Dark Energy Cosmological Models in Lyra Geometry for Bianchi-I Space Time

We construct cosmological models for dark energy based on Lyra’s geometry for the Bianchi-I space-time to understand the evolution of the universe. The specific choice of the deceleration parameter q=-α1t2+α2-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=-\frac{\alpha _1}{t^2}+\alpha _2-1$$\end{document} is used to determine the exact solutions of the field equations in Lyra geometry. The parameter α1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _1$$\end{document} has a dimension equal to the square of time. Further, we have discussed two cosmological models depending on the parameter α2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _2$$\end{document}. Several physical parameters are obtained and the results in both the models are analyzed and presented graphically for the analysis of the results.


Introduction
Physicists and cosmologists studied the future evolution of the universe as well as understanding the past and present state of the universe.Theory has been given to the anisotropic cosmological model [1][2][3][4], partly due to the realization that standard models, which agree well with our present-day universe, fail to account for the early phases of the universe's evolution.The exotic type of matter with large negative pressures may provide the necessary mechanism to overcome the gravitational pull [5,6].Our universe is dominated by an exotic form of energy named as dark energy (DE), which may be responsible for cosmic acceleration, but is still unknown.Though the nature of DE is completely unknown to us except its negative pressure, so it is repulsive and it results in the expansion of universe [5,6].Several recent studies [7][8][9] discovered that astronomical data from type-Ia supernovae [10][11][12], large-scale structures [13], and measurements of cosmic microwave background (CMB) anisotropy [14] provide convincing evidence that the present-day universe is not only expanding, but also accelerating.Further, the astronomical observational data suggest us that our universe consist of 68.3% dark energy, 26.8% dark matter and the rest 4.9% is baryonic matter [15].Moreover, two important parameters, like Hubble parameter H and deceleration parameter q, play an important role in understanding the universe [16].The Hubble parameter is useful for the study of the expansion rate of the universe, whereas the deceleration parameter signifies the acceleration and deceleration nature of the universe.
Back to Einstein, the cosmological constant has a repulsive character, which makes it an ideal candidate for the dark energy [5].The cosmological constant is subject to the fine-tuning problem and to the cosmic coincidences.Thus, dynamical cosmological constants with negative pressure appear to be the most promising choice.Also we have found different candidates for dark energy such as quintessence [17], phantom [18], k-essence [19], tachyons [20] and Chaplygin gas [21] in the literature.Modern cosmology and modern astrophysics grapple with the issue of cosmic acceleration, despite the existence of these candidates of dark energy.
We characterize dark energy models by the equation of the state parameter = p , where p is the pressure of the fluid and is the energy density [8].It is important to note that can either be a constant or a variable.Generally, one may say that depends on time t, redshift z, or scale factor a [22,23].With respect to variable , we have  > −1 , = −1 , and  < −1 , denoting respectively the quintessence, Lambda cold dark matter ( ΛCDM), and phantom dark energy models [24][25][26][27].Additionally, the equation of state p a r a m e t e r h a s l i m i t s −1.67 <  < −0.62 , −1.33 <  < −0.79 and −1.44 <  < −0.92 [13, 28-30].More details relating to dark matter, dark energy and alternate models can be found in Arun et al. [31].The constant usually has the value −1, 0, 1 3 , 1 for vacuum, dust, radia- tion, and stiff-dominated universes, respectively.
Mankind has always been curious about understanding the universe from the past to the future in a scientific manner.Einstein developed the general theory of relativity and gained considerable attention due to its success in building cosmological models.This helped in understanding the evolution and the origin of the universe.However, he was dissatisfied with the field equations of general relativity Since the right-hand side is not geometrical in nature however, the left-hand side is.Although this is the most tested theory in science [], it fails to explain the acceleration of the universe in the late time.So, alternative theories of gravitational acceleration were needed to explain why the universe accelerated late in time.Einstein's theory related to geometrizing gravitation influenced Weyl to create the Weyl's theory [], which is a geometrizing of electromagnetism and gravitation.However, this theory was not accepted by researchers due to the condition of non-integrability of length transfer under parallel displacement.There are lots of alternative theories discussed by many researchers but here we are very much interested in Lyra's geometry due to the alternative idea of Riemannian geometry suggested by Lyra [32] with the new idea of gauge function which resolves the problem of the non-integrability condition of the length of a vector under parallel transport.Lyra suggested a modification of Riemannian geometry which has a close resemblance to Weyl's geometry.The Einstein's field equations (EFEs) based on Lyra's manifold in normal gauge given by Sen and Dunn [33] are expressed as (Take c = 1 and 8 G = 1 ) In which i is the displacement vector defined through the gauge function .The displacement vector field is considered as In Lyra geometry, several models with a constant displacement vector field [34,35] and a time-dependent displacement vector field [36][37][38][39][40] have been analyzed.In these models, the universe appears to be decelerating, which is incompatible with current observations.However, in Ref [41][42][43][44][45][46][47], it is reported that the universe has been accelerating in late time in Lyra geometry.Halford [48] pointed out the constant displacement field i in Lyra's geometry plays the role of the cosmological constant in normal general relativistic treatment.A dark energy cosmological model in Lyra geometry was studied by Hova [49], which involved matter energy and a constant equation-of-state parameter interacting with the geometric displacement field.According to the study, the late-time accelerated expansion of the universe for m ≠ 1 is explained by nonzero effective time- dependent gravitational constants.By taking into account the time-varying deceleration parameter, Ram et al. [50] have presented the Kantowski-Sachs model of the universe filled with anisotropic dark energy.As part of the Lyra geometry cosmological model, Yadav et al. [51] have studied a model of the quintessential universe with dominance of dark energy.In this study, authors pointed out that is not a contributor to late time universe acceleration.In the derived model, dark energy dominates and is the cause of acceleration.Five-dimensional holographic Ricci dark energy model is analyzed by Das and Bharali [52] for anisotropic locally rotationally symmetric (LRS) Bianchi type I space-time under Lyra geometry.They have investigated the interacting and non-interacting dark energy and dark matter.Hybrid expansion law is used for the exact solutions.Their models at late time correspond to the cosmological constant and phantom dark energy depending on the non-interacting and interacting model, respectively.A five-dimensional spherically symmetric cosmological model is analyzed by Singh and Singh [53].The authors pointed out that in addition to vacuum energy, the displacement vector also acts as timedependent dark energy.Essentially, the model represents an oscillating model, evolving from a big bang to a big crunch and experiencing several bounces in between.The universe undergoes super-exponential expansion throughout the entire process of evolution.Ram et al. [54] investigate the (1) (2) i = (0, 0, 0, (t)).
Bianchi type-VI0 cosmological model through the lens of the Lyra manifold when an anisotropic dark energy is present along with a massless scalar field.Exact solutions of the field equations are obtained by utilizing the Hubble parameter, which leads to a time-varying deceleration parameter.In the framework of general relativity and Lyra's geometry, Gusu and Shanthi [55] studied anisotropic and homogeneous Bianchi type V space-time in the presence of dark matter and holographic dark energy model components.Additionally, they demonstrated that Lyra geometry converges to general relativity in late time.Moreover, the models are similar to the ΛCDM model in late time.Casana et al. [56] studied the Dirac field in the Lyra space-time background using the classical Schwinger variational principle.Further, studies include equations of motion, conservation laws and derivation of a scale relation related to energy momentum and spin tensors.
As a consequence of the above importance of dark energy and Lyra geometry, this manuscript develops dark energy cosmological models with a variable deceleration parameter of the form q = − 1 t 2 + 2 − 1.

Field Equations for Bianchi Type-I Line Element
Let us consider the Bianchi type-I line element of the form Here, A, B and C are function of t only.The energy momentum tensor T j i is considered as Here, and p represents the proper energy density and pressure, respectively.The quantity is the scalar of expansion which is given by, and i satisfies the relation We assume that coordinates to be co-moving, so that Equation ( 4) yields (4) where p x , p y , p z and x , y , z indicates the directional pressure and equation of state (EoS) parameters along x-, y-and z-axes, respectively.The EoS parameter of the fluid, which is deviation-free, is denoted by .We have parameterized the deviation from isotropy by setting x = and then introducing skewness parameter and that are the deviation from along y-and z-axes, which are y = ( + ) and z = ( + ) , respectively.The field Eq. ( 1) in Lyra's geom- etry, for the metric (2), using Eqs.( 2) and ( 4 C are the directional Hubble's parameter in directions of x-, y-and z-axis, respectively.The deceleration parameter, q, is expressed as ( 8) The following physical parameters are defined as

Solution of Field Equations
The field Eqs. ( 9) to ( 12) are the four differential equations with eight unknown parameters A, B, C, , , , and .We know that two observable parameter H and q are related as q = d dt 1 H − 1 .This can be integrated to get the scale factor of the form where 0 and are integration constants.Here, we are free to choose the form of q.The choice of q in the form leads to an explicit form of the scale factor.The details about such consideration of deceleration parameter is discussed by Abdussattar and Prajapati [57].Here,  1 > 0 is a param- eter having the dimension of square of time and  2 > 1 is a dimensionless constant.From Eqs. (20), and ( 21), we have It is possible to evaluate the integral in Eq. ( 22) to obtain: Average anisotropy parameter where a = 1 ∕ 2 , b = 0 ∕ 2 .However, the resulting expres- sion for R is too wieldy and cumbersome to be used to calculate the metric potentials A, B and C. Hence, we set 0 = 0 in Eq. ( 22) to obtain the simplest form of the scale factor as Thus, Eq. ( 23) takes the form Equation ( 22) leads to the Hubble parameter in the form Here, we observe that H > 0 for different choices of 1 , 2 .As per the classification of Bolotin et al. [58], the choice (21) of q leads to (i) expanding and decelerating models for sufficiently large value of t as q → 2 − 1 when t → ∞ .(ii) expanding and accelerating models for 2 = 1 . (iii) expand- ing with constant expansion/zero deceleration for t 2 = 1 2 −1 (See Fig. 1).In order to obtain a deterministic solution, firstly, we assume quadratic deceleration parameter of the form (21).We assume the second condition as 1  1 ∝ , which leads where m > 0 .As a third assumption, we take = which means that the deviation of along the y-and z-axes are equal.
From the differential equations ( 10) and ( 11), we deduce: Fig. 1 Profile of deceleration parameter q against time where 3 and 4 are integration constants.From Eq. ( 27), the metric potentials is written as The parameter 2 makes it difficult to find the general expression for the integral involved.Let us return to Eq. ( 21), which shows that q → 2 − 1 when t → ∞ .This puts a limit on 2 as 1 ≤ 2 ≤ 2 .Hence, we have developed two different cosmological models based on two different values for 2 , viz., 2 = 1 and 2 = 3 2 .On the other hand, for some other choices of 2 , we encountered unrealistic expressions or were unable to find the integral expressions.

Model-I, for the Case: 2 = 1
The metric potentials ( 28)- (30), in this case, take the form of As a result, the required metric (3) is . The gauge function (t) from Eqs. ( 13) and (31) takes the form , where N is constant of integration.The directional Hubble parameter H i along x-, y-and z-axes are given by The following physical parameters for model (32) are deduced as The expansion scalar decreases with cosmic time.At a initial time t, the expansion scalar is zero, and it approaches zero when the cosmic time approaches infinity.The shear scalar 2 of the model depends on the gauge function , and it has the same qualitative behavior as the expansion scalar .The shear scalar 2 evolves with constant value initially, and it approaches zero as cosmic time approaches infinity.Likewise, the average anisotropy parameter A m decreases as cosmic time progresses.Initially, it takes a large value and continues to decrease.In late time, A m has a con- stant value 2(m−0.5)(m+1) 2 , and it shows that the parameter A m van- ishes for m = 0.5 .Using Eqs.(33) to (36), in Eq. ( 12), we obtain as The energy density of the model depends on the gauge function .It begins with a negative value − 3 , changes to a positive value, and then reaches zero when t → ∞ (See Fig. 2).Using Eqs.(33) and (40) in Eq. ( 9), the EoS parameter has been calculated as (34) 3 (m + 1) 3 (m + 1) In Figs. 3 and 4, we see the profile of the EoS parameter for different N and m, respectively, where starts with a positive value and then gradually slides into a negative value as time passes.At late time t → ∞ , the EoS parameter approaches 4m−5 3(4m+1) and vanishes for m = 5 4 .In other words, as time progresses, the EoS parameter starts in the quintessence region and moves along into the phantom region and then in the quintessence region.We write the values of the skewness parameter and , using Eqs.( 9), ( 10), ( 33) and (40) as follows: (41) as its value, but it ends at − 4(2 m−1)(m+1) 3(4 m+1) .The skewness parameter vanishes for m = 0.5 throughout the evolution for the value of time t = ( 0 2 ) ± given by (42 which leads to equation of state parameter = 0 .So that dusty universe at time t = ( 0 2 ) ± is given by Eq. (43).It is to be noted that when time t lies in open interval 2 ) ± where the EoS parameter lies in the interval −1.67 <  < −0.62 .This range of follows the observational limit of Knop et al. [28].Further for the value of time t = ( 2 ) ± , given by the values of comes out to be −1 , where For flat model, the matter energy density Ω M and dark energy Ω Λ satisfy where Thus, Eqs. ( 47) and ( 48) give and then using Eqs.( 25) and (40) in Eq. ( 49), we write the values of Λ as The profile of cosmological constant Λ for different N and m is presented in Fig. 6.It shows that cosmological constant Λ is positive and decreasing function of cosmic time t for different N, whereas for certain values of m, it increases for certain period of time and then decreases.Further, a positive cosmological constant Λ indicates the negative effective mass density.The expansion of the universe should therefore tend to accelerate due to positive cosmological constants.( 48) The matter energy density Ω M and dark energy Ω Λ are expressed as A plot of the matter energy density Ω m and dark energy Ω Λ is shown in Figs.7 and 8, respectively.The dark energy Ω Λ is a decreasing function of cosmic time t, taking positive values over time for different N and m, while the matter energy density Ω m is an increasing function of cosmic time t, taking values from negative to positive for different N and m (51) The metric potentials ( 28)- (30), in this case, take the form of ( 53) As a result, the required metric (3) is . The displacement vector (t) from Eqs. ( 13) and ( 53) takes the form where N is constant of integration.The directional Hubble parameter H i along x-, y-and z-axes is given by The following physical parameters for model ( 54) are deduced as The expansion scalar increases for a certain period of time, then decreases with cosmic time t until it approaches zero in late time.Initially, the shear scalar 2 of the model has a constant value , then increases for a period of time, then decreases with cosmic time t until it approaches zero.The shear scalar 2 tends to zero as cosmic time t approaches infinity, indicating that our model is shear free at late time.
The average anisotropy parameter A m of the model decreases as cosmic time t increases.It approaches 2(m−0.5) 2 (m+1) 2 , when cosmic time t tends to infinity.Using Eqs.(55) to (58), in Eq. ( 12), we obtain as 3 ) 3(4(2m − 1) 2 6 6 t 2 + 3(m + 1) 2 2 4 ) 4 (m + 1) 2   The profile of energy density is given in Fig. 9 for different N and m.Starting from a constant value − 9(3N 2 + 4 2 ) 2 , it increases for a while and then decreases with cosmic time until it reaches zero when cosmic time reaches infinity.Using Eqs. ( 62) and (55) in Eq. ( 9), the EoS parameter has been calculated as In Figs. 10 and 11, we see the profile of the EoS parameter for different N and m, where starts with a positive value ) (3 m+3) 4 2 +(9 m+9)N 2 +16 6 6  negative value as time passes.In terms of qualitative behavior, it is similar to Model-I.We write the values of the skewness parameter and , using Eqs.( 9), ( 10), ( 55) and ( 62) as follows: The skewness parameter at the early stage t → 0 gives the constant value equals to 8 6 6 (2 m−1) 1 3(m+1)(3 N 2 + 4 2 ) and approaches when cosmic time approaches infinity (See Fig. 12).When m = 0.5 , the skewness parameter vanishes time.For the value of time t = ( 0 ) ± given by  which leads to equation of state parameter = 0 , the dusty universe at time t = 0 is given by Eq. (63).It is to be noted that when time t lies in open interval ( 1 ) ± < t < ( 2 ) ± where the EoS parameter lies in the interval −1.67 <  < −0.62 .This range of follows the observational limit of Knop et al. [28].Further for the value of time t = ( ) ± , given by (66) It is seen from Fig. 13 that the cosmological constant Λ exhibits the same qualitative behavior as Model-I.From Eqs. ( 57), ( 60) and (69), we have Mass energy density Ω M and dark energy Ω Λ exhibit the same qualitative characteristics as Model-I (See Figs. 14  and 15). (72)

Conclusion
The manuscript offers a Bianchi type-I dark energy cosmological model with variable EoS parameter under variable deceleration parameter q = − 1 t 2 + 2 − 1 in the framework of Lyra geometry.This is quite different from what is published elsewhere.According to the manuscript, the following findings are made: (a) The exact solutions of the dark energy cosmological models for Bianchi type-I space time are obtained under variable q (21).Depending upon the parameter 2 , we have discussed two cosmological models.Model-I and Model-II follow quintessence-phantom-quintessence dark energy with respect to the evolution of time.
(b) According to the recent observations, EoS parameters presented in Eqs. ( 41) and (63) for Models-I and -II are in line with observations [26,28].(c) In both models, the cosmological constant Λ is positive and decaying with respect to time, indicating a negative effective mass density.Therefore, the toy models of our universe are accelerating.(d) In both models, the energy density starts off as negative and increases over time, and then decreases until it approaches zero with the evolution of cosmic time.(e) The EoS parameter is time-dependent here; such a type of (t) was previously discussed by Pradhan et al. [59].It can also be determined by redshift z or scale factor R. The redshift dependence of can be linear or nonlinear [60,61].(f) If N = 0 ⇒ = 0 and 1 = 0 , then our solution resembles that for a constant deceleration parameter, q = 2 − 1 as in general relativity, as studied by Prad- han et al. [59] and Kumar & singh [62].(h) In light of the accelerated expansion of the universe, the Model-II is not appropriate, because we noticed the accelerating to decelerating characteristics of q for 2 = 3 2 .(i) In fact, we can show that for 1 <  2 ≤ 2 , we obtain models that are not realistic, in the sense that they are firstly accelerating, and then decelerating, as opposed to the converse case.Firstly, we see from Eq. ( 21) that q = 0 when t = 1 ∕( 2 − 1).Now, we see that 0 < t <  1 ∕( 2 − 1), q < 0 , imply- ing an early accelerating phase.On the other hand,  1 ∕( 2 − 1) < t ≤ 2, q > 0 , which means a decelerat- ing phase.Hence, for all 1 <  2 ≤ 2 , the models are not realistic.

4 A 4 B
), takes the form where A 4 = dA dt ,A 44 = d 2 A dt 2 .The energy conservation T j i,j = 0 leads to The spatial volume V for model and the average scale factor are defined as In terms of scale factor or metric potentials, the Hubble parameter may be define as or where H x = A , H y = B and H z = C 4

3 Fig. 2 Fig. 3
Fig. 2 Profile of energy density for Model-I against time for different N and m Fig. 3 Profile of EoS parameter for Model-I against time for different N

Fig. 4 Fig. 5
Fig. 4 Profile of EoS parameter for Model-I against time for different m Fig. 5 Profile of skewness parameter = for Model-I against time for different N and m

Fig. 6 Fig. 7
Fig. 6 Profile of cosmological constant Λ for Model-I against time for different N and m Fig. 7 Profile of matter energy density Ω M for Model-I against time for different N and m

Fig. 8
Fig. 8 Profile of dark energy Ω Λ for Model-I against time for different N and m

Fig. 9
Fig. 9 Profile of energy density for Model-II against time for different N and m Fig. 10 Profile of EoS parameter for Model-II against time for different N

Fig. 11 Fig. 12
Fig. 11 Profile of EoS parameter for Model-II against time for different m Fig. 12 Profile of skewness parameter = for Model-II against time for different N and m

Fig. 13
Fig. 13 Profile of cosmological constant Λ for Model-II against time for different N and m Fig. 14 Profile of matter energy density Ω M for Model-II against time for different N and m

Fig. 15
Fig. 15 Profile of dark energy Ω Λ for Model-II against time for different N and m 3 t 2 + 2 1 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.