Analysis of Bianchi Type V Holographic Dark Energy Models in General Relativity and Lyra’s Geometry

In this paper, we analyze anisotropic and homogeneous Bianchi type V spacetime in the presence of dark matter and holographic dark energy model components in the framework of general relativity and Lyra’s geometry. The solutions of differential equation fields have been obtained by considering two specific cases, namely, the expansion scalar ðθÞ in the model is proportional to the shear scalar ðσÞ and the average scale factor taken as hybrid expansion form. The solutions for field equations are obtained in general relativity and Lyra’s geometry. The energy density of dark matter in both natures was obtained and compared so that the energy density of dark matter in general relativity is slightly different from the energy density of dark matter in Lyra’s geometry. A similar behavior occurred in case of pressure and EoS parameter of holographic dark energy model in respective frameworks. Also, it is concluded that the physical parameters such as the average Hubble parameter, spatial volume, anisotropy parameter, expansion scalar, and shear scalar are the same in both frameworks. Moreover, it is observed that the gauge function βðtÞ is a decreasing function of cosmic time in Lyra’s geometry, and for late times, the gauge function ðβðtÞÞ converges to zero and Lyra’s geometry reduces to general relativity in all respects. Finally, we conclude that our models are a close resemblance to the ΛCDM cosmological model in late times and consistent with the recent observations of cosmological data.


Introduction
The relatively recent discovery of the present acceleration of our universe is strictly proved by the astronomical observation of type Ia supernova [1,2], galaxy redshift survey [3], and cosmic microwave background radiation (CMBR) data [4,5] which convincingly suggest that the rate of expansion of our universe is positive, i.e., we live in an accelerating expanding universe. This fact initiated a great number of theoretical hypothesis and inspired researchers to a diversity of different explanations of such an unusual behavior of the universe. The most surprising and counterintuitive result coming from these observations is the fact that only ≈4% of the total energy density of the universe is in the form of baryonic matter, ≈24% is nonbaryonic matter, and almost ≈72% is of completely unknown component with negative pressure. In literature, the component with negative pressure is named as dark energy (DE) that produces repulsive force which gives rise to the current accelerating expansion of the universe.
The most fundamental principle of quantum gravity, the holographic principle (HP), states that all of the information contained in a volume of space can be represented as a hologram, which corresponds to a theory locating on the boundary of that space that may play an important role in solving the DE problem. After applying the HP to the DE problem, one of the present authors [6] proposed a new DE model, called holographic dark energy (HDE) model. In this model, the dark energy density ρ Λ only relies on two physical quantities on the boundary of the universe: one is the reduced Planck mass M p ≡ ffiffiffiffiffiffiffiffiffiffiffiffi ffi 1/8πG p , where G is the gravitational constant; another is the cosmological length scale L, which is chosen as the future event horizon of the universe [6].
The HDE model is the first theoretical model of DE inspired by the HP and is in good agreement with the current cosmological observations at the same time. This makes HDE a very competitive candidate of DE. Granda and Oliveros [7] proposed a holographic density of the form ρ Λ ≈ αH 2 + γ _ H, where H is the Hubble parameter and α, γ are constants which must satisfy the restrictions imposed by the current observational data. They showed that this new model of dark energy represents the accelerated expansion of the universe and is consistent with the current observational data. Granda and Oliveros [8] have also studied the correspondence between the quintessence, tachyon, k-essence, and dilation dark energy models with this holographic dark energy model in the flat FRW universe. Chattopadhyay and Debnath [9], Farajollahi et al. [10], Debnath [11], Malekjani [12], and Sarkar [13,14] are some of the researchers who have investigated several aspects of holographic dark energy. Recently, Kiran et al. [15,16] have studied minimally interacting dark energy models in some scalar-tensor theories. Rao et al. [17] and Adhav et al. [18] have discussed interacting dark matter and HDE in Bianchi type V universe. Rao et al. [19] have discussed the five-dimensional FRW holographic dark energy in the Brans-Dicke theory.
In recent years, our knowledge of cosmology has improved remarkably by various experimental and theoretical results. Einstein [20] introduced his general theory of relativity in which gravitation is described in terms of geometry of spacetime. Motivated by it, Einstein geometrized other physical fields in general relativity. One of the first attempts in this direction was made by Weyl [21] who proposed a more general theory in which gravitation and electromagnetism are also described geometrically. However, this theory was never considered seriously as it was based on the nonintegrability of length transfer. Later, Lyra [22] suggested a modification of Riemannian geometry by introducing a gauge function which removes the nonintegrability condition of the length of a vector under parallel transport. This modified Riemannian geometry is known as Lyra's geometry. Subsequently, Sen [23] proposed a new scalar tensor theory of gravitation. They constructed an analog of Einstein's field equations based on Lyra's geometry which is in normal gauge. He found that the static model with finite density in Lyra's manifold is similar to the static model in Einstein's general relativity.
Halford [24] has shown that the constant displacement vector field in Lyra's geometry plays the role of cosmological constant Λ in general relativity. He has also shown that the scalar-tensor analysis based on Lyra's geometry suggests the same effects, within observational limits, as in Einstein's theory [20]. Katore et al. [25] studied the Einstein-Rosen bulk viscous cosmological model and zero-mass scalar field in Lyra's geometry. Ghate and Sontakke [26], Asgar and Ansari [27], and Das and Sharma [28] studied the Bianchi type V cosmological models in Lyra's geometry with dark energy, in the presence of bulk viscous string; Sahu et al. [29] studied the Bianchi type III cosmological model in Lyra's geometry, and Katore and Kapse [30] studied dynamics of Bianchi type VI 0 holographic dark energy models in general relativity and Lyra's geometry.
Motivated by the abovementioned investigations, we have considered the homogeneous Bianchi type V holographic dark energy cosmological models in general relativity and Lyra's geometry. This paper is outlined as follows. In Section 2, we discussed the metric and the field equations in general relativity. In Section 3, we have obtained the solutions of field equations. Some physical properties of the models have been studied in Section 4. In Section 5, we have obtained the field equations in Lyra's geometry; its solutions and physical properties are studied under Sections 5.1 and 5.2, respectively. Finally, the conclusions are summarized in Section 6.

Metric and Field Equations
We consider the spatially homogeneous and anisotropic Bianchi type V spacetime as where m 1 is an arbitrary constant and A, B, C are functions of cosmic time t only.
The Einstein field equations are given by where R ij , R, g ij , G, and c are the Ricci tensor, Ricci scalar, metric tensor, Newton's gravitational constant, and speed of light, respectively. Here, we consider 8πG = c = 1. T ij is the energy momentum tensor which is expressed as the sum of the energy momentum tensors of dark matter ð T ðmÞij Þ and the holographic dark energy ðT ðΛÞij Þ.
The energy momentum tensor of holographic dark energy ðT ðΛÞij Þ of the source with anisotropic pressures along different spatial directions has the form where ρ Λ and p i ði = x, y, zÞ are the energy density of holographic dark energy (HDE) and pressures of the HDE in the three different directions of the universe. A relation between ρ Λ and p i is given by equation of state (EoS) parameter as p i = ω i ρ Λ , and ω i are equation of state parameters along directions of x, y, and z.
Here, we suppose that the universe is filled with dark matter and dark energy whose energy momentum tensors can be written as where ρ m and ρ Λ are the energy densities of dark matter and holographic dark energy (HDE), respectively, and p Λ is the pressure of the HDE. For the sake of simplicity, we choose ω x = ω Λ , ω y = ω Λ + δ y , ω z = ω Λ + δ z and the skewness parameters δ y and δ z are the deviations from the equation of state parameter ω Λ on y and z directions, respectively. The physical parameters are defined as follows.

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The mean Hubble parameter ðHÞ is given by where the overhead dot denotes differentiation with respect to the cosmic time t. The average scale factor ðaÞ and spatial volume ðVÞ of the Bianchi V spacetime are defined by The scalar expansion ðθÞ, shear scalar ðσÞ, anisotropy parameter ðA m Þ, and deceleration parameter ðqÞ are defined as As suggested by Granda and Oliveros [8], the energy density of HDE with IR cut-off is given by with M −2 p = 8πG = 1, where M p is the reduced Planck mass and α and γ are the dimensionless parameters, which must satisfy the restrictions imposed by the current observational data. The energy conservation equation By adopting comoving coordinates, the field equation (2) for the metric (1) using the energy-momentum tensors (5) yields the following equations: where the overhead dot denotes ordinary differentiation with respect to cosmic time t.
Integrating equation (15) and absorbing the constant of integration into B or C, we obtain

Solution of Field Equations
We can observe that the field equations (11)-(15) are a system of five independent equations with the eight unknown parameters A, B, C, ω Λ , ρ m , ρ Λ , δ y , and δ y . In order to solve the system completely, we need more additional conditions to obtain explicit solution of the system. So first, we consider the hybrid expansion law in which the average scale factor is an increasing function of cosmic time t as follows: where k > 0 and n ≥ 0 are constants. This type of ansatz for the scale factor has already been considered by Akarsu et al. [31], Moraes et al. [32], Moraes and Sahoo [33], Ram and Chandel [34], Pradhan and Amirhashchi [35], Pradhan et al. [36], and Santhi et al. [37]. The relation (17) gives the exponential law when n = 0. This is a combination of exponential and power law, which is termed as hybrid expansion law. Also, this relation yields a time-dependent deceleration parameter which describes the transition of the universe from the early decelerating phase to the current accelerating phase. Thus, our choice of scale factor is physically acceptable. Secondly, we assume that the expansion scalar ðθÞ in the model is proportional to shear scalar ðσÞ as considered by Thorne [38] and Collins et al. [39], which leads to where A and B are the metric potentials. From equations (16), (17), and (18), we obtain

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The metric (1) can now be written as

Some Physical and Geometrical Features of the Model
The physical parameters such as the Hubble parameter ðHÞ, the anisotropic parameter ðA m Þ, the shear scalar ðσ 2 Þ, the expansion scalar ðθÞ, and the spatial volume ðVÞ of model (22), which are of cosmological importance, are, respectively, given by The directional Hubble parameters are as follows: From aðtÞ = 1/ðz + 1Þ, with z being the redshift and the present scale factor a 0 = 1. Using (17), we obtain the following redshift It is evident from the above result in equation (27) that the spatial volume is zero at t = 0. But we observe that as cosmic time t → ∞, the spatial volume expands to infinite. Therefore, the model starts evolving at t = 0 and expands with cosmic time t. The mean anisotropy parameter is constant and different from zero for l ≠ 1. From equation (24), it is observed that the mean anisotropy parameter of the present model becomes zero for l = 1 and the anisotropy of the universe vanishes. But the universe is anisotropic throughout the evolution except for l = 1. Other dynamical physical parameters such as expansion scalar ðθÞ, shear scalar ðσ 2 Þ, and Hubble parameter ðHÞ diverge as cosmic time t approaches to zero. These dynamical physical parameters are decreasing functions as cosmic time t increases. Hence, the model (22) has a big bang type of initial singularity.
The deceleration parameter ðqÞ is obtained to be which is a constant value for late time throughout the evolution of the universe that means q¬−1 as t → ∞. From equation (30), the universe will expand with decelerated rate for q > 0, i.e., t < ffiffiffiffiffi nk p − n, accelerated rate for q < 0, i.e., ffiffiffiffiffi nk p − n < t, and marginal inflation for q = 0, i.e., t = ffiffiffiffiffi nk p − n. One can explicitly observe the dependence of deceleration parameter ðqÞ on the constant parameters n and k. Thus, we can obtain a decelerated or accelerated expansion of the universe depending on the suitable choices of these parameters. According to the recent observations of type Ia supernova (SNe Ia) [2,40,41,42], the present universe is accelerating and the value of deceleration parameter is in the range −1 < q < 0. Thus, the deceleration parameter of our model (22) is consistent with the recent astronomical observations. Following Granda and Oliveros [8] and Sarkar [13] and using equation (23) in equation (9), the energy density of HDE with the IR cut-off is given by Using equation (31) in equation (14), the energy density of matter is given by From equation (11), the pressure of HDE with the IR cutoff is given by Here, from (31), we observe that the holographic dark energy density ðρ Λ Þ decreases as cosmic time t increases. The energy density of dark matter is infinite at cosmic time t = 0, and the pressure of holographic dark energy is a decreasing function of cosmic time. From (33), we concluded that the pressure of holographic dark energy has negative value as cosmic time t → ∞.
Using equations (31) and (33), the EoS parameter of HDE is obtained as Advances in High Energy Physics From equations (11), (12), and (13), the skewness parameter along y direction is From equations (11), (12), and (13), the skewness parameter along z direction is Thus, the metric (22) together with equations (31)-(36) constitutes a Bianchi type V HDE cosmological model in general relativity with hybrid expansion of the universe. The behavior of the energy densities depends on the values of constants α, γ, k, and n. Moreover, the energy density of dark matter depends on constants m 1 and l. From equations (31) and (32), we observe that the energy densities of matter and HDE are decreasing functions of time. Also, we have seen that from equation (34) it is also observed that the pressure of HDE is a decreasing function of cosmic time t. The obtained EoS parameter of HDE is time varying and it is evolving with negative sign for late times which may be attributed to the current acceleration of the expansion the universe [43]. The EoS parameter of the HDE also behaves like quintessence or phantom region based on choices of constants.
The coincidence parameter is It is observed that coincidence parameter r at very early stage of evolution varies, but after some finite time, it converges to a constant value and remains constant throughout the evolution, thereby avoiding the coincidence problem (unlike ΛCDM).
The energy densities of parameters of dark matter (Ω m ) and HDE ðΩ Λ Þ are as follows: Using equations (38) and (39), we get the sum of overall energy density parameter as From equation (40), one can observe that the overall sum of the energy density parameter approaches ð1/3Þððð2l − 1Þ/ l 2 Þ + 2Þ as t → ∞. Moreover, it can be concluded that for l = 1, this model predicts that the sum of overall energy density parameter becomes 1. Figure 1 shows the nature of total density parameter as cosmic time increases. As it was illustrated in Figure 1, total density parameter ðΩÞ tends to 1 as cosmic time increases, which is physically acceptable based on the recent cosmological data. Figure 2 shows the accelerated phase of the universe ðq ≃−1Þ for some small values of redshift. This illustrates that the findings are in accordance with the observational values of the current cosmological data of accelerated universe. Figure 3 indicates the energy density of dark matter versus cosmic time t for constants l = 1:5, n = 2, α = 0:5, k = 0:5, m 1 = 0:5, and γ = 0:6. From Figure 3, it can be seen that at early stage of the evolution, energy density dark matter dominates and at late times it approaches zero. This supports the recent observations of the cosmological data. Figure 4 indicates the plot of pressure of holographic dark energy versus cosmic time t for constants l = 1:5, n = 2, k = 0:5, and m 1 = 0:5. As it is shown in Figure 4, the negative pressure of holographic dark energy increases as cosmic time t increases. The negative pressure indicates the accelerated phase of the expansion of the universe. Figure 5 shows the equation of state parameter of dark energy with respect to cosmic time. As it was shown in Figure 5, the dark energy of equation of state lies −1/3 < w de < −1 which implies the region of quintessence. Figure 1 shows total density parameter versus cosmic time. As it was shown in Figure 1, the total density parameter ðΩÞ approaches to 1 at late time which exactly matches with the ΛCDM model of recent cosmological data.
In order to discriminate among the various DE models, Sahni et al. [44] and Alam et al. [45] introduced a new geometrical diagnostic pair for DE, which is known as statefinder pair and denoted as ðr, sÞ. The statefinder pair probes the dynamic expansion of the universe through higher derivatives of the scale factor. Hence, it is a geometrical diagnosis in the sense that it depends on the scale factor which describes the spacetime.
The statefinder pair is defined as Substituting the required values from equations (17), (23), and (30) into (41), we get 5 Advances in High Energy Physics From equations (42) and (43), it can be observed that the statefinder parameters are constant whose values depend on k and n in late times. Sahni et al. [44] and Alam et al. [45] have observed that the Lambda cold dark matter ðΛCDMÞ model and the standard cold dark matter (SCDM) model have fixed point values of statefinder parameter ðr, sÞ = ð1, 0 Þ and ðr, sÞ = ð1, 1Þ, respectively. Figure 6 shows r − s statefinder parameters. As it was shown in Figure 6, our model approaches to ðr, sÞ → ð1, 0Þ as cosmic time t approaches to infinity which would clearly show the ΛCDM model.

Lyra's Geometry and Field Equations
Lyra [22] suggested a modification of Riemannian geometry by introducing a gauge function which is a metrical concept in Weyl [21] geometry in the geometrical structureless manifold. It is a generalization of Weyl's geometry which is removing the defect of nonintegrability of length transfer that has been discussed. The Einstein modified field equation in normal gauge for Lyra's manifold obtained by Sen [23] is given by where ϕ i is the displacement vector defined as ϕ i = ð0, 0, 0, βðtÞÞ.
In a comoving coordinate system, the modified Einstein field equation (44) for Bianchi type V spacetime with the help of (5) is       Advances in High Energy Physics where the overhead dot denotes derivative with respect to cosmic time t.
The conservation of the right-hand side of equation (44) Equation (50) is reduced to leading to Integrating equation (49), we obtain where the constant of integration is absorbed in B or C. (6) and (17) in (52) and integrating, we obtain the gauge function β

Solutions of Field Equations. Using equations
where c 1 is a constant of integration.
Using the same conditions above in (9) and (21), we have solved the field equations (45)-(49). From equation (54), we observe that at early time, the gauge function β tends to infinity in this limit and also it tends to zero as cosmic time t → ∞ . It is large in the beginning and decreases fast with the evolution of the universe. Hence, βðtÞ is a decreasing function of cosmic time t. Thus, the model has singularity at t = 0.     Advances in High Energy Physics shows that the behavior of β is as decreasing function of cosmic time t. It can be concluded that as n values increase, the values of βðtÞ decrease more with cosmic time increases. The physical parameters such as Hubble parameter ðHÞ, anisotropic parameter ðA m Þ, shear scalar ðσ 2 Þ, expansion scalar ð θÞ, spatial volume ðVÞ, and deceleration parameter ðqÞ are found to be the same as in the above case of general relativity. Thus, we get the model (22).

Some Physical
Properties of the Model. Using equations (9) and (21), the solutions of the field equations (45)-(49) are obtained. Therefore, the skewness parameters, energy density of dark matter ðρ m Þ, pressure of HDE ðp Λ Þ, and EoS parameter of HDE ðω Λ Þ are given below as follows.
Here, from equations (45) and (46), we obtain Also, from equations (45) and (47), we get From equation (48), the energy density of dark matter is From equation (45), we can get the EoS parameter as From equation (54), it is noted that the gauge function β is a decreasing function of time which is corroborated with Halford [46] as well as with the recent observations [2,47,48] [1,40] leading to the conclusion that Λ (cosmological constant) is a decreasing function of cosmic time t. From equation (57), it is observed that the energy density of dark matter is a decreasing function of time. From equation (58), it is also observed that the pressure of HDE is a decreasing function of time. The energy density of holographic dark energy in Lyra's geometry is similar to the energy density of HDE in general relativity. Similarly, the skewness parameters along y and z directions in general relativity and Lyra's geometry will be the same. But the dark energy density of dark matter, the pressure of holographic dark energy, and the EoS parameter of holographic dark energy in general relativity are slightly different from those in Lyra's geometry. It can be seen that the behavior of EoS parameter strongly depends on parameters of the model. However, by choosing appropriate values for α, m 1 , n, k, and γ, we can obtain ω Λ in quintessence or in phantom phase which is consistent with the recent observations of cosmological data. The EoS parameter of holographic dark energy is obtained as time varying, and it is evolved with negative sign for late times and its range is in good agreement with large-scale structure of the universe data.
The coincidence parameter is From equation (60), it is observed that coincidence parameter r at the very early stage of evolution varies, but after some finite time, it converges to a constant value and remains constant throughout the evolution, thereby avoiding the coincidence problem (like ΛCDM).
Advances in High Energy Physics The energy density parameter of dark matter in Lyra's geometry is given by Using (39) and (61), the overall sum of density parameter ðΩÞ is obtained as The overall sum of the energy density parameter approaches ð1/3Þððð2l − 1Þ/l 2 Þ + 2Þ as cosmic time t → ∞. It converges to 1 as t → ∞ for the constant l = 1, and it holds for l > 1 which is physically acceptable. Hence, it indicates that the universe is very close to critical density or Ω = 1. Thus, the gauge function βðtÞ is large in the start of the model but decays continuously during its evolution. The energy density and the pressure tend to small positive and negative constants, respectively, for the large values of cosmic time t. The negative pressure corresponds to the accelerated phase of the universe. The gauge function βðtÞ becomes zero for large time t. In this case, the solutions reduce to general relativistic one.

Conclusions
In this paper, we have studied anisotropic and homogeneous Bianchi type V universe filled with dark matter and holographic dark energy in the framework of general relativity and Lyra's geometry. The solutions for field equations of cosmological models are obtained in general relativity and Lyra's geometry by using two specific cases: firstly by taking the expansion scalar ðθÞ in the model is proportional to the shear scalar ðσÞ as considered by Thorne [38] and Collins et al. [39] and secondly by using the average scale factor as hybrid expansion form (combination of the power law and exponential form). We have investigated the nature of deceleration parameter versus redshift as shown in Figure 2 which reveals that the accelerated nature of the universe maintained for small values of redshift. Figure 3 indicates that the energy density of dark matter versus cosmic time t is a decreasing function. It is constructed based on arbitrary constants constructed domain. Here, it can be concluded that at early stage of the evolution, energy density of dark matter dominates and at late times it approaches zero. This supports the recent observations of cosmological data of the universe. Figure 4 shows the pressure of HDE versus cosmic time t. It shows that the pressure of HDE negatively increases with time. The overall density parameter has been matched with Λ CDM as shown in Figure 1 which indicates that Ω ≃ 1. The energy density of dark matter, the pressure of HDE, the EoS parameter of HDE, the coincidence parameter, and overall density parameter in Lyra's geometry slightly differ by the term β from general relativity framework. We have observed that the gauge function βðtÞ is a decreasing function of cosmic time t in Lyra's geometry. The gauge function βðtÞ is large in the beginning and reduces fast with the evolution of the universe. It is found that for late cosmic times the gauge function β → 0 and Lyra's geometry tends to general relativity in all respects. The present model of the gauge function βðtÞ is infinite at the initial singularity. From Figure 7, we see that β tends to zero as t → ∞. Therefore, the concept of the Lyra manifold is meaningful for finite time, but does not remain for very large time. But we have observed that the dynamical parameters: the average Hubble parameter, spatial volume, anisotropy parameter, expansion scalar and shear scalar, are the same in both frameworks. In each case, the cosmological models approach to anisotropic parameter for large value of cosmic time t. For l = 1, anisotropic model vanishes. These models represent a shearing, nonrotating, and expanding universe, which approaches anisotropy for large value of time t. The EoS parameter of the HDE also behaves like quintessence region based on the selected choice of the constants and parameters which is shown in Figure 5. From geometrical diagnostic pair for DE, it can be seen that ðr, sÞ → ð1, 0Þ as cosmic time t approaches to infinity and it is clearly illustrated in Figure 6 which is consistent with the ΛCDM model. The present model is consistent with the recent observations of the present day of accelerating universe.

Data Availability
The data used to support the findings of this study are included within the article and are cited at relevant places within the text as references.