R) and f (T ) classes of models and have been also studied as f (R, T ) in [24]. ...
... In this section, we obtain the effective gravitational coupling with a scalar field non-minimally coupled to both torsion scalar and the boundary term. We restrict our attention to the following action, introduced in [22]: ...
In the present study, we consider an extended form of teleparallel Lagrangian , as function of a scalar field , its kinetic term X and the torsion scalar T. We use linear perturbations to obtain the equation of matter density perturbations on sub-Hubble scales. The gravitational coupling is modified in scalar modes with respect to the one of General Relativity, albeit vector modes decay and do not show any significant effects. We thus extend these results by involving multiple scalar field models. Further, we study conformal transformations in teleparallel gravity and we obtain the coupling as the scalar field is non-minimally coupled to both torsion and boundary terms. Finally, we propose the specific model . To check its goodness, we employ the observational Hubble data, constraining the coupling constant, , through a Monte Carlo technique based on the Metropolis-Hastings algorithm. Hence, fixing to its best-fit value got from our numerical analysis, we calculate the growth rate of matter perturbations and we compare our outcomes with the latest measurements and the predictions of the CDM model.
... So, one can modify the gravitational part of the action to allow non-linear corrections to the Lagrangian [29,30], this is the general approach followed by f (R)-theories of gravity [31,32]. Some other extensions of GR increase the number of spacetime dimensions or introduce nonminimal matter couplings to boundary and topological terms [33][34][35][36][37][38][39][40][41][42]. These are terms in the Lagrangian that describe how matter couples to geometrical quantities. ...
... In [40,49], a dynamical system analysis where teleparallel quintessence is nonminimally coupled to a boundary term is presented. In the same spirit, we study the background cosmology within this framework and apply dynamical systems tools to investigate the dynamics of the different models. ...
Cosmological models can be studied effectively using dynamical systems techniques. Starting from Brown’s formulation of the variational principle for relativistic fluids, we introduce new types of couplings involving a perfect fluid, a scalar field, and boundary terms. We describe three different coupling models, one of which turns out to be particularly relevant for cosmology. Its behaviour is similar to that of models in which dark matter decays into dark energy. In particular, for a constant coupling, the model mimics well-known dynamical dark energy models while the non-constant couplings offer a rich dynamical structure, unseen before. We are able to achieve this richness whilst working in a two-dimensional phase space. This is a significant advantage which allows us to provide a clear physical interpretation of the key features and draw analogies with previously studied models.
... Isolating G in the gravitational action makes it straightforward to consider the limit of General Relativity. The inclusion of non-minimal coupling terms [66] makes it possible to study some interesting models, for instance where a scalar field would couple to a boundary term, see for instance [98]. If we were to consider couplings of the form αφ 2 B and βφ 2 B, where α and β are coupling constants and φ is a scalar field, then the corresponding non-minimal matter action would not be a true coordinate scalar, as previously discussed. ...
... Therefore it would be unnatural to assume diffeomorphism invariance for this part of the action alone and agrees with our approach of considering models where diffeomorphism invariance only holds for the total action. Assuming the flat FLRW metric a direct calculation shows that the cosmological field equations in the presence of a non-minimally coupled scalar field indeed agree with the field equations given in [98]. This should not be a surprising result at this point. ...
Starting from the original Einstein action, sometimes called the Gamma squared action, we propose a new setup to formulate modified theories of gravity. This can yield a theory with second order field equations similar to those found in other popular modified gravity models. Using a more general setting the theory gives fourth order equations. This model is based on the metric alone and does not require more general geometries. It is possible to show that our new theory and the recently proposed f(Q) gravity models are equivalent at the level of the action and at the level of the field equations, provided that appropriate boundary terms are taken into account. Our theory can also match up with f(R) gravity which is an expected result. Perhaps more surprisingly, we can also show that this equivalence extends to f(T) gravity at the level of the action and its field equations, provided that appropriate boundary terms are taken in account. While these three theories are conceptually different and are based on different geometrical settings, we can establish the necessary conditions under which their field equations are indistinguishable. The final part requires matter to couple minimally to gravity. Through this work we emphasise the importance played by boundary terms which are at the heart of our approach.
... In [46], the authors found that this coupling allows the crossing of the phantom divide line. The new coupling between the scalar field and the boundary term is motivated from the scalar tensor theory studied in [47,48], where the authors found that, without fine-tunning, the system evolves to a late-time accelerating attractor solution. By taking variation with respect to the tetrad, we find the corresponding gravitational field equations given by ...
... The case p = 1 is a very special one which gives a 3-dimensional dynamical system of equations, but generically, does not give any interesting cosmological behaviour. One more realistic model is the one where p = 2 which gives a coupling like (1/2)χφ 2 B which was studied in [47]. In that case however, nontachyonic scalar fields were considered. ...
In this paper we propose a new dark energy model in the teleparallel alternative of general relativity, by considering a generalized non-minimal coupling of a tachyonic scalar field with the teleparallel boundary term. Within the framework of teleparallel gravity, the boundary coupling term is associated with the divergence of the torsion vector. Considering the linear stability technique for various potentials and couplings, we have analyzed the dynamical properties of the present tachyonic dark energy model in the phase space, uncovering the corresponding essential dynamical features. Our study of the phase space structure revealed that for a specific class of potential energy, this model exhibits various critical points which are related to different cosmological behaviors, such as accelerated expansion and scaling solutions, determining the existence conditions and the corresponding physical features.
... where the torsion scalar is replaced by a nonlinear function f (T ) [16][17][18][19][20][21], represent a viable framework naively inspired by f (R)gravity. This prescription can be even extended by a more general form based on f (T, B) functions in order to include both torsion and the boundary term [22,23]. This leads to a generalization of both f (R) and f (T ) classes of models and have been also studied as f (R, T ) in [24]. ...
... In this section, we obtain the effective gravitational coupling with a scalar field non-minimally coupled to both torsion scalar and the boundary term. We restrict our attention to the following action, introduced in [22]: ...
In the present study, we consider an extended form of teleparallel Lagrangian , as function of a scalar field , its kinetic term X and the torsion scalar T. We use linear perturbations to obtain the equation of matter density perturbations on sub-Hubble scales. The gravitational coupling is modified in scalar modes with respect to the one of General Relativity, albeit vector modes decay and do not show any significant effects. We thus extend these results by involving multiple scalar field models. Further, we study conformal transformations in teleparallel gravity and we obtain the coupling as the scalar field is non-minimally coupled to both torsion and boundary terms. Finally, we propose the specific model . To check its goodness, we employ the observational Hubble data, constraining the coupling constant, , through a Monte Carlo technique based on the Metropolis-Hastings algorithm. Hence, fixing to its best-fit value got from our numerical analysis, we calculate the growth rate of matter perturbations and we compare our outcomes with the latest measurements and the predictions of the CDM model.
... The advantage in the teleparallel approach is that it can be interpreted as a gauge theory of gravity [22][23][24] and that it allows for numerous extension of general relativity without introducing higher than second order derivative field equations [25][26][27][28][29][30]. Modifying teleparallel theories of gravity with additional non-minimally coupled scalar fields has been studied throughout the literature [31][32][33][34][35][36][37][38], and we will extend this class in this article by considering a non-minimal coupling to a pseudo-scalar field. This vast variety of possible teleparallel theories of gravity is possible since their building block is the torsion tensor of the flat connection, which contains only first derivatives of the tetrad. ...
We consider the most general teleparallel theory of gravity whose action is a linear combination of the five scalar invariants which are quadratic in the torsion tensor. Since two of these invariants possess odd parity, they naturally allow for a coupling to pseudo-scalar fields, thus yielding a Lagrangian which is even under parity transformations. In analogy to similar fields in gauge theories, we call these pseudo-scalar fields teleparallel axions . For the most general coupling of a single axion field, we derive the cosmological field equations. We find that for a family of cosmologically symmetric teleparallel geometries, which possess non-vanishing axial torsion, the axion coupling contributes to the cosmological dynamics in the early universe. Most remarkably, this contribution is also present when the axion is coupled to the teleparallel equivalent of general relativity, hence allowing for a canonical coupling of a pseudo-scalar to general relativity. For this case we schematically present the influence of the axion coupling on the fixed points in the cosmological dynamics understood as dynamical system. Finally, we display possible generalizations and similar extensions in other geometric frameworks to model gravity.
... We therefore conclude that, unlike the Lagrangians introduced in Refs. [54][55][56][57] to be used in scalar-tensor (or even "Brans-Dicke") teleparallel gravity theories, these extensions of teleparallel gravity would be more accurately described by letting their Lagrangian acquire a coupling between the gradient of the scalar field and torsion. ...
The Noether charge associated to diffeomorphism invariance in teleparallel gravity is derived. It is shown that the latter yields the ADM mass of an asymptotically flat spacetime. The black hole entropy is then investigated based on Wald's prescription that relies on the Noether charge. It is shown that, like in general relativity, the surface gravity can be factored out from such a charge. Consequently, the similarity with the first law of thermodynamics implied by such an approach in general relativity does show up also in teleparallel gravity. It is found that, based on the expression of the first law of black hole mechanics, which is preserved in teleparallel gravity, entropy can thus be extracted from such a Noether charge. The resulting entropy can very naturally be expressed as a {\it volume integral}, though. As such, it is shown that the conformal issue that plagues the entropy-area law within general relativity does not arise in teleparallel gravity based on Wald's approach. The physics behind these features is discussed.
... When a phase space is not compact, one needs to study if there are critical points at infinity. To do this, one can use the method available in [33,50,51] by introducing compactified Poincaré variables. We follow this approach for the two models studied in this paper. ...
In this work we have investigated the dynamics of a recent modification to the general theory of relativity, the energy-momentum squared gravity model f(R,T2), where R represents the scalar curvature and T2 the square of the energy-momentum tensor. By using dynamical system analysis for various types of gravity functions f(R,T2), we have studied the structure of the phase space and the physical implications of the energy–momentum squared coupling. In the first case of functional where f(R,T2)=f0Rn(T2)m, with f0 constant, we have shown that the phase space structure has a reduced complexity, with a high sensitivity to the values of the m and n parameters. Depending on the values of the m and n parameters, the model exhibits various cosmological epochs, corresponding to matter eras, solutions associated to an accelerated expansion, or decelerated periods. The second model studied corresponds to the f(R,T2)=αRn+β(T2)m form with α, β constant parameters. In this case a richer phase space structure is obtained which can recover different cosmological scenarios, associated to matter eras, de–Sitter solutions, and dark energy epochs. Hence, this model represent an interesting cosmological model which can explain the current evolution of the Universe and the emergence of the accelerated expansion as a geometrical consequence.
... In standard flat spacetimes, a vector v i remains constant along a line if it satisfies 27) where λ is an affine parameter used to characterised the curve. Since GR is based on curved spacetimes, the notion of transporting vectors parallel along a line needs to be changed. ...
Teleparallel gravity is an alternative formulation of gravity which has the same field equations as General Relativity (GR), therefore, it is also known as the Teleparallel equivalent of General Relativity (TEGR). This theory is a gauge theory of the translations with the torsion tensor being non-zero but with a vanishing curvature tensor, hence, the manifold is globally flat. An interesting approach for understanding the late-time accelerating behaviour of the Universe is called modified gravity where GR is extended or modified. In the same spirit, since TEGR is equivalent to GR, one can consider its modifications and study if they can describe the current cosmological observations. This thesis is devoted to studying several modified Teleparallel theories of gravity with emphasis on late-time cosmology. Those Teleparallel theories are in general different to the modified theories based on GR, but one can relate and classify them accordingly. Various Teleparallel theories are presented and studied such as Teleparallel scalar-tensor theories, quintom models, Teleparallel non-local gravity, and f(T,B) gravity and its extensions (coupled with matter, extensions of new GR and Gauss-Bonnet) where T is the scalar torsion and B is the boundary term which is related with the Ricci scalar via R=-T+B.
... Introducing this term with an arbitrary function would give a theory which interpolates between scalarcurvature and scalar-nonmetricity theories. This is similar to the case of scalar-torsion theories and their generalisations [35][36][37], where one has to include the boundary term relating the Ricci and torsion scalars in order to obtain a conformally invariant action. ...
Einstein's celebrated theory of gravitation can be presented in three forms: general relativity, teleparallel gravity, and the rarely considered before symmetric teleparallel gravity. These formulations differ in the underlying geometric structure and interpretations, but are equivalent in observational predictions. The theories truly part when one extends them by e.g. nonminimally coupling a scalar field to the metric tensor degree of freedom. Thus extending symmetric teleparallel gravity, we introduce a new class of theories where a scalar field is coupled nonminimally to nonmetricity Q, which here encodes the gravitational effects like curvature R in general relativity or torsion T in teleparallel gravity. We derive the field equations and point out the similarities and differences with analogous scalar-curvature and scalar-torsion theories. We show that while scalar-nonmetricity gravity lacks invariance under conformal transformations, a suitable extra term can restore this; and also establish that f(Q) gravity forms a particular subclass of scalar-nonmetricity theories. We illustrate the theory with an example of flat Friedmann-Lema\^{\i}tre-Robertson-Walker spacetime.
Teleparallel gravity has significantly increased in popularity in recent decades, bringing attention to Einstein’s other theory of gravity. In this Review, we give a comprehensive introduction to how teleparallel geometry is developed as a gauge theory of translations together with all the other properties of gauge field theory. We also related this form of geometry to the broader metric-affine approach to forming gravitational theories where we describe a systematic way of constructing consistent teleparallel theories that respect certain physical conditions such as local Lorentz invariance. We first use teleparallel gravity to formulate a teleparallel equivalent of general relativity which is dynamically equivalent to general relativity but which may have different behaviors for other scenarios, such as quantum gravity. After setting this foundation, we describe the plethora of modified teleparallel theories of gravity that have been proposed in the literature. We attempt to connect them together into general classes of covariant gravitational theories. Of particular interest, we highlight the recent proposal of a teleparallel analogue of Horndeski gravity which offers the possibility of reviving all of the regular Horndeski contributions. In the second part of the Review, we first survey works in teleparallel astrophysics literature where we focus on the open questions in this regime of physics. We then discuss the cosmological consequences for the various formulations of teleparallel gravity. We do this at background level by exploring works using various approaches ranging from dynamical systems to Noether symmetries, and more. Naturally, we then discuss perturbation theory, firstly by giving a concise approach in which this can be applied in teleparallel gravity theories and then apply it to a number of important theories in the literature. Finally, we examine works in observational and precision cosmology across the plethora of proposal theories. This is done using some of the latest observations and is used to tackle cosmological tensions which may be alleviated in teleparallel cosmology. We also introduce a number of recent works in the application of machine learning to gravity, we do this through deep learning and Gaussian processes, together with discussions about other approaches in the literature.
We analyze the cosmological parameters and thermodynamics of a newly proposed generalized tachyonic teleparallel gravity whose action contains a non-minimal coupling of the teleparallel boundary term with a tachyonic scalar field in the generalized form. Three different Hubble parametric H(z) models are directly obtained through parameterized deceleration parameter (q) in terms of redshift parameter (z). For these models, we investigate the fractional matter density and fractional dark energy density through graphs and found their consistent values with recent Planck 2018 data. We also examine the equation of state parameter for each model which shows phantom phase of the universe in far future era, whereas lies in the range of cosmological constant and quintessence region of the universe for near future, present and past era. The phase plane lies in the freezing region of the accelerated universe, and indicates the stability of each model. In thermodynamic aspects, we examine the validity of generalized second law of thermodynamics by choosing logarithmic and power law entropy corrections for the Hubble horizon. We find out that this law is verified for each case of every model in each era.
Among the large class of modified gravity theories, teleparallel gravity theories are distinguished by the fact that they express gravity by the torsion of a flat (curvature-free), metric-compatible connection. This approach, which offers both an alternative formulation of general relativity itself, as well as modifications thereof, is appealing under several aspects. Teleparallel gravity theories can be understood as gauge theories of the translation group, whose Lagrangian is of first derivative order in the fundamental fields, and thus has more in common with the Yang-Mills Lagrangian known from particle physics. Further, they offer a rich phenomenology, giving rise to numerous viable candidate theories addressing the open questions in cosmology. This chapter gives an introduction to the foundations of teleparallel gravity, an overview of modified teleparallel theories, and a summary of the phenomenology of various such theories in cosmology and beyond.
Starting from the original Einstein action, sometimes called the Gamma squared action, we propose a new setup to formulate modified theories of gravity. This can yield a theory with second order field equations similar to those found in other popular modified gravity models. Using a more general setting the theory gives fourth order equations. This model is based on the metric alone and does not require more general geometries. It is possible to show that our new theory and the recently proposed f(Q) gravity models are equivalent at the level of the action and at the level of the field equations, provided that appropriate boundary terms are taken into account. Our theory can also reproduce f(R) gravity, which is an expected result. Perhaps more surprisingly, we show that this equivalence extends to f(T) gravity at the level of the action and its field equations, provided that appropriate boundary terms are taken in account. While these three theories are conceptually different and are based on different geometrical settings, we can establish the necessary conditions under which their field equations are the same. The final part requires matter to couple minimally to gravity. Through this work we emphasize the importance played by boundary terms which are at the heart of our approach.
We consider a recently proposed class of extended teleparallel theories of gravity, which entail a scalar field which is nonminimally coupled to the torsion of a flat, metric-compatible connection. This class of scalar-torsion theories of gravity is constructed in analogy to and as a direct extension of the well-studied class of scalar-curvature gravity theories, and has various common features, such as the conformal frame freedom. For this class we determine the parametrized post-Newtonian limit, both for a massive and for a massless scalar field. In the massive case, we determine the effective gravitational constant and the post-Newtonian parameter γ, both of which depend on the distance between the gravitating and test masses. In the massless case, we calculate the full set of parameters and find that only γ and β potentially deviate from their general relativity values. In particular, we find that for a minimally coupled scalar field, the theory becomes indistinguishable from general relativity at this level of the post-Newtonian approximation.
In this article we derive the post-Newtonian limit of a class of teleparallel theories of gravity, where the action is a free function L(T,X,Y,ϕ) of the torsion scalar T and scalar quantities X and Y built from the dynamical scalar field ϕ. We restrict the analysis to a massless scalar field in order to use the parametrized post-Newtonian formalism without modifications, such as introducing an effective gravitational constant which depends on the distance between the interacting masses. In particular the results show a class of fully conservative theories of gravity, where the only nonvanishing parameters are γ and β. For a particular choice of the function L(T,X,Y,ϕ) the theory cannot be distinguished from general relativity in its post-Newtonian approximation.
The Noether charge associated to diffeomorphism invariance in teleparallel gravity is derived. It is shown that the latter yields the Arnovitt-Deser-Misner mass of an asymptotically flat spacetime. The black hole entropy is then investigated based on Wald’s prescription that relies on the Noether charge. It is shown that, like in general relativity, the surface gravity can be factored out from such a charge. Consequently, the similarity with the first law of thermodynamics implied by such an approach in general relativity does show up also in teleparallel gravity. It is found that, based on the expression of the first law of black hole mechanics, which is preserved in teleparallel gravity, entropy can thus be extracted from such a Noether charge. The resulting entropy can very naturally be expressed as a volume integral, though. As such, it is shown that the conformal issue that plagues the entropy-area law within general relativity does not arise in teleparallel gravity based on Wald’s approach. The physics behind these features is discussed.
We present a physically plausible solution representing Einstein's cluster mimicking the behaviors of compact star in the context of teleparallel equivalent of general relativity. The Teleparallel gravity (TEGR) is an alternative formulation of gravity which uses tetrads as the dynamical variables. We focus on two particularly interesting scenarios. First, we develop the Einstein clusters in TEGR field equations using effective energy-momentum tensor for diagonal as well as off-diagonal tetrad. We then study the clusters in modified f (T)−gravity for anisotropic fluid distribution. Based on these two theories, we further study the solution without net electric charge and then for charged solution. For charge parameter k → 0, the charged solution reduces to neutral one. Our calculations show that when charge increases, the stiffness of the EoS also increases. This is due to increase in adiabatic index and sound speed approaching speed of light. When the charge increase beyond a certain limit (0 ≤ k ≤ 1.3 × 10 −5 and 0 ≤ k ≤ 1 × 10 −6), the compactness parameter crosses the Buchdahl limit i.e. 2M/R > 8/9 and the solution start violating the causality condition. We test the Tolman-Oppenheimer-Volkoff (TOV) limit for such compact objects. We analyze the static stability criterion of the Einstein clusters for both charged and uncharged case, and the stability of such compact objects is enhanced by the presence some net electric charge. In addition, we present and discuss the energy conditions, causality condition and the adiabatic index close to the stability limit. After analyzing these problems, we conclude that the Einstein clusters do exists only if f (T) is a linear function of the torsion scalar T , that is in the case of Teleparallel Equivalent of General Relativity. Finally, we compare our solution for pure general relativity. As a result, we concluded that the Einstein cluster solution do exist in pure GR, however, physically unfit to mimic compact stars. We have also extend our findings by assuming the diagonal or off-diagonal tetrad and specific case of f (T). In such models, Einstein's cluster solutions do exist however can't mimic the properties of a compact star.
We discuss a class of teleparallel scalar-torsion theories of gravity, which is parametrized by five free functions of the scalar field. The theories are formulated covariantly using a flat, but non-vanishing spin connection. We show how the actions of different theories within this class are related via conformal transformations of the tetrad and redefinitions of the scalar field, and derive the corresponding transformation laws for the free function in the action. From these we construct a number of quantities which are invariant under these transformations, and use them to write the action and field equations in different conformal frames. These results generalize a similar formalism for scalar-tensor theories of gravity, where the invariants have been used to express observables independently of the conformal frame.
We consider a generalized teleparallel theory of gravitation, where the action contains an arbitrary function of the torsion scalar and a scalar field, , thus encompassing the cases of f(T) gravity and nonminimally coupled scalar field as subclasses. The action is manifestly Lorentz invariant when besides the tetrad one allows for flat but nontrivial spin connection. We derive the field equations and demonstrate how the antisymmetric part of the tetrad equations is automatically satisfied when the spin connection equation holds. The spin connection equation is a vital part of the covariant formulation, since it determines the spin connection compatible with a given tetrad. We discuss how the spin connection equation can be solved in general, and provide the cosmological and spherically symmetric examples. Finally we generalize the theory to an arbitrary number of scalar fields.
We present a systematic analysis of the dynamics of flat Friedmann-Lema\^{i}tre-Robertson-Walker cosmological models with radiation and dust matter in generalized teleparallel f(T) gravity. We show that the cosmological dynamics of this model is fully described by a function W(H) of the Hubble parameter, which is constructed from the function f(T). After reducing the phase space to two dimensions we derive the conditions on W(H) for the occurrence of de Sitter fixed points, accelerated expansion, crossing the phantom divide, and finite time singularities. Depending on the model parameters it is possible to have a bounce (from contraction to expansion) or a turnaround (from expansion to contraction), but cyclic or oscillating scenarios are prohibited. As an illustration of the formalism we consider power law models, and show that these allow only one period of acceleration and no phantom divide crossing.
The dynamical aspects of scaling solutions for the dark energy component in the theoretical framework of teleparallel gravity are considered, where dark energy is represented by a scalar field nonminimally coupled with the torsion and with a boundary term, where the boundary coupling term represents the divergence of the torsion vector. The behavior and stability of the scaling solutions are studied for scalar fields endowed with inverse power law potentials and with exponential potentials. It is shown that for scalar fields endowed with inverse power-law potentials, the stability conditions are not affected by the coupling coefficients. For the scalar fields endowed with exponential potentials, two cases are studied: at first, we have considered an infinitesimal deviation from the scaling solution in the corresponding Klein–Gordon equation, and the impact of distinct coupling coefficients on the stability of the solution are analyzed. Secondly, the potential-free case is considered where the dominance of the coupling terms over the potential term is analyzed, discussing the validity of the corresponding particular solution.
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