A STUDY OF EVOLUTION OF COSMOLOGICAL PARAMETERS BASED ON A DARK ENERGY MODEL IN THE FRAMEWORK OF BRANS-DICKE GRAVITY

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INTRODUCTION
Based on the observations regarding the Type 1a supernova, it was established that the universe is expanding with acceleration and the present phase of acceleration was preceded by a phase of deceleration [1][2][3][4][5].It has been a great challenge since then to explain the cosmic acceleration.According to the general theory of relativity (GTR), there is a strange form of energy, named Dark Energy (DE), which causes this acceleration.One of the parameters representing DE is the cosmological constant (Λ) in ΛCDM model which is known to be consistent with several important astrophysical observations related with supernova, baryon acoustic oscillation and cosmic microwave background.But it cannot account for the structure formation at small scales.DE models with scalar fields, such as phantom, quintessence and tachyon models were formulated to account for the phenomenon of late-time acceleration by Copeland et al. [6].
Several models were formulated to explain the dynamical behavior of DE [7][8][9].One of the most significant among the theories which are modified versions of GTR is the one formulated by Brans and Dicke, which supports Mach's principle and weak equivalence principle [10].The theory of  ,  gravity is another such important theory of modified gravity [11,12].Brans and Dicke made a pioneering contribution to the theoretical exploration of the scalar tensor theories, and the elegant theoretical framework built up by them is known as Brans-Dicke theory of gravity.The scalar field () in this theory evolves with time and we have  1/ where  stands for the gravitational constant.A dimensionless constant  in BD theory couples the scalar field with gravity.BD theory can generate the results of GTR if  is constant and  is infinitely large [13].An important role is played by the scalar field  in explaining the characteristics of the inflationary universe [10,14,15].Based on BD theory, various cosmological models have been formulated by several researchers to account for the observed features [16][17][18][19][20].The dynamics of Bianchi Type-V universe have been studied by Prasad et al. under the framework of BD theory [21].DE models in BD theory have also been constructed where the dimensionless parameter  is a function of the scalar field () [22].
The main objective of the present study is to construct a cosmological model, in the framework of Brans-Dicke theory of gravity, to find and analyze the features of time evolution of different cosmological quantities.For this purpose, we have built a model starting from an empirical expression for the deceleration parameter () which is based upon the fact that  undergoes a signature flip as time goes on.Expressions for Hubble parameter () and scale factor () have EEJP.3 (2023) been derived from that ansatz for  which clearly demonstrates (as a function of time) the change of the mode of expansion of the universe from deceleration to acceleration.The values of the arbitrary constants involved in this model have been calculated by using the presently accepted values of some cosmological quantities obtained from observational data.Considering the homogeneity and isotropy of the universe at large scale, we have used FLRW metric to obtain the field equations.To extract information from these equations, a power-law relation between the scalar field () and the scale factor () has been used in the present study.Employing these equations, we have derived expressions for energy density, cosmic pressure, cosmological constant, gravitational constant, equation of state (EoS) parameter and shown their time variation graphically.For a cosmographic analysis, we have shown the variation of jerk, snap and lerk parameters in terms of redshift () graphically.To analyze the DE model characteristics, we have employed the theoretical tools named Statefinder diagnostic and Om diagnostic.It is observed that there is a gradual transition from quintessence dark energy regime to a phantom dark energy regime in the universe.A novel finding is that, the signature flip of the cosmological constant is almost simultaneous with the signature flip of the deceleration parameter, pointing towards a role of dark energy (represented by the cosmological constant) in causing cosmic acceleration.

FIELD EQUATIONS
The action for the Brans-Dicke theory of gravity is expressed as, where,  represents the BD scalar field, which is reciprocal of the gravitational constant ( In the above equation,  denotes the energy-momentum tensor,  is the metric tensor and  stands for the Ricci tensor.The semicolon (;) in this equation stands for the covariant derivative and the symbol □ represents the d'Alembert operator.
Through a variation of the scalar field () in BD action (i.e.,  in eqn. 1) we get the following equation.
where,    is the trace of energy-momentum tensor  .The matter content of the universe is considered to be that of a perfect fluid distribution, which is given by, where,  and  are cosmic fluid's energy density and pressure respectively.ℎ    where  represents the four-velocity vector of cosmic fluid with    1.It has been concluded from recent cosmological findings that the observable universe is homogeneous and isotropic at large scales.To take into account this aspect of the cosmos, we have chosen FLRW metric to represent the space-time geometry of the universe.This metric is given by, In equation (5), the symbol  denotes the scale factor.The coordinates (co-moving) of the spherical polar system are , ,  .The symbol  is regarded as the curvature parameter which has three values 1, 0, 1, denoting respectively three characteristics of the expanding universe, namely open, flat and closed.
In equations ( 6), ( 7) and ( 8),  represents the Hubble parameter (given by,  ).A dot over any parameter represents the conventional derivative of that parameter with respect to time ().
Based on the three above equations, we have obtained the following expressions for the parameters Λ,  and , represented as functions of ,  and their time derivatives.

SOLUTION OF FIELD EQUATIONS
Considering the change of phase of the cosmic expansion from deceleration to acceleration [1][2][3][4][5], we assume an ansatz regarding the time evolution of the deceleration parameter ().It is given by, where , ,  0. This function of time,   , undergoes a change of sign (with time) from positive to negative.
Since   1 , where  is the Hubble parameter, we may write equation ( 12) as, Integrating equation ( 13), we get the following expression for the Hubble parameter.
where  is a constant of integration.Substituting  in equation ( 14) we get the following differential equation.
Equation ( 15) can be solved analytically for  0, which corresponds to its simplest solution.That solution leads to the following expression for the scale factor ().
where  is a constant of integration.The expressions for ,  and  (in terms of time) are dependent upon the parameters A, B, b and n.The interdependence of these parameters can be determined by using the values of the cosmological quantities ( ,  ,  ) obtainable from observational data.The symbols,  ,  ,  denote respectively the values of , ,  at the present time (i.e.,   ) where  is the present age of the universe.
Using the fact that   at   in equation ( 12), we get, Similarly, using the fact that   at   in equation ( 14), we get, EEJP. 3 (2023) Solving equations ( 17) and (18) for  and  one gets, Thus,  and  are found to be dependent upon the parameter .
Using the fact that   at   , in equation ( 16), we get, where,  is the value of the scale factor at the present time.Using equation ( 21), we obtain  in terms of  and  , as given below.
Substituting the equations ( 19) and (20) in equation ( 22) we get, Thus, among the four arbitrary parameters of this model (A, B, b and n), it is observed from equations ( 19), ( 20) and (22A) that A, B, b can be expressed as functions of n.We have chosen  1 in the present study.Hence ,  and  are dependent on  only.Taking   1 [18], we obtain  from equation (20), satisfying the condition  0, since  0 for an accelerating universe and  0 by definition.For  to be positive, we must have    0 (according to eqn.17).It means  0, leading to  1 (using  and  0).Thus, the range of values for  is obtained as, 0  1.
An ansatz for the scalar field parameter (), where it has a power-law relation with the scale factor (), has been used in the present formulation, based on some recent studies [17][18][19][20].This ansatz is, where,  is an arbitrary constant and  , where  is the present value of the gravitational constant .In BD theory, the gravitational term   1/   .Based on equation (23), the first and the second order time-derivatives of  are given by the following two equations.
Putting   ,   ,   ,   in equation ( 27), we get the following expression for  in terms of .

𝑚 (29)
Hence, along with ,  is also a free parameter in this model.Λ,  and  are dependent on both  and , while ,  and  are dependent on  only.
4. COSMOGRAPHIC ANALYSIS For a cosmographic analysis, we have determined the time dependence of jerk  , snap  and lerk  parameters, which are defined as,    ,    and    respectively [23,24].These are dimensionless quantities which depend upon the scale factor () and its third, fourth and fifth order time derivatives.They allow us to determine the rate of cosmic expansion more accurately by a model independent analysis of the evolution of the universe.Using equation (16) we get the following expressions (eqns.35, 36 and 37) for j, s and l respectively.

STATEFINDER AND OM DIAGNOSTICS
Sahni et al. introduced two parameters  and  which are defined as [25], The expression for  is the same as the expression for the jerk parameter   .Like  and , these dimensionless parameters are functions of the scale factor () and its higher order derivatives.These parameters help us to differentiate between any DE model and the  model.As per statefinder diagnostic, (, ) and (, ) trajectories are plotted in   and   planes, respectively, to analyze the evolution of the universe under the frameworks of different models of dark energy.Since ,  and  involve only the scale factor () and its time derivatives of higher orders, this method is independent of the framework of gravity.Thus, this diagnostic is model independent.
Using equations ( 12) and ( 16) in equations ( 38) and ( 39),  and  are expressed as,   [26].In this theory, a parameter called   is defined as where,   and  ≡ 1 is the redshift parameter.The positive curvatures of   trajectories imply phantom behaviour while negative curvatures of   trajectories indicate quintessence behaviour of dark energy.Constant value of   for a model indicates that its behaviour is the same as that of the  model.

RESULTS AND DISCUSSION
To plot some cosmological quantities graphically with respect to redshift (), we have derived the following relation (redshift versus time) based on equation (16).
Since the behaviour of the deceleration parameter () determines how the phase of decelerated expansion of the universe changes into the phase of accelerated expansion, we have derived the following expression for Λ as a function of , using equations ( 12), ( 14) and (26).
To validate the present model, the arbitrary constants associated with the formulation have been so adjusted that the values of  ,  ,  ,  ,  are obtained correctly from the model, as discussed in Section-3 of this article.For this purpose, we have used the following values of these parameters [18].For all calculations we have used  2, leading to  0.61, which are consistent with a recent study in the framework of BD theory [19].Another study by Goswami et al, in BD framework, also used positive values of  [18].As per equation (23),  increases with time if  0, implying that  (≡ 1/) decreases with time, as shown by Figure 5, which is in agreement with some recent studies based on different theoretical models and experimental observations [18,27,28].Figure 3 shows that  decreases with time, as obtained from many other studies [9,17,19,29,30].Figure 4 shows that Λ rises very steeply in the early universe, becoming positive from negative and then changes slowly.This behavior is consistent with the findings of various other studies [31][32][33][34].As per Figure 6,  is negative and decreases gradually with time, with    0.8, which is consistent with values obtained from observational data [35,36].According to the plots of Figure 6, the universe presently has a quintessence dark energy regime ( 1) and it is making a gradual transition towards a phantom dark energy regime ( 1).It is negative and it becomes less negative with time.Negative pressure is associated with DE, causing the accelerated expansion of the universe.This behavior is consistent with the findings of several studies [37,38].Positive values for  and  and negative values for  represent accelerated expansion [39].Figures 8 and 10 show  and  to have positive values for all values of .As per Figure 9, the values of the snap parameter () undergo a transition with time from negative to positive (based on the fact that  decreases as  increases).From equation (36) it is found that the value of the snap parameter at the present time (i.e., at   ) is negative for  0.227, which is consistent with the plot for  0.3 in Figure 9 where  is negative at   (i.e.,  0).It provides a clear guideline for choosing the values of  for an accelerating universe.It has been shown in Section-3 of this article that 0  1, based on the requirements for a proper parameterization of the ansatz that we have chosen for the deceleration parameter (represented by eqn.12). Figure 8 shows that, the rate of increase of  with time (i.e., as  decreases) is larger for smaller values of the parameter .We observe almost the same behaviour for the plots of  in Figure 9.In Figure 10, the values of  initially decrease with time and, at some point of time in future (i.e.,  0) the values increase with time, having the largest rate of rise for -1.0x10 the lowest value of the parameter .For plotting the graphs in Figures 8-10, we did not have to express jerk (), snap () and lerk () parameters as functions of redshift ().Expressions for these parameters, in terms of , would have been extremely complicated and difficult to handle.We generated datasets for , ,  and  as functions of time (using eqns.35-37, 43 respectively) for three different values of the parameter , using Microsoft Excel.Based on these datasets, we have plotted , ,  as functions of .
Figures 11 and 12 show the plots of (, ) and (, ) trajectories.Their natures are found to be close to those obtained from a different model in the BD framework [19].In Figure 11, trajectories begin in the Chaplygin gas (CG) region ( 0,  1), and enter the quintessence region ( 0,  1).Then they merge together and reach the Chaplygin gas region again after passing through the point (0, 1) which stands for the ΛCDM model, for all values of .In Figure 12, the line  1 represents the evolution of ΛCDM model.The trajectories, starting from the region of decelerated expansion ( 0), are found to reach and cross the  1 line for all three values of .Here,  0.5 line represents matter-dominated era.These two figures show that the constructed model is presently behaving as a quintessence dark energy model.Its predictions for the future evolution of the universe will be like the Chaplygin gas model, after passing through an intermediate stage having the behavior which is consistent with that of the ΛCDM model.Figure 14 shows the variation of the cosmological constant (Λ) as a function of the deceleration parameter ().These plots are based on equation (44).It is observed that, as Λ changes its sign from negative to positive,  undergoes a signature flip from positive to negative, indicating the phase transition (i.e., deceleration to acceleration) to be associated with some phenomena involving dark energy which is represented by Λ.

CONCLUSION
In the present article, we have constructed a cosmological model using FLRW metric for zero spatial curvature, in the Brans-Dicke framework with cosmological constant (Λ).Solutions of the field equations have been obtained from a proper parameterization of the deceleration parameter     with , ,  0. The choice of this expression is based on the phenomenon of signature flip of the deceleration parameter as obtained from astrophysical observations.The characteristics of the physical and geometrical parameters have been depicted graphically.We have a detailed interpretation of these graphs in Section-6 of this article.A significant finding of the present study is that the time at which the deceleration parameter changes its sign (from positive to negative) is almost the same as the time at which the cosmological parameter changes its sign (from negative to positive), indicating clearly that the change of phase from decelerated expansion to accelerated expansion is governed by some dark energy dynamics which is generally regarded as being represented by the characteristics of the cosmological parameter (Λ), in calculations under different gravitational frameworks.The plots based on the statefinder parameters show that, for all values of the parameter , the (, ) trajectories enter the quintessence region from Chaplygin gas region, ending up finally in the Chaplygin gas region after passing through the point representing Λ.Thus, the future characteristics of the universe, based on this model, is like those obtained from Chaplygin gas model.The (, ) trajectories start evolving from a region close to SCDM and move ahead crossing the line representing Λ.The findings of the present study are sufficiently consistent with the findings of models constructed under various other theoretical frameworks, and they are also in reasonable agreement with observational data.As a future extension of the present work, we have plans to use some new ansatzes representing deceleration parameter to determine the time evolution of various cosmological quantities and find their average behaviour under different theoretical frameworks of gravitation.The construction of the present model might be helpful to the researchers in studying the evolution of the universe under the Brans-Dicke framework by formulating more such models in future.

Figure 1 .Figure 2 .
Figure 1.Plots of Hubble parameter () versus time Figure 2. Plots of deceleration parameter () versus time Figures 1 and 2 show the time evolution of Hubble parameter () and deceleration parameter () respectively, for three values of the parameter n.It is observed that  decreases with time, which is consistent with recent studies based on various models[7,[17][18][19].The deceleration parameter shows a signature flip indicating a change from decelerated expansion to accelerated expansion, in accordance with the observed features[7,[17][18][19].

Figure 3 .Figure 4 .
Figure 3. Plots of energy density () versus time Figure 4. Plots of cosmological Constant (Λ) versus time Figures 3, 4, 5 and 6 show respectively the behavior of energy density (), cosmological constant (Λ), Gravitational constant () and EoS parameter () with respect to time, for three values of the parameter .These four cosmological quantities depend also on the Brans-Dicke parameter .It is found from equation (29) that  0 for  1.11.Using equation (32) we have found that, for some values of , in the range of  1.11,  becomes negative.Due to this discrepancy regarding the sign of  values, we have chosen to use  values belonging to the range of  1.11.For all calculations we have used  2, leading to  0.61, which are consistent with a recent study in the framework of BD theory[19].Another study by Goswami et al, in BD framework, also used positive values of [18].As per equation(23),  increases with time if  0, implying that  (≡ 1/) decreases with time, as shown by Figure5, which is in agreement with some recent studies based on different theoretical models and experimental observations[18,27,28].Figure3shows that  decreases with time, as obtained from many other studies[9,17,19,29,30].Figure4shows that Λ rises very steeply in the early universe, becoming positive from negative and then changes slowly.This behavior is consistent with the findings of various other studies[31][32][33][34].As per Figure6,  is negative and decreases gradually with time, with    0.8, which is consistent with values obtained from observational data[35,36].According to the plots of Figure6, the universe presently has a quintessence dark energy regime ( 1) and it is making a gradual transition towards a phantom dark energy regime ( 1).

2 Figure 5 .Figure 6 .Figure 7
Figure 5. Plots of gravitational constant () versus time Figure 6.Plots of EoS Parameter () versus time Figure 7 depicts the variation of cosmic pressure () with respect to time for three values of .It is negative and it becomes less negative with time.Negative pressure is associated with DE, causing the accelerated expansion of the universe.This behavior is consistent with the findings of several studies[37,38].

Figure 11 .Figure 12 .
Figure 11.Plots of (, ) trajectories for statefinder diagnostic Figure 12.Plots of (, ) trajectories for statefinder diagnostic Figure 13 depicts the variation of   as a function of  for different values of .It is known that, if the curvature of   is positive with respect to , the model is a phantom dark energy model ( 1) and, if the curvature is negative, it is a quintessence dark energy model ( 1).For zero curvature, it represents the  model[40][41][42].It is observed that,   rises steeply as  increases and it decreases slowly beyond  0.75 (approximately).Its decreasing behavior at  0 (i.e., the present time) indicates that our model has the characteristics of a quintessence DE model at the present time.Since  decreases with time, Figure13shows a transition of the model characteristics from those of a quintessence DE model to those of a phantom DE model, which is consistent with the inferences drawn from Figure6.This transition takes more time to occur for greater values of the parameter .

Figure 13 .Figure 14 .
Figure 13.Plots of Om () versus redshift () Figure 14.Plots of cosmological constant (Λ) versus deceleration parameter () 1/).R stands for the Ricci scalar.The symbol Λ represents the cosmological constant.ω is called the BD parameter which represents a dimensionless coupling constant.The symbol  denotes the ordinary derivative of  with respect to  .The matter Lagrangian density is denoted by the symbol  .The field equation, which is given below, is obtained by the variation of action (i.e.,  in eqn. 1) through infinitesimal changes in the metric tensor  .
diagnostic has been used in recent cosmological studies to distinguish between the standard  model and various other DE models model is represented by the point ,  0, 1 in   plane.Standard cold dark matter (SCDM) model is represented by the point ,  1, 1 in FLRW background.The point ,  1, 1 stands for steady state (SS) model and ,  0.5, 1 stands for SCDM model in   plane.