A Built-in Inflation in the $f(T)$-Cosmology

In the present work we derive an exact solution of an isotropic and homogeneous Universe governed by $f(T)$ gravity. We show how the torsion contribution to the FRW cosmology can provide a \textit{unique} origin for both early and late acceleration phases of the Universe. The three models ($k=0, \pm 1$) show a \textit{built-in} inflationary behavior at some early Universe time; they restore suitable conditions for the hot big bang nucleosynthesis to begin. Unlike the standard cosmology, we show that even if the Universe initially started with positive or negative sectional curvatures, the curvature density parameter enforces evolution to a flat Universe. The solution constrains the torsion scalar $T$ to be a constant function at all time $t$, for the three models. This eliminates the need for the dark energy (DE). Moreover, when the continuity equation is assumed for the torsion fluid, we show that the flat and closed Universe models \textit{violate} the conservation principle, while the open one does not. The evolution of the effective equation of state (EoS) of the torsion fluid implies a peculiar trace from a quintessence-like DE to a phantom-like one crossing a matter and radiation EoS in between; then it asymptotically approaches a de Sitter fate.


Introduction
Our Universe seems to be accelerating from the recent observational data. This acceleration can be demonstrated in terms of what is called DE, in which modified gravity models may explain it [1]. Cosmological constant, from an ideal fluid having different shapes of EoS and negative pressure, a scalar field with quintessence-like or phantomlike behavior can also explain DE [2]. The true nature and origin of DE has not been persuasively explained due to the existence of many possibilities which explained it. It is not obvious what kind of DE is more suitable to explain the present epoch of the universe. Observational data point to some type of DE having an EoS parameter which is near to ω = −1, or even less than −1 (which is the phantom case). Modification of General Relativity (GR) seems to be quite attractive possibility to resolve the above mentioned problem. Modifications of the Hilbert-Einstein action through introducing several functions of the Ricci scalar, R, have been methodically explained what is called f (R) gravity models [3]- [9]. f (R) gravity can be rewritten using a scalar field, quintessence or phantom like, through redefining the function f (R) by help of a scalar field, and then carrying a conformal transformation.
Attention to the reformulation of f (R) gravity through scalar tensor theory has been paid. It has been shown that when starting from f (R) gravity, the phantom case in scalar tensor theory does not exist. However, when the conformal transformation becomes complex the phantom barrier is crossed, and therefore the resulting f (R) function becomes complex. These cases are studied [8] in more detail, in which to avoid this handicap, a dark fluid was used to produce the phantom behavior such that f (R) function reconstructed from the scalar tensor theory continues to be real.
Actually the idea of teleparallelism theory has been proposed firstly by Einstein in order to unify gravity and electromagnetism [10,11]. Later Einstein left the teleparallelism theory not because his failure attempt of unification only, but also because of the vanishing of the curvature tensor of the Weitzenböck connection. But the non vanishing torsion tensor shows recently a great interest in astrophysical and cosmological applications in the so called f (T ) gravitational theory. The main motivations of such theory were: (1) GR can be viewed as a certain theory of teleparallelism and, thus, it could be regarded at least as a different perspective that could lead to the same results [12].
(2) In such context, one can define energy and momentum tensor of the gravitational field which is the true tensor under all general coordinate transformations but not under local Lorentz transformation.
(3) This theory is interesting because it can be seen as gauge theories of the translation group (not the full Poincaré group) consequently, they provide an alternative interpretation of GR [13,14]. Most recently Teleparallel Equivalent of General Relativity (TEGR) has been generalized to f (T ) theory, a theory of modified gravity formed in the same spirit as generalizing GR to f (R) gravity [15,16]. A main merit of f (T ) gravity theory is that its gravitational field equation is of second order, same as of GR, while it is of fourth order in metric f (R) gravity. This merit makes the analysis of the cosmological expansion of the universe in f (T ) gravity much easier than in f (R) gravity. f (T ) gravity has gained a significant attention in the literature, and proves to exhibit interesting cosmological implications [17]- [62].
The main target of this work is to investigate other useful applications of f (T ) gravity theories in the early Universe cosmology as well as its role in the different phases of the late Universe cosmology. It is well known that the flat Universe model is most suitable model matching the present Universe observations. This is because the curvature density evolves much faster than other density parameters, so we must observe a highly curved space Universe at the present time which is not fulfilled. The observations of the present Universe indicates that our Universe now is almost spatially flat. This leads to exclude the closed and open Universe models. On the other hand the initial flat space assumption contradicts the presence of the strong gravitational field (i.e. the Riemann curvature) as it should be! This contradiction might be explained as the flatness problem of the standard cosmology. Actually this problem has been overcome by the idea of an inflationary scenario at 10 −34 sec from the Big Bang. Lots of inflationary models have been proposed by adding scalar fields. But to gain the benefits of both inflation and the standard cosmology the inflation should end at 10 −36 sec from the Big Bang. This needs slow-roll conditions so that the inflationary Universe ends with a vacuum dominant epoch allowing Universe to restore the Big Bang scenario. So the inflation can be considered as an add-on tool rather a replacement of the Big Bang. Until now there are no satisfactory reasons for the transition from inflation to Big Bang. Our trail here to treat these problems starts by diagnosing the problem from the core. We found that the curvature within the framework of the GR may lead to these conflicts, while introducing new qualities to the space-time, as torsion, might give a different insight of these problems.
The work is arranged as follows: In Section 2, we describe the fundamentals of the f (T ) gravity theory. We next show the contribution of the torsion scalar field to the density and the pressure of the FRW models and necessary modifications in Section 3. Also, we obtain a model dependent scale factor R(t) and f (T ) as a solution of the continuity equation. In Section 4, we investigate the cosmological behaviour of the flat, close and open Universe models due to f (T ) modifications. Moreover, we give the physical descriptions for the obtained results. In the flat Universe the teleparallel torsion scalar field T and the f (T ) appear as constant functions and the later might replace the cosmological constant, the Universe shows an inflationary behaviour as the scale factor R(t) ∝ e Ht , where the Hubble parameter H is a constant. The flat Universe shows no evolution with time. Moreover, we investigate the close Universe model which shows an inflationary behaviour as well. In spite of the torsion scalar field T appears as a constant function similar to the flat case, but the f (T ) of the close Universe appears as a function of time. This allows the cosmological parameters to evolve. In particular the evolution of the curvature density parameter Ω k shows a clear tendency to vanish at late time, which explains how the Universe can start with initial curvature then it goes naturally to flat behaviour. Combining the curvature density parameter in the total density parameter Ω Tot with the matter Ω m and the torsion Ω T density parameters gives a very restrictive range for the total density parameters |Ω Tot − 1| ≤ 10 −16 at some early time. This is a suitable value to begin the primordial nucleosynthesis epoch. The late accelerating expansion of the Universe is also recognized as the Hubble parameter H > 0 and the deceleration parameter q → −1. Furthermore, the investigation of the open Universe shows a behaviour similar to the close model. So both close and open models suggest a unique source for early and late acceleration phases of the Universe. While the open model, uniquely, implies an time dependent effective EoS of the torsion fluid. Its evolution started initially with a quintessence-like energy to asymptotical de Setter crosses radiation, dust and phantomlike energies. In addition, the open Universe uniquely preserves the conservation energy principle. Section 5 is devoted for summarizing and concluding the results.

ABC of f (T )
In the Weitzenböck spacetime the fundamental field variables describing gravity are a quadruplet of parallel vector fields [63] h i µ , which we call the tetrad field characterized by where Γ µ λν define the nonsymmetric affine connection with h iµ,ν = ∂ ν h iµ 1 . Equation (2.1) leads to the metricity condition and the identically vanishing of curvature tensor defined by Γ λ µν , given by Equation (2.2). The metric tensor g µν is defined by with η ij = (+1, −1, −1, −1) is the metric of Minkowski spacetime. We note that, associated with any tetrad field h i µ there is a metric field defined uniquely by (2.3), while a given metric g µν does not determine the tetrad field completely; for any local Lorentz transformation of the tetrads h i µ leads to a new set of tetrads which also satisfy (2.3). Defining the torsion components and the contortion as where the contortion equals the difference between Weitzenöck and Levi-Civita connection, i.e., K µ νρ = Γ µ νρ − µ νρ . The tensor S α µν is defined as which is skew symmetric in the last two indices. The torsion scalar is defined as Similar to the f (R) theory, one can define the action of f (T ) theory as and we assumed the units in which G = c = 1 and Φ A are the matter fields. Considering the action (2.7) as a function of the fields h a µ and putting the variation of the function with respect to the field h a µ to be vanishing one can obtain the following equations of motion [64,15].
∂T 2 and T ν µ is the energy momentum tensor.

Cosmological Modifications of f (T )
Recent cosmic observations support that the Universe is expanding with an acceleration. In this paper we attempted to apply the f (T ) field equations to the universe. In this cosmological model the Universe is taken as homogeneous and isotropic in space, which directly gives rise to the tetrad given by Robertson [67]. This tetrad has the same metric as FRW metric; it can be written in spherical polar coordinate (t, r, θ, φ) as follows: where R(t) is the scale factor, L 1 = 4 + kr 2 and L 2 = 4 − kr 2 . Substituting from the vierbein (3.1) into (2.6), we get the torsion scalar is the curvature energy density parameter. The field equations (2.8) read where the EoS is taken for a perfect fluid so that the energy-momentum tensor is T µ ν = diag(ρ, −p, −p, −p). Using (3.3), the perfect fluid density ρ is given by and using (3.4), the proper pressure p of the perfect fluid is given by Equations (3.5) and (3.6) are the modified Friedmann equations in the f (T )-gravity in its generalized form. Then, the EoS parameter ω = p ρ of the perfect fluid is given by Considering the total energy density and pressure of the Universe behaves as the DE. Assuming the EoS of the DE, i.e., p = −ρ, we get from Eq. (3.7) an explicit form of f (T ) as: where a and b are constants of the integration. The above equation indicates that there is a certain code relating the f (T ) to the scale factor R(t) so that we should investigate possible compatibilities of these two functions. Later, in §3.3, we will show that enforcing the universal density to produce a DE, as we have just done in (3.8), is not a functional code for the universe!

The FRW dynamical equations
Let us assume that the background is a non-viscous fluid. As we have mentioned, we can not enforce the total density and pressure to be a DE. Alternatively, we can study the torsion contribution to both ρ and p in the Friedmann dynamical equations by replacing ρ → ρ + ρ T and p → p + p T , where ρ, ρ T , p and p T are the matter density, the torsion density, the matter pressure and the torsion pressure respectively.

Ṙ
where q(= − RṘ R 2 ) is the deceleration parameter. In the above equation we take the general case of a non-vanishing pressure p = 0. It is clear that when ρ T = 0 and p T = 0 the above equations reduce to the usual Friedmann equations in GR. We take ρ = ρ c where ρ c is the critical density of the Universe when it is full of matter and spatially flat (k = 0), then ρ c = 3H 2 8π . Substituting in equations (3.9) and (3.10) we get where Ω m = ρ ρc = ρ 3H 2 /8π represents the matter density parameter and Ω T = ρT ρc = ρT 3H 2 /8π represents the torsion density parameter.

The torsion contribution
In order to obtain the torsion contribution ρ T and p T , we rewrite equations (3.5) and (3.6), in terms of the Hubble parameter, as below Also, the EoS-parameter (3.7) can be rewritten as Substituting the matter density that is obtained by the f (T ) field equation (3.13) into the FRW dynamical equation (3.9), we get the torsion density The above equation can be written in the form so that the torsion density parameter is comparing the above equation to equation (3.11) we get the modified matter density parameter as Similarly we substitute from (3.13), (3.14) and (3.16) into (3.10) we get The EoS parameter due to the torsion contribution is thus It is clear that ω T = −1 for the case of flat Universe (k = 0 andḢ = 0), c.f. [45]. The torsion contributes to the FRW model in a way similar to the cosmological constant.

A generalized R(t) and f (T ) as an ordered pair
The scale factor R(t) plays the key role in the Universe evolution and composition. Most of the cosmological applications leaves the scale factor to be chosen! In this section, we aim to get a generalized form for a model dependent f (T ) and R(t). In this case some solutions will be rejected due to incompatibility. This can be done as follows, we assume that matter content is conserved so we substitute the matter density (3.5) and pressure (3.6) into the continuity equationρ the continuity equation readsṘ the solution of the above differential equation has many possible cases: We exclude the case of R(t) is a constant as it gives a steady Universe. We interested to examine the case of the vanishing of the first and second brackets simultaneously. So we first takeRR −Ṙ 2 − k = 0, by solving for the scale factor R(t) we get where c 1 and c 2 are constants of integrations. We next examine the vanishing of the second bracket of (3.22) so that 12f T TṘ 2 + f T R 2 = 0, by substituting from (3.23) and solving for f (T ) we get

World Models
One of the benefits of the obtained solution that it is a generalized f (T ) and R(t) formula valid for the three world models, the spatially flat Universe (k = 0), the pseudo sphere, open, Universe (k = −1) and the sphere, closed, Universe (k = +1). This enables us to examine the behavior of the DE and its effects on the cosmological parameters in these different models as follows.

Flat Universe
In the case of spatially-flat FRW universe, k = 0, the scale factor (3.23) becomes  can easily conclude that the torsion scalar plays the role of the cosmological constant during the inflation period. We next evaluate the critical density, using (4.3), for flat space is the matter density (3.13) and pressure (3.14) read the torsion density (3.16) and pressure (3.19) are One can easily find that the total density is at its critical value exactly ρ c = ρ + ρ T . Also, it should be mentioned that assuming the torsion fluid fulfills the continuity equation gives a case similar to the steady state cosmology, where theρ = 0 andρ T = 0. The EoS parameter for both matter and torsion are the curvature density parameter for the flat Universe Ω k = 0, while the matter density parameter (3.18) is 8) and the torsion energy parameter (3.17) is (4.9) The above cosmological parameters show that the scale factor (4.1) growths exponentially with time. But the Universe constituents do not change with time. This does not allow the Universe to evolute. However, the Universe shows an accelerated expansion. The Eqs. (4.5), (4.6) and the continuity (3.21) lead to the conclusion that the total density has a constant value, nevertheless the universe is expanding! This leads directly to a violation of the conservation principle of energy. In the following two Sections, we are going to examine similar cases in both the closed and open Universes.

Sphere, close, Universe
In the case of the close FRW Universe, k = +1, the scale factor (3.23) becomes The Hubble parameter H appears in the closed Universe as a function of time not a constant as given in the flat case, but keeping the same exponential behaviour of the scale factor with time as the flat Universe. We find this case is more suitable to describe the evolution of the constituents of the Universe. Another cosmological parameter which is related to the Universe evolution is the deceleration parameter, this parameter appears for the close Universe as a function of time as q = − tanh 2 t + c 2 c 1 , (4.14) In order to show the cosmological behavior, the deceleration parameter (4.14) versus redshift z = R0 R − 1, ehere R 0 is the scale factor at the present time, is plotted in Figure 1(a). The graph shows that the deceleration parameter q → 0 as z → ∞, then q → −1 as z → 0 at late Universe. The plot shows that the accelerating phases of the close Universe from early to late time. Also, the graph shows that the deceleration parameter is −1 when the torsion scalar field is dominant. The curvature density parameter for the closed Universe is In the standard Cosmology it is will known that if there is a slight deviation from the flat Universe, it goes to be more and more curved one very quickly. The the curvature density parameter in the closed Universe model initially chosen to produce a closed Universe. The cosmological parameter Ω k , given by (4.15), is plotted versus the redshift z in Figure 1 One should note that in spite of the Hubble and the curvature density parameters are functions of time they combine in a way to rule out the evolution of the torsion scalar field with the expansion. Also, it is clear that the torsion scalar field is dominant T → ∞ as c 1 → 0. The critical density, for a closed universe, is generalized to be a function of time the matter density (3.13) and pressure (3.14) are while the torsion density (3.16) and pressure (3.19) read The cosmological parameters in equations (4.15), (4.21) and (4.22) are plotted versus the redshift z in Figure 2(a) to provide information about the evolution of the cosmos components during the expansion for the close Universe model. In spite of all the Universe compositions vary with time, we found a rapid change at early Universe, then it converges all compositions to act as a steady behaviour of the flat Universe at late Universe. This leads to investigate the global behaviour of the Universe compositions. Thus we define the the total density parameter Ω Tot := Ω m + Ω T + Ω k , where it includes the curvature one. According to the FRW dynamical equation (3.9), the total density parameter Ω Tot initially equals to 1. The early variation of the densities parameters in Figure 2(a) is reflected on the total density parameter, see Figure 2(b). The plot shows very high frequency variations which is explained by recognizing the rapid, but smooth, variation of the densities parameters at early time, then it turns back to 1 at late Universe when the parameters become steady. The inflationary behaviour of (4.10) combined with the violent variations shown in Fig 2(b) of an amplitude of |Ω Tot − 1| ≤ 10 −16 restores the most outstanding success of the Hot Big Bang, the nucleosynthesis. Also, the obtained closed Universe model shows a behaviour different from the standard cosmology. It is well known that when the Universe is slightly shifted from the flat case it goes further away to be more curved which is inconsistent with the present observation. This leads to assume an initial flat Universe model. Here we show that the Universe might start initially with a positive curvature then it turns to a flat Universe behaviour. This reopens the closed Universe model for more investigations.
In addition, the calculations of the cosmological parameters for the closed Universe model (4.18) and (4.19) show that a case similar to the flat Universe. Where the total density of the Universe is constant,ρ = 0 andρ T = 0, while the Universe expands! Again, we get a violation to the energy conservation principle.

Pseudo sphere, open, Universe
In the case of the open FRW Universe, k = −1, the scale factor (3.23) becomes the Hubble parameter appears as a function of time whose gradual change in time aṡ Also here in the open Universe case we got an exponential scale factor but a varying Hubble parameter. This case is more suitable to find the evolution of the Universe. The deceleration parameter will be we plot the deceleration parameter versus the redshift z in Figure 3(a). The evolution of the deceleration parameter versus the redshift z shows a possible deceleration epoch when q > 0 before going to be negative allowing accelerated expansion of the open Universe. Also, the curvature density parameter is given by (4.28) The evolution of the curvature density parameter is plotted versus the redshift z in Figure 3(b). The plot shows that the curvature density parameter started initially with an arbitrary value then it converges naturally to the flat case, which agrees with the present Universe observations. This encourages to reconsider the curved open Universe model. We next evaluate the torsion scalar field (3.2) in the open Universe, we get  We recognize that the open Universe case, uniquely, gives a dynamical behaviour of the EoS of the torsion fluid. The evolution of the EoS parameter, (4.35) shows an initial quintessence-like DE, crossing ω = 0 dust like epoch to a radiation one at ω ∼ 1 3 then it turns back crossing ω = 0 very quickly to cross ω = −1 implying that a phantom-like DE (ω T < −1), then it asymptotically approaches a de setter fate. It is well known that the density of the phantom-like dark torsion fluid ρ T ∝ R(t) n , where n is positive, which implies an increasing of the density as the Universe expands. The phantom energy epoch might be created as a result of the curvature density parameter decay in order to preserve the energy conservation principle. We next write the matter density parameter  Fig 5(a). However, the Universe compositions vary with time very quickly, they combine later in a way to give a flat Universe behaviour. The investigation of the global behaviour of the Universe compositions shows that the total density parameter Ω Tot is extremely close to 1. Figure 5(b) shows that a very restrictive variation range of the total density parameter |Ω Tot − 1| ≤ 10 −15 at early Universe, which is similar to the close Universe case but slightly less. Then it turns to 1 at some late Universe time. However, the Universe shows an inflationary behaviour, (4.23), it restores the critical value of Ω Tot for the nucleosynthesis to begin. We must mention here that the open Universe model is the most accurate model, in the present work, as the nucleosynthesis epoch is from ∼ 1 sec → 200 sec, while similar case of the close Universe takes much longer time.
In addition, the calculations of the cosmological parameters for the open Universe model (4.31) and (4.33) show that a case similar to the flat Universe for the matter content where the matter density of the Universe is constant,ρ = 0. Again, by assuming that the torsion fluid fulfills the continuity equation, we find a behaviour different from the flat or the close models. Since, the torsion density is not constant,ρ T = 0, while the Universe expands! We get a unique behaviour of the open Universe model prevents the violation of the energy conservation principle. This leads to the conclusion that the torsion density might decay reproducing a matter density as the Universe expands. Moreover, the open Universe uniquely implies an initial quintessence-like and later phantomlike energy and a de Setter in the future. For the above mentioned reasons we find that the open Universe model is the most accurate and consistent model in the present work. We summarize the evaluated cosmological parameters of the three models in the next subsection.

Cosmological parameters summary in the three world models
In the following table we summarize the calculated cosmological parameters for the three world models k = −1, k = 0 and k = +1. These values are useful to discuss the standard problems of cosmology, i.e. the particle horizon, the flatness and the singularity problems. We may split these cosmological parameters to two different sets: the first is to describe the composition of the universe, which contains three parameters Ω m , Ω k and Ω T . The second set is to describe the expansion of the universe, which contains two parameters H and q. In the following Section we are going to study the possible interpretations for the cosmological parameters, Table 1, and their effects on the evolutionary scenario of the Universe.

Concluding Remarks
In this work we evaluated the matter density and pressure of the f (T ) field equations. We modified the FRW models due to the torsion contribution by replacing ρ → ρ + ρ T and p → p + p T . Most of the cosmological models choose the scale factor R(t) independent of the model. In this work we got a model dependent R(t) and f (T ) as order pairs, when applying the continuity equation to the Universe matter. The obtained solution allows us to study the three world models, i.e. k = 0, ±1. The calculations show that the torsion scalar (3.2) can be written as a combination of the Hubble parameter H and the curvature density parameter Ω k . These two parameters always combine to keep the constant torsion scalar at all the time t.
The study of the flat Universe model produces an inflationary cosmological model R(t) ∝ e Ht , H = const. But the Universe compositions have no evolution where the matter density is constant during the expansionρ = 0. Applying the continuity equation to the torsion fluid leads to a constant torsion density during the expansion. The total density of the Universe is equivalent to the Universe critical density. Then we conclude that the flat Universe model violates the conservation principle.
The cosmological parameters for the close Universe model are founded as functions of time. These parameters show a quick evolution at some early Universe, then they show a steady behaviour at later time. Although the Universe in the closed model is chosen to be curved initially, the Universe composition enforces the Universe to be flat at some late time as Ω Tot → 1 and Ω k → 0. We applied the continuity equation to the torsion fluid, this gives a steady state Universe. The obtained results for the closed Universe model violate the conservation principle.
In the case of the open Universe model we found a quick evolution of the cosmological parameters at some early time. The Universe in this model has been chosen to be initially curved, while the evolution of the cosmological parameters turns the Universe to be flat at some later time. The calculations show that the evolution of the open Universe prevents the violation of the conservation principle. This makes the open Universe model the most acceptable one.
The inflationary Universe has been started as a speculative idea to solve some problems of the Big Bang cosmology. The inflation has been considered as an add-on extra tool to the standard Big Bang during some very early Universe. In this model, we get a built-in inflationary behaviour at early time then the model enables the Big Bang to be restored naturally.
In the standard Big Bang cosmology it is known that the Universe becomes more and more curved very quickly, if it has been chosen to be initially curved, i.e. Ω Tot diverges away from the unity. But the current cosmological observations show that our present Universe is almost flat. This requires a flat Universe initial condition. In our model, unlike the standard cosmology, we found that the even if the Universe has started with an initial curvature, the evolution of Ω Tot converges to the unity. This tells that the Universe in the case of k = ±1 models are enforced to be a flat. This solves many of the hot big bang cosmology problems. The close Universe model shows that an extremely restrictive range for the total density parameter |Ω Tot − 1| ≤ 10 −16 at early Universe time, which is required for the nucleosynthesis epoch to begin and restoring the Big Bang scenario. The open Universe shows almost the same restrictive range but much lesser time, which perfectly agrees with the BBN period (∼ 1 sec -200 sec). This again supports the open Universe model.
In this model we found that the teleparallel torsion density explains both early and late cosmic acceleration. This eliminates the need for the DE, in addition, it does not address the cosmological constant problem. Also, the use of the torsion scalar instead of the cosmological constant gives a conservative Universe in the case of the open Universe. In addition, the torsion contribution to the open Universe gives a built-in inflationary behaviour at a very early time then the evolution of the total parameter density Ω Tot shows a good agreement with later stages of the Big Bang scenario. Moreover, the open Universe converges to a flat one which agrees perfectly with the current observations. Furthermore, The evolution of the torsion fluid EoS, Fig. 4, shows a peculiar dynamical behaviour during different phases of the cosmic expansion. There are many other details of these models that need further investigations. In particular, the torsion density and pressure in the open Universe model and their possible justifications from quantum cosmology.