The Chaplygin Gas as a Model for f(T) Gravitation?

This paper explores the possibility of treating the exotic Chaplygin-gas (CG) fluid model as some manifestation of an f(T) gravitation. To this end, we use the different cosmological CG equations of state, compare them with the equation of state for the modified teleparallel, i.e. f(T), gravity and reconstruct the corresponding f (T) Lagrangian densities. We then explicitly derive the equation of state parameter of the torsion fluid w T and study its evolution for vacuum-torsion, radiation-torsion, dust-torsion, stiff fluid-torsion and radiation-dust-torsion multi-fluid systems. The preliminary results show that, in addition to providing Chaplygin-gas-like solutions to f (T), which naturally behave like dark matter and dark energy at early and late times respectively, the technique can be used to overcome some of the challenges attributed to the CG cosmological alternative.


Introduction
The discovery of the accelerated expansion of the Universe has posed one of the biggest challenges for observational and theoretical cosmology and, two decades on, remains an open problem. Several suggestions have been put forward as the possible causes behind this cosmic acceleration, including the dark energy (DE) hypothesis and modified gravity theories. Dark energy is a hypothetical form of energy with negative pressure, acting against gravity, and believed to permeate all of space, accounting for about 68% of the entire energy content of the Universe [1]. Among the most well known natural candidates for dark energy are: a cosmological constant [2], a self-interacting scalar field [3] and the cosmological CG as a non-interacting fluid [4]. In the modified gravity approach, several theories have been constructed by modifying the Einstein-Hilbert action such as f (T ) -T being the torsion scalar, f (R) -R being the Ricci scalar, a combination of the two named f (R, T ); f (G) -G being the universal gravitational constant, etc. Among the earliest such attempts was by Einstein himself, when he used the teleparallel gravity (TG) theory [5] to unify the theory of electromagnetism with gravity theory [6].
In the general relativity (GR) approach, T is assumed to vanish and in TG theory, R is assumed to vanish [7]. Fortunately, the two basic theories of gravity describe the gravitational interaction equivalently. So, torsion is an alternative direction of describing the gravitational field interaction. The energy-momentum tensor (EMT) is the source of curvature in GR and torsion in TG theory [7,8,9,10,11,12,6,13]. More recently, generalizations to the TG theory have been introduced in the form of f (T ) gravity, where the action is now a generic function of T , rather than T itself. It is a second-order modified gravity theory and understanding its cosmological implications is an active area of research [14,15,16,17,18,19,20].
The Chaplygin Gas (CG), first introduced by Chaplygin [21], had a non-cosmological origin but has recently gained new attention in cosmology due to its negative pressure. Together with a Friedman-Lemaître-Robertson-Walker (FLRW) background in the GR framework, this model can explain the cosmic expansion history for a Universe filled with an exotic background fluid [22,23,24,25,26,27,28], but has recently been studied in the context of modified gravity theories as well [26,29,27]. Although the model mimics early-and late-time cosmic evolution scenarios in the asymptotic limits, certain technical issues remain unresolved for it to be the final candidate for a dark fluid, such as the issues of large-scale structure formation and the violation of causality due to a negative speed of sound associated with the fluid.
In this work, our aim is to reconstruct f (T ) gravity models from the different variants of the CG model, namely the original CG (OCG), the Generalized CG (GCG) and the modified generalized CG (MGCG) models. The idea is that, in that way, we will be able to mimic the same expansion behavior (as the CG) with a theory of gravity, in this case f (T ), but without having to worry about the exoticity of the fluid with the above-mentioned problems. To do this, we assume the usual torsion fluid description with effective pressure p T and energy density ρ T related via the CG equation of state. After obtaining the reconstructed f (T ) gravity models, we study the expansion dynamics of the Universe by calculating the energy density, pressure and equation of state parameter for the torsion fluid in the non-interacting fluid systems such as vacuum-torsion, radiation-torsion, dust-torsion and radiation-dust-torsion systems for the three variants of the CG model.
The layout of the manuscript is as follows: in the following section, we review the cosmology of f (T ) gravity together with the simplest CG cosmological models and relate the two through some sort of a master equation, which we then solve in Sec. (3) for different cases. In Secs. (4) and (5), we go a step or two further and reconstruct different f (T ) gravity models from the GCG and MGCG models, respectively. We then devote Sec. (6) for discussions of our results and the conclusions.

f (T ) gravity and CG cosmology
We consider a covariant version of the field equations of f (T ) gravity with a clear analogy to Einstein's field equations as [30,31,32] where g ab is the metric tensor, S d ab denotes a super-potential term K c ab is the contortion tensor and T d db is the torsion tensor, f ≡ df /dT , f ≡ d 2 f /dT 2 , T ab denotes the usual EMT of the matter fluid expressed as and the coupling constant κ 2 ≡ c 4 8πG . In the limiting case f (T ) = T (cf. [30,31]) the field equations reduce to those of GR. 1 . From this generalizd form of the gravitational field equation of motion in f (T ) gravity we obtain the modified Friedmann and Raychaudhuri equations in f (T ) for FLRW space-times as follows 2 : where H(t) is the Hubble (expansion) parameter defined from the scale factor a(t) and the cosmic time t as H ≡ȧ a . One can directly find the torsion contribution thermodynamical quantities such as energy density ρ T and pressure p T of the torsion can be extracted from the above given field equation as follows: where ρ m is the energy density, p m is the pressure of the matter fluid, torsion scalar T = −6H 2 . Here we assume a slowly changing torsion fluid i.e.,Ṫ ≈ 0 such that the pressure of the torsion fluid is given by 1 Here we assume geometric units where κ = 1 = 8πG = c, where c is the speed of light and we use the (+, −, −, −) metric convention for this manuscript. 2 Overhead dots denote differentiation w.r.t. cosmic time.
Since the effective energy density of the fluid ρ ef f is the sum of the two non-interacting fluid components (matter and torsion), we have the effective (total) energy density and the effective pressure of the total fluid is Then, the corresponding conservation equation of the fluid iṡ Then, from the modified Friedmann equation (4), the effective energy density of the total fluids as and the corresponding effective pressure of the fluids In the special case f (T ) = T , Eqs. (1) -(13) all reduce to the GR limit, which in turn describes similar cosmic dynamics as GR. A cosmological fluid of particular interest in recent years is the so-called CG. It is a fluid model proposed as a candidate for a unified description of dark matter and dark energy [33,28,4], and its cosmological scenarios are widely presented in the literature. In this work, we consider the torsion fluid as an exotic fluid with equations of state similar to the ones for three variants of the CG model and we reconstruct f (T ) gravity toy models corresponding to the original, generalized and modified generalized models. The characteristic equation of state for the CG model is given by [4] where 0 < α ≤ 1, A is a positive constant and the energy density ρ > 0. Here we consider the possibility of the torsion fluid mimicking the CG with the characteristic equation of state given by Now we substitute the energy density ρ T and pressure p T of the torsion fluid from Eqs. (6) and (8) into Eq. (15) to obtain the master equation from which our solution process starts. This is a general expression of the equation of state to reconstruct different f (T ) gravity models from the given two paradigmatic models of CG (original and generalized) and in the following two sections we reconstruct different f (T ) gravity models based on Eq. (16). Let us discuss the scheme that we follow in the original and generalized CG models. We first reconstruct the f (T ) gravity model for five different cases: vacuum, radiation-torsion, dust-torsion, stiff matter-torsion and radiation-dust-torsion systems for both CG models. Then, secondly, we substitute the reconstructed f (T ) gravity models into Eq. (6) and Eq. (8) and compute the corresponding energy density, pressure and the equation of state parameter of the torsion fluid for each case in these CG models. The energy density of each fluid apart from that of CG is expressed as where ρ d , ρ r and ρ s are the energy density for dust, radiation and stiff matter fluids respectively and ρ i0 denotes the present-day value of the energy density of fluid type i = {d, r, s} 3 . Consequently, we compute the effective Here we also present the other dimensionless parameter χ to represent the fraction of the effective energy density of the radiation-torsion fluid: the explicit values of which are given by All the above thermodynamic quantities namely ρ T , p T , w T and ρ ef f are depend on the energy density of the radiation fluid ρ r and ρ r is proportional to the cosmological scale factor. Consequently, the energy density parameters such as ξ and χ also depend the scale factor of the Universe. One can obtain the growth factor of the energy densities of the fluids ξ and χ; here we present numerical results in Fig. (2).

Dust-torsion system
After the era of radiation, the Universe was predominantly filled by pressureless matter (dust) fluid. Thus, in a dust-dominated Universe, p d ≈ 0, and the equation of state parameter is w d ≈ 0. In our current model, we consider the non-interacting dust-torsion system and briefly discuss its implications for cosmic expansion. The general expression of Eq. (16) in the dust-torsion system becomes and this equation admits four different f (T ) gravity models as solutions: and the corresponding isotropic pressures 8 The reconstructed EoS parameters of the torsion fluid during the dust-dominated phase w T = p T /ρ T are then given by: We use Eq. (17)   Moreover, the fractional energy densities of torsion in the dust-torsion system give as follows: and for the effective fluid, these become: In Fig. (4), we present the growth of the fractional energy densities for torsion and effective fluids versus the scale factor. Figure 4: In all panels, we present ξ 1,2,3,4 associated with χ 1,2,3,4 versus a for dust-torsion system for OCG. We use c = 1.9 and A = 0.5 for these numerical plots.

Stiff matter-torsion system
Here, we consider a Universe composed of stiff matter ( p s = ρ s with w = 1) and torsion. The general expression of Eq. (16) for such a system is given as From this expression we reconstruct four different f (T ) gravity models as follows: Here also we see that setting ρ s = 0 reduces two of the f (T ) gravity models, Eqs. (78) and (79), to the vacuum cases, Eqs. (22) and Eq. (23) we studied earlier, whereas the other two solutions, Eq. (80) and (81), both vanish. By substituting the above f (T ) solutions into Eq. (6), we obtain the corresponding energy densities and isotropic pressures of the torsion fluid in the era of stiff matter as follows: We also compute the equation of state parameter of the torsion fluid in the stiff fluid-torsion system as follows:

Radiation-dust-torsion system
Here, we consider a non-interacting multi-fluid system, namely radiation and dust with torsion, and those fluids acts as an exotic fluid for cosmic expansion. In this context, all thermodynamical quantities are the mixture of the individual species. For instance, the pressure of the effective fluid is given as p ef f = p m + p T . The general form of Eq. (16) for radiation-dust-torsion system is given as where the energy density of matter ρ = ρ r + ρ d . We reconstruct four different f (T ) gravity models through this system as follows: If ρ d = ρ r = 0, the above to f (T ) gravity models Eqs. (95) and (96) reduce to the vacuum case Eq. (22) and (23) respectively. The other two solutions Eq. (97) and (98) go to zero in the vacuum limiting case. We substitute f 1 (T ), f 2 (T ), f 3 (T ) and f 4 (T ) into Eq. (6). Then, the energy density of the torsion fluid in radiation-dust-torsion system: where And we also substitute f 1 (T ), f 2 (T ), f 3 (T ) and f 4 (T ) into Eq. (8), and reconstruct the correspond pressure of the fluid given as follows: And the equation of state parameter of the torsion fluid express as follows: We present the numerical plots of the evolution of effective equation of state parameter in Fig. (5) for noninteracting fluids (radiation-dust-torsion systems). Here we reconstruct the fraction of energy densities for torsion fluid in radiation-dust-torsion system as follows: , (115 , (117 We present the behavior of the fractional energy densities in Fig. (6).

Reconstructing f (T ) gravity from the GCG model
The frame-work of the GCG in the modified theory of gravity was first proposed by Rastall [37]. Here, we consider the general model of CG model to reconstruct f (T ) gravity model and the corresponding thermodynamical quantities of the torsion fluid. Similar to our previous discussions for the OCG model, we consider five cases, namely vacuum, radiation-torsion, dust-torsion, stiff matter-torsion and radiation-dust-torsion systems.

Vacuum case
Here we assume that the vacuum Universe and the energy density and pressure of the matter fluid are negligible, i,e., p m = ρ m = 0. The general expression of Eq. (16) in vacuum system is given as and we can reconstruct the f (T ) gravity model from this expression and it is given by In most cases, we have α as a positive constant, with a value between 0 to 1. However, in the literature [38,39,40,41,42,43], the value of α can be a free parameter and larger than −1. If, in our case, α = 1/2, the reconstructed f (T ) gravity model in Eq. (120) becomes imaginary, f (T ) = cT − 2ic √ A; on the other hand, if α = −1/2, then f (T ) = cT − 2c √ A and this solution is exactly the same as the selected solution in the OCG model, Eq. (22). This is our motivation to account for the parameter α as being either positive or negative. Based on the claim of [38,40,41,42,43] and the above motivation, α can be a negative number α ≥ −1. Then, we choose the parameter α = −1/2, the reconstructed f (T ) function in GCG in Eq. (120) is reduced to OCG in Eq. (22). Based on this suggestion, we have set the value of α = −1/2 in this paper for further investigation of different f (T ) gravity models, to manifest the cosmological implication of the corresponding thermodynamical quantities. Therefore, this f (T ) which is presented in Eq. (120) is the general of the Eq. (22). By substituting Eq. (120) into Eq. (6) and Eq. (8), it could be reconstruct the energy density, pressure and the corresponding equation of state for torsion fluid in the vacuum system.

Radiation-torsion system
Here we consider the radiation-torsion system in GCG model, and radiation with torsion component is considered as an exotic fluid to lead the cosmic expansion. The energy density and pressure of the fluid has been involving from the early to late Universe as a constant and which are presented in Eqs. (6) and (8). The general expression of Eq. (16) for radiation-torsion system is given by Then, by applying the same reasoning as vacuum case, we set the value of α = −1/2, to reconstruct different f (T ) gravity models in the below: By substituting f 1 (T ) and f 2 (T ) into Eq. (6), we obtain the energy density of the torsion fluid ρ 1 T and ρ 2 T . These the energy density of the torsion fluid express as follows: In the limiting case of ρ r = 0, the function f 1 (T ) in Eq. (122) and the energy density of the torsion fluid ρ T 1 in Eq. (124) are reduced to vacuum case while, the function f 2 (T ) in Eq. (123) and the energy density of the torsion fluid ρ T 2 in Eq. (125) go to zero. Here, we also reconstruct the corresponding pressure of the torsion fluid during radiation-torsion system by substituting f 1 (T ) and f 2 (T ) into Eq. (8) and we have and the equation of state parameter w T of the torsion fluid is given as We use the definition of Eq. (17) and the numerical plots of the equation of state parameter for the radiationtorsion system is presented in Fig. (7) for GCG. Figure 7: w ef f (1,2) versus a for radiation-torsion system for GCG. We use c = 2.5 and A = 1 for all numerical plotting.
In the similar mathematical manipulation as OCG in GCG we also obtain the growth factor parameters by taking the ratio of the energy density of torsion fluid ξ and effective fluid χ. These the growth factor parameters are given as follows: The Eqs. (130) and (132)

Dust-torsion system
In this section, we also consider that dust-torsion system in GCG model and dust and torsion component are considering as an exotic fluids to reconstruct f (T ) gravity model. Then, Eq. (16) is given as By setting the α = −1/2, the solutions of Eq. (134) are given as follows: Based on the above reconstructed functions f 1 (T ) and f 2 (T ), we also reconstruct the energy density of the torsion fluid as follow. and By substituting f 1 (T ) and f 2 (T ) in Eq. (8) we reconstruct the pressure of the torsion fluid as follows: From the reconstructed energy density and pressure of the fluid we have to obtain the following equation of state parameters of torsion fluid in dust dominated case We use the definition of Eq. (17) and the numerical plots of the equation of state parameter for the dust-torsion system is presented in Fig. (9) for GCG. Figure 9: w ef f (1,2) versus a for dust-torsion system for GCG. We use c = 2 and A = 0.98 for all numerical plotting.
By applying the same reason as radiation dominated case, the growth factor parameters of the fluid are given as follows: If ρ d = 0, all thermodynamical quantities are reduced to the vacuum case. The Eqs. (143) and (145) are presented on the left side of Fig. (10), and Eqs. (144) and (146) are presented on the right side of Fig. (8).

Stiff matter-torsion system
Here, we consider the stiff matter component is considered behind the exotic fluid and the energy density ρ m = ρ s has been involving from the early to late Universe as a constant and which are presented in Eqs. (6) - (8). Then, Eq. (16) is given as which gives It can be reconstructed different f (T ) gravity models by setting α = − 1 2 and we have and also by substituting Eqs. (148) and (149) into Eq. (6) we reconstruct the corresponding energy density of torsion. Then we have By substituting Eqs. (148) and (149) into Eq. (8) we can reconstruct the corresponding pressure of torsion. Then we have and we also obtain the equation of state parameters as follows: Figure 11: w ef f (1,2) versus a for radiation-dust-torsion system for GCG. We use c = 1.3 and A = 1 for all

Reconstructing f (T ) gravity from the MGCG model
In this section, we consider the generalization of the GCG [26,44,45,46] in the form: In analogy with previous sections, the pressure of the torsion fluid in MGCG is given by We substitute p T and ρ T from Eqs. (8) and (6) into Eq. (170), obtaining Then, this equation is the general expression of Eq. (16); it reduces to Eq. (16) by eliminating the parameter β. We now reconstruct different f (T ) gravity models in the following cases.

Vacuum case
In vacuum case, the energy density and pressure of the matter fluid are neglected, p m = ρ m = 0, and Eq. (171) yields: 1 the solutions of which are For β = 0, the above function f 2 (T ) is equal f 1 (T ) and it also coincide with the selected solution of vacuum case in Eq. (22). In order to obtain the energy density of the torsion fluid for this model we substitute Eqs.
(173) and (173) into Eq. (6) and it is given by and the corresponding pressures can be found by substitute Eqs. (173) and (173) into Eq. (8) as: It can be shown that the EoS parameter of the fluid asymptotically approaches that of DE.

Dust-torsion system
As we indicated in the earlier in dust dominated Universe, the energy density of the pressure less fluid acts as a constant behind the exotic fluid from the early to late Universe [43]. In this limit, Eq. (171) will look like For α = − 1 2 , the solutions as and the energy density of the fluid Fig. (15) shows the effective equation of state parameter for dust-torsion system for MGCG model. The ratio Figure 15: w ef f (1,2) versus a for dust-torsion system for MGCG for β = 4, c = 1.2 and A = 1.
of the energy density in torsion fluid expressed as follows: where The ratio of the effective energy density of the fluid expressed as follows where where

Stiff matter -torsion system
Here also we consider the stiff fluid as an exotic fluid in MGCG and the general express of Eq. (171) and we obtain with the solutions of We reconstruct the energy density by substituting f 1 (T ) and f 2 (T ) into Eq. (6) and we obtain Here also we reconstruct the pressure of the torsion fluid by substituting f 1 (T ) and f 2 (T ) into Eq. (8) and we obtain

Radiation-dust-torsion system
Here is the general expression of radiation-dust-torsion system as an exotic fluid in MGCG model with solutions given as From this constructed f 1 (T ) and f 2 (T ) gravity model we can reconstruct the energy density of the torsion fluid in the non-interacting fluid. By substituting f 1 (T ) and f 2 (T ) into Eq. (6), we obtain ρ 1 T = Figure 17: w ef f (1,2) versus a for radiation-dust-torsion system for MGCG for β = 4, c = 1.2 and A = 1. The implication for this behavior is that once the stiff fluid is small, the two solutions grow in opposite directions based on the presence of a negative sign on the last solution.
The consideration of the generalized CG brings the f (T ) Lagrangian with a couple of features. For instance, the vacuum universe has the f (T ) Lagrangian with a dependence on α, a parameter that constrains the behavior of the cosmological constant if one is interested in the reduction of this Lagrangian to that of GR. Both radiationtorsion and dust-torsion systems possess f (T ) Lagrangians spanned by two solutions and there is no presence of the quadratic terms in those solutions. However, using these Lagrangians, the obtained ρ T and p T are independent of the torsion T . The stiff-torsion fluid also has two f (T ) solutions with features that are similar to those of the radiation and dust systems. The consideration of the system that has all components (radiation, dust, and torsion), has f (T ) solutions that have a dependence on the total pressure of standard matter.
For the third type of the CG, the modified generalized one, the vacuum case is considered where the f (T ) solution comes with no presence of quadratic terms in T . Both radiation and dust systems have two f (T ) two solutions, and the same story is manifested for both stiff and the system that has radiation, dust and torsion as components. However, when the consideration of both generalized Chaplygin and modified Chaplygin models is done, we followed the suggestion presented in the literature that the value of α has to be negative in favour of the observations. We consequently selected α = − 1 2 for our purpose. The choice of other values of α in the suggested range is needed to constrain the f (T ) Lagrangians.
We obtained the expressions for the equation of state parameter for the respective considered f (T ) solutions and the corresponding effective equation of state parameters for all cases in each CG model and plotted the results. From the plots, we observed that our reconstructed f (T ) gravity models replacing exotic Chaplygin-gas (CG) fluid models can, in principle, explain the currently observed cosmic evolutionary dynamics. For instance, in Fig. (1) for radiation-torsion for OCG, in Fig. (5) for radiation-dust-torsion system for OCG, in Fig. (7) a radiation-torsion system for GCG models (in the limits of 1 ≥ a > 0.2), in Fig. (11) for a radiation-dust-torsion system for GCG, in Fig. (13) for a radiation-torsion system for MGCG, in Fig. (15) for a dust-torsion system for GCG and in (17) for a radiation-dust-torsion system for MGCG models, the evolution equation of state parameters is less than negative one-third ( w ef f (1,2) < − 1 3 ) and asymptotically behaves like that of the equation of state parameter of the cosmological constant fluid w = −1. This indicates that torsion fluid replacing the Chaplygin fluid can explain the accelerating expansion of the Universe without cosmological constant scenario. On the other hand, w ef f (1,2,3,4) in Fig. (3) for a dust-torsion system for OCG, in Fig. (9) for a dust-torsion system for GCG, w ef f (3,4) in Fig. (1) for a radiation-torsion system for OCG, in Fig. (5) for a radiationdust-torsion system for OCG, in Fig. (9) for dust-torsion system for GCG, the effective evolution equation of state parameters found in the intervals of − 1 3 < w ef f show that the Universe undergoes a decelerated cosmic expansion such as is the case in the early expansion phases of a Chaplygin-gas-dominated universe.
A key assumption made in this work in order to obtain the exact solutions presented is that the torsion scalar T changes slowly over time. The general case of a time-dependent torsion requires more rigorous com-putational solving techniques, with carefully selected initial conditions. This task and the constraining of the free parameters appearing in our models with existing and future observational data is left for a forthcoming investigation.