Reconstructions of $f(T)$ Gravity from Entropy Corrected Holographic and New Agegraphic Dark Energy Models in Power-law and Logarithmic Versions

Here, we peruse cosmological usage of the most promising candidates of dark energy in the framework of $f(T)$ gravity theory. We reconstruct the different $f(T)$ modifed gravity models in the spatially flat FRW universe according to entropy-corrected versions of the holographic and new agegraphic dark energy models in power-law and logarithmic corrections, which describe accelerated expansion history of the universe. We conclude that the equation of state parameter of the entropy-corrected models can transit from quintessence state to phantom regime as indicated by recent observations or can lie entirely in the phantom region. Also, using these models, we investigate the different erase of the stability with the help of the squared speed of sound.


Introduction
The type Ia Supernovae and Cosmic Microwave Background (CMB) [1,2] observations point out that our universe is precisely accelerating which is caused by some unknown fluid having positive energy density and negative pressure, called as "Dark Energy" (DE). Observations indicate that dark energy occupies about 70% of the total energy of the universe, whereas the contribution of dark matter is 26% and rest 4% is the baryonic matter. For related review works see the references [3,4]. Although a long-time argument has been made on this interesting issue of modern cosmology, we still have a few knowledge about DE. The cosmological constant Λ is the most appealing and simplest candidate for DE which obeys the equation of state parameter w = −1. However, the cosmological constant suffers from two serious theoretical problems, i.e., the cosmological constant problem and the coincidence problem. In this respect, different dynamical DE models and different modified theories of gravity have been developed. Moreover, the reconstruction phenomenon of different DE models [5,6,7,8] gains great attention to discuss the accelerated expansion of the universe.
In recent years, an interest has been proposed to study the dark energy in the new form i.e., Holographic Dark Energy (HDE) model [9,10,11] which arises from the holographic principle [12] stating that the number of degrees of freedom of a physical model must be finite [13] and an infrared cut-off should constrain it [14]. In quantum field theory [14], for developing a black hole , the UV cut-off Λ should relate with the IR cut off L due to limit set. In the reference [15] by Li, he debated a relation L 3 ρ Λ ≤ LM 2 P , where ρ Λ is the quantum zero point energy density and M P = 1 √ 8πG is the reduced Plank Mass i.e., the mass of a black hole of the size L should not be exceeded by the total energy in a region of same size. The HDE models have been discussed in [16,17,18,19,20]. The black hole entropy S BH plays an important role in the simplification of HDE, given as usually, S BH = A 4G , where A ∼ L 2 is the area of horizon.
The power-law corrections arise in dealing with the entanglement of quantum fields moving into and out of the horizon [21,22,23] for which the entropy-area relation for power-law correction can be given as where Here, r c is the crossover scale and ǫ is the dimensionless constant. Motivated by this corrected entropy-area relation (1) in the setup of LQG (loop quantum gravity), Wei [24] suggested the energy density of the ECHDE in Power-law Correction.
Also the of entropy-area relation for a logarithmic correction can be improved to [25,26,27,28] where α and β are dimensionless constants of order unity. Recently, inspired by the corrected entropy-area relation (2) in the setup of LQG, Wei [24] propounded the energy density of the entropycorrected HDE (ECHDE)in Logarithmic Correction.
From quantum mechanics along with the gravitational purpose in General Relativity, the another type of dark energy is the agegraphic DE (ADE) model. The original agegraphic DE model was brought by Cai [29] to study the accelerating expansion of the universe where the age (T ) of the universe is present in the expression of energy density, given by The numerical factor 3c 2 is used to recover some uncertainties. Subsequently, Wei and Cai [30] suggested a new kind of ADE model by removing the age of the universe and placing with the conformal time (η), called as new agegraphic DE (NADE) model. Recently, Wei [24] initiated the energy density of the entropy-corrected NADE (ECNADE) in power-law and logarithmic corrections like the entropycorrected HDE (ECHDE) in power law and logarithmic corrections model and details of these were discussed in [31,32,33,34,35].
There is an another discussion for the cosmic acceleration of the Universe (predict from observational data) , so-called "modified gravity" where we do not require any additional components like DE (for review see [36]) for acceleration of the Universe. Various kinds of modified theories have been proposed such as f (R) [38], f (G) [39,40], Horava-Lifshitz [41] and Gauss-Bonnet [42] theories of gravity. Recently, [43,44] formulate a new kind of theory of gravity known as f (T ) gravity in a space-time possessing absolute parallelism. f (T ) gravity have been recently studied in [45,46]. In the f (T ) theory of gravity, the teleparallel Lagrangian density gave a description of the torsion scalar T , evoked to be a function of T , i.e., f (T ), for the late time cosmic acceleration [47]. In a recent work, Jamil et al [48,49] investigated the interacting DE model and state-finder diagnostic in f (T ) cosmology. entropy-corrected (m, n) type pilgrim dark energy. Motivated by these works, with the help of the modified f (T ) gravity and considering the entropy-corrected versions of the HDE and NADE scenarios, it is interesting to investigate how the f (T )-gravity can describe ECHDE and ECNADE densities in power-law and logarithmic versions as effective theories of DE models. This paper is arranged as follows. In section 2, we give a brief idea of the theory of f (T ) gravity and corresponding solutions for FRW background. In sections 3 and 4, we reconstruct the different f (T ) gravity models i.e., find unknown function f (T ) corresponding to the ECHDE and ECNADE models in power-law and logarithmic versions, respectively and analyze the EoS parameter for the corresponding models. Karami et al [58] also investigated the modified teleparallel gravity models as an alternative for holographic and new agegraphic dark energy models. In section 5, we provide the analysis and comparison of the reconstructed models. Section 6 is invoked to our conclusions. 2 The brief idea of f (T ) gravity and ECHDE in power-law and logarithmic correction: Teleparallel gravity is correlated with a gauge theory for the translation group. For unusual character of this translations, any gauge theory with these translations is different from the usual gauge theory in many ways, mostly in the background of tetrad field whereas this field is used to define a linear Weitzenbock connection, presenting torsion without no curvature. For the details of this gravity theory see the review [59]. We consider here to generalize the teleparallel Lagrangian T to a function f (T ) = T + g(T ), which is same as the generalization of the Ricci scalar in Einstein-Hilbert action to the modified f (R) gravity. We can write the action of f (T ) gravity, coupled with matter L m by [60,61,62,63,64] where e = det(e i µ ) = √ −g. Now we will take the units 8πG = c = 1. Here, the teleparallel Lagrangian T , known as the torsion scalar, is defined as follows: where and K µν ρ is the contorsion tensor Making a variation of the action with respect to vierbein e i µ , we get the field equations as where g T and g T T are the first and second derivatives of g with respect to T . Here Υ ρν is the stress tensor. Now we assume the usual spatially flat metric of the Friedmann-Robertson-Walker (FRW) universe giving the line element written as where a(t) is the scalar factor , a function of the cosmic time t. Moreover, we consider the background to be a perfect fluid. Using the FRW metric and the perfect fluid matter in the Lagrangian (5) and the field equation (9), we obtain where ρ and p are the energy density and pressure of ordinary matter content of the universe, respectively. The Hubble parameter (H) is defined as H =ȧ a , where the "dot" denotes the derivative with respect to the cosmic time. Equation (11) shows that T < 0.
The equation of state (EoS) parameter due to the torsion contribution is defined as which shows that for the phantom, w Λ < −1, and quintessence, w Λ > −1, dominated universe.
We define the redshift z as where a 0 (t)=1 for the present epoch.
For a given a(t), by the help of equations (12) and (13) one can reconstruct the f (T ) gravity according to any DE model given by the EoS p Λ = p Λ (ρ Λ ) i.e., ρ Λ = ρ Λ (a). There are two classes of scale factors which usually people consider them for describing the accelerating universe in f (R), f (T ), etc.
Class I: The first class of scale factor is given by [65], where a 0 , n are constants and t s defines the future singularity time. Hence, Class II: For the second class of scale factor defined as [65], One can obtain, For the both cases we get 3 f (T ) reconstruction from ECHDE in power-law and logarithmic corrections models 3.1 ECHDE in power-law correction [24] proposed the energy density of the ECHDE in power-law correction using the relation (1) as [24] where λ is a constant related with ǫ and K ǫ , δ is a constant.In the special case λ = 0, the above equation reduces to the well-known HDE density. Also R h is the future event horizon defined as For the first class (class I) of scale factor (15) and using equation (16), the future event horizon Replacing equation (24) into (22) one can get Substituting equation (25) in the differential equation (12) i.e., ρ = ρ Λ , gives the following solution where c is the integration constant to be determined from the necessary boundary condition. In figure 1, we understand that f (T ) 0 as T → 0 for the solution obtained from equation (26). We also observe that f (T ) first decreases and then increases as T increases keeping in the mind that f (T ) takes always negative value for all values of negative T . Replacing equation (26) into (13) and using (25) we obtain the EoS parameter of the ECHDE f (T ) gravity in power-law correction model as w Λ = p Λ ρ Λ graphically. In figures 3 and 5, we see that the EoS parameter can justify the transition from quintessence state w Λ > −1, to the phantom regime, w Λ < −1, i.e., it crosses the phantom divide line w Λ = −1 if we draw the graph of EoS parameter with T and z using the equation (21) respectively. So in this case, f (T ) gravity generates phantom crossing.
For the second class (class II) of scale factor (18) and using equation (19), the future event horizon Replacing equation (27) into (22) one can get Substituting equation (28) in the differential equation (12) i.e., ρ = ρ Λ , gives the following solution where c is the integration constant to be determined from the necessary boundary condition. In figure 2, we understand that f (T ) 0 as T → 0 for the solution obtained from equation (29). We also observe that f (T ) first increases and then decreases as T increases keeping in the mind that f (T ) takes always positive value for all values of negative T .It may be stated that the solutions obtained in equations (26) and (29) are not so-realistic models. Replacing equation (29) into (13) and using (28) we obtain the EoS parameter of the ECHDE f (T ) gravity in power-law correction model as w Λ = p Λ ρ Λ graphically. In figures 4 and 6, we see that the EoS parameter wholly lies in the phantom region i.e., w Λ < −1 always if we draw the graph of EoS parameter with T and z using the equation (21) respectively. So in this case, f (T ) gravity does not generate phantom crossing. [24] described the energy density of the ECHDE in logarithmic version using the corrected entropy-area relation (2) as [24]

ECHDE in logarithmic correction
where α and β are dimensionless constants of order unity and δ is a constant.In the special case α = β = 0, the above equation becomes the well-known HDE density. Since for only R h being very small, the last two terms in equation (30) can be comparable to the first term , the corrections is sensible only at the early stage of the universe. When the universe becomes large, ECHDE converts to the ordinary HDE [24].
For the first class (class I) of scale factor (15) and using equation (16), the future event horizon R h (24) into (30) one can get Substituting equation (31) in the differential equation (12) i.e., ρ = ρ Λ , gives the following solution where c is the integration constant to be determined from the necessary boundary condition. In figure 7, we understand that f (T ) 0 as T → 0 for the solution obtained from equation (32). We also observe that f (T ) first increases and then decreases as T increases keeping in the mind that f (T ) takes always negative value for all values of negative T . Replacing equation (32) into (13) and using (31) we obtain the EoS parameter of the ECHDE f (T ) gravity model in logarithmic version as w Λ = p Λ ρ Λ graphically. In figures 9 and 11, we see that the EoS parameter can justify the transition from phantom state w Λ < −1, to the quintessence regime, w Λ > −1, i.e., it crosses the phantom divide line w Λ = −1 if we draw the graph of EoS parameter with T and z using the equation (21) respectively. So in this case, f (T ) gravity generates phantom crossing.
For the second class (class II) of scale factor (18) and using equation (19), the future event horizon R h (27) into (30) one can get Substituting equation (33) in the differential equation (12) i.e., ρ = ρ Λ , gives the following solution where c is the integration constant to be determined from the necessary boundary condition. In figure 8, we understand that f (T ) 0 as T → 0 for the solution obtained from equation (34).   We also observe that f (T ) decreases from some positive value to negative value as T increases from negative value to zero. It may be stated that the solutions obtained in equations (32) and (34) are not so-realistic models. Replacing equation (34) into (13) and using (33) we obtain the EoS parameter of ECHDE f (T ) gravity model in logarithmic version as w Λ = p Λ ρ Λ graphically. In figures 10 and 12, we see that the EoS parameter can justify the transition from quintessence state w Λ > −1, to the phantom regime, w Λ < −1, i.e., it crosses the phantom divide line w Λ = −1 if we draw the graph of EoS parameter with T and z using the equation (21) respectively i.e., it crosses the line w Λ = −1.
4 f (T ) reconstruction from ECNADE model in power-law and logarithmic corrections models 4.1 ECNADE in power-law correction [24] gives the energy density of the ECNADE in power-law correction with the help of quantum corrections to the relation (1) in the setup of LQG given as which are very similar to that of ECHDE in power-law correction density (22) and R h is replaced with the conformal time η which is given by Here ξ and ζ are dimensionless constants of order unity.
For the first class (class I) of scale factor (15), the conformal time η by the help of equation (36) Substituting equation (37) into (35) one can obtain ρ Λ = 3δ 2 a 2 0 (n + 1) 2 (−T ) n+1 (6n 2 ) n+1 − λ{ a 0 (n + 1) Solving the differential equation (12) for the energy density (38) reduces to i.e., ρ = ρ Λ , gives the following solution where c is the integration constant to be determined from the necessary boundary condition. In figure 13, we understand that f (T ) 0 as T → 0 for the solution obtained from equation (39). The function f (T ) decreases as T increases to zero. Replacing equation (39) into (13) and using (38) we obtain the EoS parameter of the ECNADE f (T ) gravity model in power-law version as w Λ = p Λ ρ Λ graphically. In figures 15 and 17, we see that the EoS parameter can justify the transition from phantom state w Λ < −1, to the quintessence regime, w Λ > −1, i.e., it crosses the phantom divide line w Λ = −1 if we draw the graph of EoS parameter with T and z using the equation (21) Solving the differential equation (12) for the energy density (41) reduces to i.e., ρ = ρ Λ , gives the following solution where c is the integration constant to be determined from the necessary boundary condition. In figure 14, we understand that f (T ) 0 as T → 0 for the solution obtained from equation (42). The function f (T ) decreases but keeps negative value as T increases to zero.It may be stated that the solutions obtained in equation (39) and (42) both are not realistic models. Replacing equation (42) into (13) and using (41) we obtain the EoS parameter of the ECNADE f (T ) gravity model in power-law version as w Λ = p Λ ρ Λ graphically. In figures 16 and 18, we see that the EoS parameter entirely lies in the phantom region, w Λ < −1 if we draw the graph of EoS parameter with T and z using the equation (21) respectively. So in this case, f (T ) gravity generates phantom crossing i.e., it does not cross the line w Λ = −1. [24] gives the energy density of the ECNADE with the help of quantum corrections to the entropy-area relation (2) in the setup of LQG given as

ECNADE in logarithmic correction
which are very similar to that of ECHDE density in logarithmic version (30) and R h is replaced with the conformal time η.
For the first class (class I) of scale factor (15), using the conformal time η (40), equation (43) gives 44) Solving the differential equation (12) for the energy density (44) reduces to i.e., ρ = ρ Λ , gives the following solution where c is the integration constant to be determined from the necessary boundary condition. In figure 19, we understand that f (T ) → 0 as T → 0 for the solution obtained from equation (45). The function f (T ) increases but keeps negative value as T increases to zero. Replacing equation (45) into (13) and using (44) we obtain the EoS parameter of the ECNADE f (T ) gravity model in logarithmic version as w Λ = p Λ ρ Λ graphically. In figures 21 and 23, we see that the EoS parameter can justify the transition from quintessence state w Λ > −1, to the phantom regime, w Λ < −1, i.e., it crosses the phantom divide line w Λ = −1 if we draw the graph of EoS parameter with T and z using the equation (21) respectively i.e., it crosses the line w Λ = −1.     (43) gives ρ Λ = 3α 2 a 2 0 (1 − n) 2 (−T ) 1−n 6 1−n n 2(1−n) + ξa 4 0 (1 − n) 4 T 2−2n 6 2(1−n) n 4(1−n) ln( Solving the differential equation (12) for the energy density (46) reduces to where c is the integration constant to be determined from the necessary boundary condition. In figure 20, we understand that f (T ) → 0 as T → 0 for the solution obtained from equation (47). The function f (T ) decreases from some positive value to some negative value as T increases upto certain negative value and after that f (T ) increases keeping in negative sign. It may be stated that the solutions obtained in equation (45) and (47) are both realistic model. Replacing equation (47) into (13) and using (46) we obtain the EoS parameter of the ECNADE f (T ) gravity model in logarithmic version as w Λ = p Λ ρ Λ graphically. In figures 22 and 24, we see that the EoS parameter can justify the transition from quintessence state w Λ > −1, to the phantom regime, w Λ < −1, i.e., it crosses the phantom divide line w Λ = −1 if we draw the graph of EoS parameter with T and z using the equation (21) respectively i.e., it crosses the line w Λ = −1.

Analysis and comparison of the reconstructed models
We now analyze an important quantity to verify the stability of ECHDE f (T ) in power-law and logarithmic corrections model and ECNADE f (T ) in power-law and logarithmic corrections model, named as the squared speed of sound v 2 s : v 2 s = dp dρ = dp dT dρ dT The sign of v 2 s is very important for checking the stability of a background evolution of the universe. In general relativity a negative sign implies a classical instability of a given perturbation [66,67]. Myung [67] has observed the always negative sign of v 2 s for HDE for the future event horizon as IR cutoff, while for Chaplygin gas and tachyon, there is non-negativity. Kim et al [66] found always negative squared speed of sound for agegraphic DE leading to the instability of the perfect fluid for the model. Also, [68] found the ghost QCD DE model as unstable model. Recently, Sharif and Jawad [69] have shown negative v 2 s for the interacting new HDE.

Investigation of stability of ECHDE in power-law and logarithmic corrections:
For ECHDE f (T ) model in power-law version there are two cases. For the first class (class I scale factor) we see from figure 25 that v 2 s > 0 for T −2 and v 2 s < 0 for T −2 and from 27 that

Concluding Remarks
In this work, we have assumed the f (T ) modified gravity theory in the background of flat FRW universe. We found the modified Friedmann equations and then from the equations, we found the effective energy density and pressure for f (T ) modified gravity theory. Modified gravity gives a natural unification of the early-time inflation and late-time acceleration. We have assumed two types of power law forms of scale factor, the first class (class I) has the future singularity and the second class (class II) has the initial singularity. In the framework of f (T ) modified gravity model, four types of dark energy have been considered, they are (i) entropy-corrected holographic dark energy (ECHDE) in power-law version, (ii) entropy-corrected holographic dark energy (ECHDE) in logarithmic version,(iii) entropy-corrected new agegraphic dark energy (ECNADE) in power-law version and (iv) entropy-corrected new agegraphic dark energy (ECNADE) in logarithmic version, where, R h is assumed to be the future event horizon and η is assumed to be conformal time. Using the two classes of scale factors, the unknown function f (T ) has been found in term of T for ECHDE and ECNADE models in power-law and logarithmic versions. For the cases of ECHDE in power-law version (class II) and ECNADE in power-law version (class II) the equation of state parameter w Λ has been shown in figures 4, 6, 16, 18 and from the figures we have seen that these models lie entirely in the phantom region. It should be mentioned that Karami et al [58] have investigated the f (T ) reconstructions for HDE, NADE models and logarithmic versions of ECHDE, ECNADE models only and for these models we got the similar expressions of f (T ) but we have details studied the results graphically. To examine the stability test for all the reconstructing models, we have investigated the signs of the square of the velocity of sound. For ECHDE model in power-law version , we have concluded from figures 25 and 27, that the corresponding model is a classical stable for T ≤ −2, z ≥ 0.1 and classically unstable for T ≥ −2, z ≤ 0.1 for the first class and from figures 26 and 28 the corresponding model is a classically unstable for second class of a given perturbation in general relativity. For ECHDE model in logarithmic version , we have concluded from figures 29, 30, 31, 32 that the corresponding models are unstable for class I and class II both. On the other hand, for ECNADE model in power-law version (class I), we have seen from figures 33, 35 that         the corresponding model is stable and for ECNADE model in power-law version (class II) we have seen from figures 34, 36 that the corresponding model is unstable. Again for ECNADE in logarithmic version (class I and II), we have seen from figures 37, 38, 39, 40 the corresponding models are stable. Thus we may conclude that our reconstructing ECHDE model (class I), ECNADE model in power-law version (class I) and logarithmic version (class I and II) are more realistic (and classically stable) than the discussed other models (classically unstable).