COSMIC ASPECTS OF SHARMA-MITTAL HOLOGRAPHIC DARK ENERGY MODEL IN BRANS-DICKE THEORY OF GRAVITY

We investigate the cosmological scenario involving spatially homogeneous and anisotropic Bianchi type-V I 0 space-time in the context of the Sharma-Mittal holographic dark energy model within the framework of Brans-Dicke’s theory of gravitation. In order to achieve this objective, the Hubble, deceleration, equation-of-state parameters have been discussed. The deceleration parameter ( q ) is used to measure the pace at which the expansion of the universe is accelerating. The equation-of-state parameter ( ω smhde ) characterizes the quintessence and vacuum areas of the universe. All the parameters demonstrate consistent behaviour following the Planck 2018 data. We assess the dynamical stability by defining the squared speed of sound and examining its behaviour. In addition, the energy conditions and the variation of ω smhde and ω ′ smhde in the model indicate the present accelerating expansion of the universe.


INTRODUCTION
The phenomenon of accelerated expansion of the universe has been thoroughly demonstrated by several observations [1]- [3].Studies imply that the universe is spatially flat and consists of two main components: dark energy (DE) with negative pressure, dust matter composed of cold dark matter (CDM) and baryons.To understand the nature of DE, it is necessary to determine whether it arises from a cosmological constant (Λ) or a dynamical model.The dynamical DE models may be distinguished from the cosmological constant by using the equation of state (EoS) parameter ω DE = p DE ρ DE , where p DE represents the pressure and ρ DE represents the energy density of DE.Multiple possibilities for differential evolution have been suggested (Copeland et al. [4]).The data analysis of SNeIa demonstrates that these dynamical models are more consistent with the current understanding of the universe compared to Λ.An alternative approach involves altering the geometric component of the Einstein-Hilbert action, which is referred to as modified theories of gravity, to analyze the expansion phenomena.For a comprehensive examination of DE and modified theories of gravity, please refer to the sources cited as [5]- [7].The developments in the exploration of black hole theory and string theory have led to the formulation of the holographic principle.This principle suggests that the number of possible configurations of a physical system should be limited as well as that this limitation should be determined by the system's surface area rather than its volume.Additionally, the holographic principle suggests that there should be a restriction on the system's lowest energy state.
The holographic DE (HDE) is a very intriguing dynamical concept that is founded upon the holographic principle.The validity of HDE has been evaluated and verified by many astronomical methodologies, including using the anthropic principle (Huang and Li [8]).Incorporating the holographic principle into cosmology allows for determining the maximum amount of entropy present in the universe.Li [9] put the following limit on the DE density, as stated by Cohen et al. [10] the symbol m p represents the decreased Planck mass, 3d 2 indicates a numerical constant, and L represents the IR-cutoff.Several types of IR-cutoff have been investigated in academic studies, including the Hubble horizon H −1 , event horizon, particle horizon, conformal universe age, Ricci scalar radius, and Granda-Oliveros cutoff [11]- [12].The HDE models, using various infrared cutoffs, provide a modern understanding of the universe's acceleration.They additionally demonstrate that the transition redshift value, which marks the transfer from a previous deceleration phase (q>0) to the current acceleration phase (q<0), corresponds with contemporary EEJP.1(2024)Y. Aditya, et al.
observable data.Nojiri and Odintsov [13] proposed a technique to merge the initial and final phases of the universe by using generalized HDE and phantom cosmology.They have now expanded this notion to what they refer to as Hinflation [14].Various formulations of entropy have been used in recent decades to construct and analyze cosmological models.Multiple innovative models of HDE have been developed, such as the Tsallis HDE [15,16], the Sharma-Mittal HDE (SMHDE) [17], and the Renyi HDE model [18].Numerous researchers have evaluated different cosmological models of new HDE models [19]- [30].Jawad and Sultan [31], Sharma [32], and Drepanou et al. [33] have investigated SMHDE models within different gravitational theories.The researchers Sadeghi et al. [34] examined the dynamic formations of HDE within the framework of Brans-Dicke's theory of gravity, using the Tsallis and Kaniadakis methodologies.
In the last few decades, universes that are both spatially homogeneous and anisotropic have attracted a lot of attention from theoretical cosmologists.According to Akarsu and Kilinc [35], the main empirical data from CMBR (Bennett et al. [36]) supports the idea that the universe is transitioning from a non-uniform to a uniform phase.Moreover, it is believed that the isotropic FRW model may not provide a complete and correct picture of matter in the early universe.It is essential to take into account anisotropic space-times to objectively evaluate cosmological models for their ability to attain the observed degree of homogeneity and isotropy.Because of its anisotropic character, researchers have given a lot of attention to homogeneous but not necessarily isotropic Bianchi type (BT) cosmological models.In the framework of anisotropic Bianchi type space-times, several renowned researchers have recently proposed fascinating cosmological models that include DE. several researchers (Ref.[37]- [49]) have looked at anisotropic cosmological models in various scenarios.
In this work, we take into account the BT-V I 0 universe filled with matter and SMHDE within the framework of Brans-Dicke's theory of gravitation, motivated by the previously mentioned findings and discussion.This paper's work is organized in the following way: Section-2 contains the BT-V I 0 metric and field equations of the model using anisotropic SMHDE fluid and matter.In addition, we constructed the SMHDE model and found the solutions to the field equations in section-3.Various cosmological parameters that constitute our model are presented in section-4.In the last section, we derive a few conclusions based on our results.

METRIC AND FIELD EQUATIONS
The Brans-Dicke [50] field equations in the presence of matter and DE are given by and ϕ ,k ;k = 8π T + T (3 + 2w) −1 .Also, the energy conservation equation is The energy-momentum tensor for the matter and DE are respectively defined as Here ρ m , ρ smhde are the energy densities of matter and the dark energy respectively.p smhde is the pressure of the dark energy.In a comoving coordinate system, from equations (4), we get where ρ m , ρ smhde and p smhde are the functions of cosmic time t only.
We consider the geometry of the universe as spatially homogeneous and anisotropic BT-V I 0 line element which can be written as where A, B and C are functions of cosmic time t only.The following are the some of physical parameters which are useful in finding the solution of field equations for the BT-V I 0 space-time given by Eq. ( 6).The average scale factor a(t) and volume V of the BT-V I 0 space-time are defined as Anisotropic parameter A h is given by where is mean Hubble's parameter.Here and after an overhead dot denotes differentiation concerning cosmic time t.Expansion scalar (θ) and shear scalar (σ 2 ) are defined as where u i = (1, 0, 0, 0) is the four-velocity in the comoving coordinates.The deceleration parameter is given by In the comoving coordinate system, with the help of ( 5), the field equations (2) for the metric ( 6) can be written as Ä Ȧ Ḃ Also the energy conservation equation ( 3) can be written as where an overhead dot denotes differentiation with respect to time t and EoS parameter of DE is ω smhde = p smhde ρ smhde .From ( 16) by taking the integrating constant as unity, we get Using equation (19), the field equations ( 12) to (17) reduce to EEJP.1(2024) Y. Aditya, et al.

MODEL AND COSMOLOGICAL PARAMETERS
The field equations ( 20)-( 23) form a set of four distinct equations involving six variables: A, B, ρ smhde , ω smhde , ρ m , and ϕ.To get a deterministic outcome for the nonlinear field equations in our model, we impose the following reasonable physical constraints.we consider the fact that expansion scalar θ is directly proportional to shear scalar σ which leads to a relation between the metric potentials as follows k represents a positive constant that accounts for the anisotropy of space-time.Collins et al. [51] have determined that in a spatially homogeneous space-time, the normal congruence to the homogeneous expansion adheres to the constraint that the ratio of the shear stress (σ) to the Hubble parameter (H) remains constant.
In addition, it is common in the literature to employ a power-law relationship between scalar field ϕ and average scale factor a(t) of the form (Johri and Sudharsan [52]; Johri and Desikan [53]) ϕ ∝ [a (t)] n where n denotes a power index.Many authors have looked into different aspects of this type of scalar field ϕ.Given the physical significance of preceding relationship, we employ the following assumption to reduce the mathematical complexity of the system Using the relations ( 24) and ( 25) in Eqs. ( 20) and ( 21), we obtain the metric potentials as where The scalar field of the model is Now metric (6), with the help of metric potentials in Eq. ( 26), can be written as Eq. ( 28) represents a spatially homogeneous and anisotropic BT-V I 0 SMHDE model within the framework of Brans-Dicke theory of gravitation with the following physical parameters.The average scale factor a(t) and volume V (t) of the model are, respectively, given by The average Hubble's parameter H and expansion scalar θ are obtained as The shear scalar σ 2 and average anisotropic parameter A h are given by From the above parameters it is observed that both the spatial volume and average scale factor of the universe exhibit the expansion of the universe.Furthermore, during the first epoch, which is when t= 0, all values become finite.However, as t approaches infinity, they diverge.It is worth mentioning that when k= 1, the model becomes shear-free and isotropic, as shown by the conditions σ 2 = 0 and A h = 0.
As a dynamical DE component, we assume Sharma-Mittal holographic DE.It is defined by (Sharma and Mittal [54]) and is formulated using Sharma-two-parametric Mittal's entropy where κ = 4πL 2 and L represents the IR cutoff.R and δ are two free parameters in this case.At the appropriate d 1 limits, Renyi and Tsallis entropies can be recovered.Sharma-Mittal entropy is transformed into Renyi entropy in the limit d 1 → 0, and Tsallis entropy in the limit d 1 → 1 − δ.According to Cohen et al. [10], the relationship between the system entropy and the IR and UV cutoffs yields the energy density Using the above equation and the Hubble horizon cutoff L = 1 H , we can calculate the energy density of the Sharma-Mittal HDE model (Jahromi et al. [17]) as follows: where C 2 denotes the free parameter.Using Hubble parameter H(t) in the above Eq.( 34), we get the energy density of SMHDE of our model as Using Eqs. ( 34), ( 27) and ( 26) in Eq. ( 15), we get the energy density of matter as Using Eqs. ( 34), ( 27) and ( 26), from Eq. ( 20) and ( 21) we obtain the EoS parameter of SMHDE as 4. PHYSICAL DISCUSSION This section examines the expansion of the universe by analyzing cosmological parameters such as the scalar field, EoS ω smhde , squared sound speed v 2 s , deceleration q parameters, and the ω smhde − ω ′ smhde plane for the anisotropic SMHDE model.Scalar field: Figure 1 indicates the behavior of the scalar field in terms of cosmic time for various values of parameter k.The scalar field exhibits a positive value and experiences a steady decline throughout its development.The decreasing nature of the scalar field indicates the increasing pattern of kinetic energy in this model.Furthermore, we have seen that when the parameter k increases, the scalar field decreases.

EoS parameter:
The equation of state parameter (ω) is often used for classifying the many phases of the expanding universe.The EoS parameter, represented as ω= p ρ , is a measure of the relationship between pressure (p) and energy density (ρ) of a given matter distribution.The decelerated and accelerated phases consist of the following time intervals: Decelerated phases, such as those involving cold dark matter or dust fluid (ω is zero), indicating the radiation era for (ω is between 0 and 1/3), and the fluid is characterized as stiff for ω = 1.Accelerating phase, such as the cosmic constant/vacuum period (ω is -1), which corresponds to the quintessence period (when −1 < ω < −1/3), it is known as the phantom era (ω < −1), indicating a quintom period characterized by the mixture of both quintessence and phantom components.
The EoS parameter of SMHDE with Hubble horizon cutoff is given in Eq. (37).In Fig. 2, we investigate the evolution of EoS parameter ω smhde in terms of cosmic time t for different values of k.Fig. 2 shows that initially ω smhde starts from DE era, varies in quintessence region −1 <ω smhde < − 1/3 and phantom region ω smhde < −1.As the parameter k increases the EoS parameter of our model enters into phantom region.Squared sound speed: The squared speed of the sound parameter is defined as The sign of this parameter is essential when considering the stability of DE models.The stability of the DE model is determined by the positive signature of v 2 s .If the signature is negative, the model becomes unstable.By substituting the energy density and EoS parameter from equations ( 35) and (37) into the equation for squared sound speed (v 2 s ) provided by equation ( 38), we do a graphical analysis of v 2 s for our model.Figure 2 illustrates the relationship between the square of the velocity, denoted as v 2 s , and the time t.The trajectories exhibit positive behaviour throughout the model's development.Therefore, this demonstrates the stability of our model.ω smhde −ω ′ smhde plane: We examine the ω smhde − ω ′ smhde plane, where ω ′ smhde represents the rate of change of the EoS parameter ω smhde concerning the natural logarithm of the scale factor 'ln a'.Caldwell and Linder [55] propose using this framework to examine the cosmic evolution of the quintessence DE scenario.Moreover, it has been observed that the ω smhde − ω ′ smhde plane can be separated into two distinct areas: thawing (ω smhde < 0, ω ′ smhde > 0) and freezing (ω smhde < 0, ω ′ smhde < 0).The freezing zone exhibits a more accelerated phase of cosmic expansion in comparison to the thawing area.
Figure 4 depicts the relationship between the ω smhde − ω ′ smhde plane and various values of k.Strong energy conditions (SEC) : ρ smhde + p smhde ≥ 0, ρ smhde + 3p smhde ≥ 0, Null energy conditions (NEC): ρ smhde + p smhde ≥ 0, Weak energy conditions (WEC): ρ smhde ≥ 0, ρ smhde + p smhde ≥ 0, Figure 5 illustrates the energy conditions of our SMHDE model.It is seen that the WEC satisfies the condition ρ smhde ≥ 0. Also, Fig. 5 demonstrates that the SEC ρ smhde + 3p smhde ≥ 0 is not fulfilled.This phenomenon, resulting from the universe's acceleration in its latter stages, aligns with current observational findings.Deceleration parameter q(t) Deceleration parameter: The expansion of the universe may be determined by using the dimensionless cosmological parameter referred to as the deceleration parameter (DP).When DP has positive values, the model slows down in the usual manner.However, when q is equal to zero, the model grows at a consistent pace.The model demonstrates accelerated expansion when the value of q is between −1 and 0, and a super-exponential expansion

Figure 6 .
Figure 6.Plot of deceleration parameter versus cosmic time t for A 0 = −3900.69.