Higher Dimensional Bianchi Type-I Cosmological Model in Lyra’s Geometry

Higher Dimensional Bianchi Type-I Cosmological Model in Lyras Geometry

S.C. Wankhade

Shri Dr. R.G. Rathod Arts and Science College, Murtizapur, Dist. Akola, (MS), India Corresponding Author

Abstract In the present study, we explore a five- dimensional Bianchi type-I cosmological model incorporating both dark matter and holographic dark energy within the framework of Lyras geometry. By applying suitable transformations under appropriate assumptions, we derive a deterministic solution describing the evolution of the universe. Various physical and kinematical properties of the derived cosmological model have been examined in detail.

A.S. Nimkar

Shri Dr. R.G. Rathod Arts and Science College,

Murtizapur, Dist. Akola, (MS), India

inWeyls geometry, Lyra [4] introduced a gauge function into the structureless manifold and recommended an adjustment to it.Einsteins modified field equation in normal gauge for Lyras manifold obtained by Sen [5] is given by

R 1 g R 3 3 T (1)

ij 2 ij 2 i j 4 ij

Where, Rij is Ricci tensor, g ij is metric tensor, R is

Furthermore, the expansion dynamics of the universe are analyzed, and it is noted that the model exhibits a

Ricci curvature,

i is displacement vector,

Tij is

singularity at the initial epoch, i.e., when cosmic time is

zero. Additionally, key parameters of cosmology such as redshift, Omz diagnostic are studied along with

thermodynamics of the universe.

Keywords Five-dimensional Bianchi type-I, Lyras geometry, Dark matter, Holographic Dark Energy.

Energy Momentum Tensor.

The proposed work contains a study of a five- dimensional Bianchi Type-I universe filled with Dark Matter and Holographic Dark Energy (HDE) in the framework of Lyras manifold.

The energy-momentum tensor for Dark Matter and HDE are respectively given by, (Singh et. al.[6])

1. INTRODUCTION

   Higher Dimensional Cosmologywas introduced by Theodor Kaluza [1], in an attempt to unifygravitywith electromagnetism. Later Oskar Klein [2] extended this

   Tij muiu j

   Tij p uiuj gij p

   (3)

   (2)

   theory by proposing the idea of compactification of extra dimension. The extra spatial dimension gets curled up in a circle of a very small radius; so that a

   particle moving along that axis returns to its initial

   The field equation in equation (1) for Energy Momentum Tensor of Dark Matter and Dark Energy becomes,

   position after traveling only a short distance. This

   R 1 g

   R 3 3 T

   T

   (4)

   curled extra spatial dimension is a compact set, and this phenomenon is referred to as compactification.

   ij 2 ij

   1. i j 4

      ij ij

      The solution of Einsteins Field Equations in higher- dimensional space-times is believed to be of physical relevance possibly at extremely early stages of the universe before the universe underwent compactification. It is argued that the extra dimensions are observable at the present time.

      Einstein geometrizes gravitation in his theory of general relativity which is treated as a basis for a model of the universe. After that, many cosmologists and astrophysicists attempted to study gravitation in different contexts. For the illustration of both gravitation and electromagnetism, Weyl [3] tried to formulate a novel gauge theory containing a metric tensor. But, due to the non-integrability condition, it does not get importance in the cosmological society. In order to eliminate the non-integrability condition

      In recent years many researchers have studied higher-

      dimensional cosmology. P.K. Sahoo et. al. [7] analysed Higher-dimensional Bianchi type-III universe with strange quark matter attached to string cloud in general relativity. A. Pradhan et. al. [8] studied A new class of holographic dark energy models in LRS Bianchi Type-I. D.D pawar et.al. [9] examined Magnetized Dark Energy Cosmological Modes with Time dependent Cosmological Term in Lyra geometry. G.U. Khapekar et.al.[10] studied Characteristic values of Six-Dimensional Symetric Tensor for generalized Peres space-time. J. Daimary et. al. [11] have studied five-dimensional Bianchi type-I string cosmological model with electromagnetic field. M.R. Mollah et. al. [12] proposed a work On Bianchi type III Cosmological Model with Quadratic

      EoS in Lyra Geometry. M.K. Singh Ranawat et. al.

      A A

      B

      B

      C

      C

      A B

      AB

      A C

      AC

      BC

      BC

      3 2

      4

      p

      [13] analyzed Five-Dimensional Bianchi Type I String

      Cosmological Model in General Relativity. J.K. Singh et. al. [14] studied The Bianchi type-V Dark Energy Cosmology in Self Interacting Brans Dicke Theory of Gravity. M.V. Santi et. al. [15] studied Bianchi type-

      (10)

      Here overhead dots denote the ordinary differentiation concerning time t .

      T

      T

      By the use of the energy conservation equation

      III holographic dark energy model with quintessence.

      M.P.V.V. Bhaskara Rao et. al. [16] studied Five-

      j i; j

      j i; j

      0 , the right-hand side of equation (4)

      dimensional FRW Modified Holographic Ricci Dark Energy Cosmological Models with Hybrid Expansion Law in a Scalar-Tensor Theory of Gravitation. F.

      leads to,

   2. j k j 3 j

      k

      Rahaman et. al. [17, 18] proposed work on higher dimensional homogeneous cosmology in Lyra Geometry and Higher-dimensional string theory in Lyra geometry. J. Baro et. al. [19] studied

      1. i , j k j 2

         3 k 3

   3. , j k j 4

   i, j k i j

   j

   n, j k k 0

   (11)

   Mathematical analysis on anisotropic Bianchi type-III inflationary string cosmological models in Lyra geometry, K.P. Singh et. al. [20] has studied higher

   Above equation (11) satisfies identically for

   i 1, 2,3, 4 and for i 0 , we get

   dimensional Bianchi type-III string cosmological

   1. 3

      2 A

      B C

      D

      models in Lyra geometry. G. Mohanty [21]et. al. examined Higher-dimensional String Cosmological model with bulk viscous fluid in Lyra geometry.

      2 2

      A B

      0

      C D

      (12)

      D.R.K. Reddy [22] studied Five-Dimensional spherically symmetric perfect fluid cosmological model in the Lyra manifold.K. S. Adhav [23] has studied LRS Bianchi type-I anisotropic dark energy in Lyra geometry.

2. METRIC AND FIELD EQUATIONS

3. COSMOLOGICAL SOLUTION

   Field equations (6) to (10) is a system of five highly non-linear differential equations in eight unknown functions hence to derive the exact solution of field equations, we use the following

   transformations,

   A exp a, B exp b, C exp c,

   Five-dimensional Bianchi type-I space-time is given as,

   D exp d and dt ABCD dT

   (13)

   ds2 dt 2 A2dx2 B2dy2 C 2dz 2 D2dm2

   (5)

   Where, a,b, c, d are functions of T only.

   The transformed field equations are given by,

   ab ac ad bc bd cd

   Where, A, B,C, D are functions of time t only.

   3 2 exp2a b c d

   1. m

   Displacement vector defined as i

   is the gauge function.

   t ,0,0,0,0

   (14)

   b c d ab ac ad bc bd cd

   In a co-moving coordinate system, the modified field equation (4) for five-dimensional Bianchi type-I space-timein (5) with equations (2) and (3) are

   3 2 exp2a b c d p exp2a b c d

   4

   (15)

   a c d ab ac ad bc bd cd

   A B AB

   A C

   AC

   A D AD

   BC BC

   BD

   BD

   CD CD

   3 2

   4 m

   (6)

   3 2 exp2a b c d p exp2a b c d

   4

   (16)

   a b d ab ac ad bc bd cd

   B C D BC BD CD 3 2 p

   (7)

   3 2 exp2a b c d p exp2a b c d

   1. C D BC BD CD 4 4

      (17)

      A C D A C A D CD 3 2 p

      (8)

      a b c ab ac ad bc bd cd

      A

      A

      C

      B

      D

      D

      AC

      A B

      AD

      A D

      CD 4

      BD 3 2

      3 2 exp2a b c d p

      4

      exp2a b c d

      (18)

      A B D AB

      AD BD 4

      p

      (9)

      Here, a dash denotes the differentiation with respects to T .

      Equating equations (15) and (16), we get

      b a C1T C2

      Equating equations (16) and (17), we get

      c a C1 C3 T C2 C4

      (19)

      (20)

      Similarly, equating equations (16) and (18), we get tends to infinity. Thus, the model has singularity

      d a C1 C5 T C2 C6

      (21)

      when the cosmic time is zero.

      Using equations (19), (20), and (21) in the condition a b c d 0 , (Ranawat et. al. [8])

      Space-time in equation (5) reduces to

      ds2 exp2C10T C11 dT 2 exp2C7T C8 dx2

      we obtain

      a C7T C8

      (22)

      exp2C1 C7 T 2C2 C8 dy2

      Where,

      exp2C1 C3 C7 T 2C2 C4 C8 dz2

   2. 3C1 C3 C7 and C

   – constant of integration

   exp2C1 C5 C7 T 2C2 C6 C8 dm2

   7 4 8

   (31)

   Using the value of a from equation (22), in (19), (20), and (21) we get

4. PHYSICAL AND KINEMATICAL

   PROPERTIES

   b C1 C7 T C2 C8

   c C1 C3 C7 T C2 C4 C8

   d C1 C5 C7 T C2 C6 C8

   (23)

   (24)

   (25)

   Physical properties of the model are studied including Energy density and pressure of Dark matter and HDE.

   Hence, we obtain

   A exp a exp C7T C8

   B exp b exp C1 C7 T C2 C8

   (26)

   (27)

   Energy Density of Dark matter

   m C12 exp C10T C11

   Where, C12 is resolved constants.

   (32)

   C exp c exp C1 C3 C7 T C2 C4 C8

   (28)

   D exp d exp C1 C5 C7 T C2 C6 C8

   (29)

   Now from equation (12) displacement vector can

   be obtained as

   C9 exp C10T C11

   Where,

   C9 – constant of integration,

   (30)

   C10 3C1 C3 C5 4C7 , C11 3C2 C4 C6 4C8

   Figure 2: Plot of Energy density of Dark matter vs.

   Time for C10 C11 C12 1.01, 1.02, 1.03

   Figure 2 gives the plot of Energy density of Dark matter versus Cosmic time for the constants C10 C11 C12 1.01, 1.02, 1.03 .

   It can be seen from figure 2 that, at an early stage of evolution of the universe, energy density of dark matter dominates and at late times it approaches zero.

   Figure 1: Plot of Gauge function vs. Time for

   Pressure of HDE

   p 3 C 2 exp 2C

   T C C

   (33)

   C C C 1.01, 1.02, 1.03

   4 9 10

   11 13

   10 11 12

   Figure 1 shows the behavior of gauge function with change in cosmic time for C10 C11 C12 1.01, 1.02, 1.03 . It is observed

   that the gauge function is a decreasing function of the cosmic time. At an early stage, the gauge function is infinite and decreases with the

   evolution of the universe and vanishes as time

   Where, C13 is resolved constants.

   Figure 3 shows the plot of pressure of Holographic Dark Energy (HDE) versus cosmic time T for

   constants C9 C10 C11 0.01,1.02,1.03, C13 0 . As it is indicated in figure 3, the negative pressure of HDE increases with increase in time. The

   negative pressure indicates the accelerated phase of expansion of universe.

   Figure 3: Plot of Pressure of HDE vs. Time for

   C9 C10 C11 0.01,1.02,1.03, C13 0

   Energy density of HDE

   C15 exp 2C10T C11 C12 exp C10T C11

   (34)

   Figure 5: Plot of EoS parameter of HDE vs. Time for C9 C10 C11 C12 C15 1.01, 1.02, 1.03

   and C13 0

   Coincidence Parameter

   r m

   Here C C 3 C 2

   15 14 4 9

   C exp C T C

   Where, C14 is resolved constants.

   12

   10 11

   C15 exp

   2 C10 T C11

   C12 exp C10 T C11

   (36)

   Figure 4: Plot of Energy density of HDE vs. Time for C10 C11 C12 C15 1.01, 1.02, 1.03

   Equation (34) gives the energy density of HDE and figure 4 shows the plot of energy density of HDE versus cosmic time. It is seen from the figure 4 that

   Figure 6: Plot of Coincidence Parameter vs. Time for C10 C11 C12 C15 1.01, 1.02, 1.03

   Figure 6 gives the behavior of the Coincidence

   energy density of HDE is decreasing function of cosmic time.

   parameter r m

   against time. It shows that

   EoS Parameter of Holographic Dark Energy

   p

   3

   C92 exp 2C10 T C11 C13

   4

   C15 exp 2C10 T C11 C12 exp C10 T C11

   (35)

   Coincidence parameter changes at initial stage of evolution of universe. But after some time, it converges to some constant value and does not vary throughout the evolution. Hence, it can be stated that there is proper kind of interaction between energy densities of the Dark Matter and Holographic Dark Energy since their ratio attains constant value during evolution.

   Figure 5 gives the graph of EoS parameter of HDE as a function of time for s

   Spatial Volume V exp C10T C11

   2

   (37)

   C9 C10 C11 C12 C15 1.01, 1.02, 1.03 and

   C13

   0 . It is observed from the graph that EoS

   Equation (37) represents the spatial volume of the model, which is also depicted in figure 7. From

   parameter decreases as time increases.

   figure 7, it is observed that initiallythe spatial volume of the model is constant and expands exponentially. Volume tends to infinity as time

   tends to infinity. That means the universe must be expanding.

   Hubble parameter is calculated as in equation (39) and a graph of the Hubble parameter is shown in figure 9. The graph shows that the Hubble parameter decreases with time, it suggests that the expansion of the universe is accelerating.

   Expansion Scalar 3C10 exp C

   T C

   8 10

   11

   (40)

   Figure 7: Plot of Spatial volume vs. Time for

   C10 C11 1.01, 1.02, 1.03

   Average Scale Factor a exp C10T C11 (38)

   8

   Figure 10: Plot of Expansion Scalar vs. Time for

   C10 C11 1.01, 1.02, 1.03

   Equation (40) gives the expression for Expansion Scalar and figure 10 gives the graph of Expansion Scalar versus time. It is observed from the graph that the Expansion Scalar reduces as time increases and vanishes as time tends to infinity.

   Anisotropic Parameter A

   1C16

   (41)

   2

   m

   C10

   Figure 8: Plot of Average Scale Factor vs. Time for C10 C11 1.01, 1.02, 1.03

   Equation (38) gives the Average Scale factor. Figure 8 shows the plot of average scale factor with respects to cosmic time. It is observed from the figure that average scale factor is increasing function of time.

   The anisotropic parameter of the cosmological model given in equation (41) is obtained to be

   constant C10 0. It is observed that, the

   anisotropy parameter is constant and different from zero for C16 0 and vanishes at C16 0 . That

   means the universe is anisotropic throughout the evolution except for C16 0 .

   Hubble Parameter

   H C10 exp C T C

   8 10 11

   (39)

   Shear Scalar 2 C16 exp 2C T C

   2 10 11

   (42)

   Figure 9: Plot of Hubble Parameter vs. Time for

   C10 C11 1.01, 1.02, 1.03

   Figure 11: Plot of Shear Scalar vs. Time for

   C10 C11 C16 1.01, 1.02, 1.03

   Shear Scalar is obtained as expressed in equation

   (42) which is plotted against time in figure 11 and it is observed from the graph that the shear scalar decreases as time increases and vanishes as time tends to infinity.

   Directional Hubble parameters are obtained as

   H1 C7 exp C10T C11

   in understanding the dynamics of strong gravitational fields. The commonly used or standard energy conditions are outlined as follows:

   Null Energy Condition (NEC)-

   p 0

   Weak Energy Condition (WEC)-

   H2 C1 C7 exp C10T C11

   0, p 0

   H3 C1 C3 C7 exp C10T C11

   Strong Energy Condition (SEC)-

   H C C C exp C T C

   p 0, 3 p 0

   4 1 5 7

   10 11

   Dominant Energy Condition (DEC)-

   p

   Red shift-

   For red shift, average scale factor a is related to

   a0 by the relation,

   1 z a0

   a

   Where, subscript 0 denotes the present phase and

   a0 is the present scale factor

   Hence, we get,

   Red shift z 1 exp C10 T0 T

   (43)

   8

   Here, T0 is the age of universe at present time.

   Figure 12: Plot of Energy conditions for HDE vs.

   T 8

   C10

   Omz

   1 z C11

   C10

   diagnostic- In the context of the

   (44)

   Omz

   Time

   From figure 12, it is observed that the obtained cosmological model satisfies all the energy conditions.

   diagnostic, the Hubble parameter serves as the foundational element. This diagnostic is independent of the model and helps to distinguish between different dark energy models, such as the

   CDM model and its alternatives. In terms of

   observations and cosmological parameters, the

   Omz diagnostic is then defined as,

   Hz 2

5. THERMODYNAMICAL BEHAVIOUR AND ENTROPY OF THE UNIVERSE

   The total entropy associated with the universe, including both the interior and the horizon boundary, increases monotonically with time [24]. By applying the first and second laws of thermodynamics to a system enclosed by a horizon

   with volume V , we get,

   2

   H

   1

   3

   Omz o

   dS dV pdV

   (46)

   1 z 1

   2

   Where, m , and S are the temperature

   C10 1 z8

   H 2

   and entropy respectively [24]. Again, above

   8

   Omz

   1 z3

   o

   2

   1 Ho

   (45)

   equation can be written as,

   dS dp V pdV

   From equation (45), it can be observed that Omz increase as z decrease, indicating that dark energy is dynamically increasing and is playing a more significant role in the recent expansion of the universe.

   Energy Conditions- The examination of energy

   Since dark matter and dark energy are often represented as perfect fluids, this assumption helps simplify the analysis of their behaviour in cosmological models. The perfect fluid in thermodynamics can be defined as,

   dp p d

   conditions plays a crucial role in analyzing the

   behavior of both null and time-like geodesic congruence. These conditions are also fundamental

   Hence, from above we get,

   dS 1 dp V p V d

   This leads to,

   p V

   Universe increases continuously throughout cosmic

   dS d

   evolution since

   S 0 .

   S

   On integrating above, we get

   S p V

   The thermodynamics of the Universe, particularly its entropy, does not depend on the properties of individual fluid components. Instead, it is determined by the total matter density and the isotropic pressure of the fluid. Let us denote the

   entropy density by S so that

   S S p 1

   (47)

   Figure 14: Plot of Entropy vs. Time for 0 1

   V

   By expressing the entropy density in terms of temperature, the first law of thermodynamics can be reformulated as follows:

6. CONCLUSION

d V dV 1 d V

(48)

In this paper, we have obtained a cosmological

model by solving field equations using a

This yields the expressions for the temperature and entropy density as follows,

transformation. Along with the gauge function, Energy density and pressure of both Dark matter

1 and HDE are calculated. Gauge function is found to

1 , S 1 1

(49)

be a decreasing function of time. At an early stage,

Thus, from equations (32), (34), the Hawking temperature and Entropy density S can be written as,

the gauge function is infinite and decreases with the evolution of the universe and vanishes as time tends to infinity. Hence, the model has singularity

when the cosmic time is zero. It is observed that

C15 exp 2C10T C11 1

1

S 1 C15 exp 2C10T C11 1

(50)

(51)

Energy density of DM is decreasing function of time. At an early stage of evolution of the universe, energy density of DM was dominating and it vanishes as time tends to infinity. Negative pressure of HDE is obtained to be increasing as cosmic time increases which indicates accelerated expansion of the universe. Some Physical and Kinematical properties of the obtained model are calculated and discussed. Plot of EoS parameter is observed to be decreasing function of cosmic time. Also, Coincidence parameter of HDE is obtained which shows that there is proper kind of interaction between energy densities of the Dark Matter and Holographic Dark Energy since their ratio approaches to a constant value during evolution. It

Figure 13: Plot of Thermodynamic Temperature vs. Time for 0 1

In the derived model, the thermodynamic temperature is found to be a decreasing function of time (Figure 13). It starts with an infinite value at the initial stage of the Universe and gradually approaches a constant value as the Universe expands. This behavior supports the second law of thermodynamics.

Furthermore, Equation (51) describes the entropy density of the proposed dark energy model. The entropy density is positive and decreases over time, mirroring the behavior of the energy density (as shown in Figure 14. The total entropy of the

is found that the universe must be expanding and this expansion of the universe must be accelerating since spatial volume increases exponentially as time increases and Hubble parameter decreases with increase in time. This idea stems from the observation that all galaxies seem to be receding from each other at an accelerating pace. Other properties like Average Scale Factor, Expansion Scalar, Anisotropic Parameter, Shear Scalar, Red shift are studied. It is observed from the value of Aniotropic parameter that the universe is anisotropic throughout the evolution except for C16 0 . Also, energy conditions for HDE are

studied and plotted against cosmic time. It is concluded that each energy condition is satisfied.

Study Omz diagnostic shows that, Omz

increases as redshift z decreases, suggesting that dark energy is evolving dynamically and becoming increasingly dominant in driving the universe's accelerated expansion at late times. The thermodynamic behavior of the derived dark energy model aligns with the second law of thermodynamics. The temperature decreases over time, approaching a constant value, while the entropy density remains positive and decreases similarly to the energy density. Despite this, the total entropy increases continuously, confirming the model's physical consistency and thermodynamic validity.

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