Dynamics of the Universe with Variable Parameters that Govern the Gravitational Interactions

The present study investigates the Friedmann-Lemaitre-Robertson-Walker models (often FLRW models) with time varying G and Λ in the general theory of relativity. In this communication the Einstein ﬁeld equations have been solved by considering the deceleration parameter q ( t ) as a varying function of time t and the Hubble parameter H deﬁned as q = − l ( t − t 0 ) + β − 1 and q = − 1 + 2 α ( H − 1) H , where l, t 0 , β, α are non-negative constants. We have analyzed the values of diﬀerent constants that will generate a transition for the universe from an early decelerating phase to a recent acceleration phase. The physical and graphic behaviour have also been planned to study in this communication.


Introduction
We all know that Einstein's field equations in the general theory of relativity, which governs the large-scale structure and emergence of the universe, have computational solutions.Such solutions are designated as Friedmann-Lemaitre-Robertson-Walker (FLRW) models.The FLRW metric also offers a foundation for understanding how to investigate the geometry and growth of the universe on a cosmic scale.Alexandra Friedmann Friedman (1922), one of the outstanding scientists who independently came up with the mathematical equation for the expanding cosmos in 1922, collaborated on this model.His solutions lay the groundwork for contemporary cosmology by suggesting that the universe might be dynamic and developing.In 1927, Georges Lemaitre Lemaître (1927) advanced the Big Bang theory, which postulated that the cosmos initially existed in an extremely hot and dense state and has since been expanding.He also came up with Hubble's law, which explains the connection between galaxies ' distance and recessional velocities, in 1927.In 1929, Edwin Hubble Hubble (1929), an American astronomer, performed revolutionary observations of far-off galaxies and discovered proof of the universe's expansion.His findings supported the FLRW model's predictions.He also offered evidence from observations to back up the Big Bang theory.While American mathematicians H. Robertson and A. Walker Robertson (1935); Walker (1935) expanded FLRW models further in 1935, they suggested the possible geometries of the expanding universe under three categories: open, flat, and closed.The density of matter and energy in the universe determines these geometries.In 1980, FLRW models were further refined with the emergence of inflammatory cosmology, as postulated by eminent physicists Alan Guth and Andrei Linde Guth (1981); Linde (1982).The cosmological constant, which stands for the energy density of black-worms and dark energy, is said to be variable according to the hypothesis of the variable cosmological constant, also known as the fluctuating cosmological constant.Through measurements of far-off supernova in the late 1990s, it was discovered that the universe was expanding at an accelerating rate.This realization led to the discovery that a non-zero cosmological constant was necessary to explain this phenomenon.Recent research findings indicate that the theory of variable cosmological constants is still a topic of active research and ongoing debate.Two of the most important unresolved issues in cosmology are the traits of dark energy and the cosmological constant.Several authors, including our own peer research group, have created several cosmological models by using varying cosmological constant because of the significance of the topic.Pande et al (2000); Mishra and Chandra (2004); Oli (2008); Chawla et al (2012b); Chand et al (2016).We are aware that, the deceleration parameter (DP) is an important quantity in cosmology that is crucial to the solution of the Einstein field equation, especially when looking at the dynamics and evolution of the cosmos.It also provides important insights into the universe's acceleration and expansion rate, which aids in understanding its past, present, and future behaviour.DP is denoted by q(t) and is defined as q(t) = − aä ȧ2 where a(t) is the scale factor of the universe and t is the cosmic time.The distribution of energy and matter in the universe are closely correlated with DP.It relies on how the energy density of the universe is affected by different elements like matter, radiation, and dark energy.Many cosmologists have built their cosmological models with the assumption that the deceleration parameter is constant (CDP) Akarsu and Dereli (2012); Berman and de Mello Gomide (1988).While many other research groups, including our own built their cosmological models using the linearly variable deceleration parameter (LVDP) Mishra and Chand (2012); Chawla et al (2012a); Mishra et al (2013) in General theory and Alternative theory of gravity.It is worth mentioning that a negative DP i.e., q < 0 suggests an acceleration rather than deceleration of the observed expansion.This indicates that the idea of CDP is typically applied in streamlined circumstances or for particular applications where it is anticipated that deceleration will be consistent.The deceleration may be manipulated or altered to fluctuate linearly with time in order to facilitate greater investigation into the mechanics of the cosmos.It indicates that LVDP may occur when particles are exposed to changing electric and magnetic fields that cause a linear fluctuation in their rate of deceleration over time.This could be significant in regulating the movement of the particles in an accelerated system.Due to the importance of the topic, we have carried out this research study entitled "Dynamics of the Universe with Variable Parameters that Govern the Gravitational Interactions" by assuming two additional requirements of DP, i.e.,q = −l(t − t 0 ) + β − 1 and −1 + 2α(H−1) H where l, t 0 , α, β are non-negative constants and H is the Hubble parameter.This paper is divided into seven different sections: In Sect.2, the equations governing the Gravitational interactions are described.Sect.3 deals with the model with DP as a varying function of time t and the Hubble parameter H, whereas in Sect.4 the expression for Jerk parameter has been derived.Sect.5 describes the physical and geometrical properties of the constructed model, and the results and discussions have been made in Sect.6.Sect.7 is the concluding section of the study.

Equation governing the gravitation interactions
As discussed in introduction the FLRW metric is a mathematical equation that describes the configuration of space-time in a cosmological principle-based model.Since at sufficiently large cosmic distances, our Universe is both homogeneous and isotropic which means that at present epoch, the geometry of the space is the geometry of homogeneous and isotropic manifold.Thus for the construction of the cosmological model, we define the expression for FLRW metric as dr 2 1 − kr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) . ( where k denotes the curvature.For various values of k, i.e., k = −1, 0, 1, we can define an open, flat and closed universe.Here, the coordinates (r, θ, φ) are spherical, t is the proper time and a ≡ a(t) denotes the scale factor.For c = 1, we consider the field equation as where R μν is the Ricci curvature tensor, g μν is Metric tensor, R is Ricci scalar, G is Gravitational constant, T μν is Energy momentum tensor and Λ is a cosmological constant.For a perfect fluid with isotropic pressure p and energy density ρ, we may explain the energy-momentum tensor by using c = 1 as where u ν is the four-velocity vector with Since in co-moving coordinate system, the particle is at rest, we have Now by substituting the values from equation ( 1), ( 3) and ( 5), the field equation ( 2) where over dot represents the derivative w.r.t.cosmic time t.On addition, we get Differentiating equation ( 7) w.r.t.′ t ′ and on simplification by using equation ( 8), we have On simplification, we get and 8π Ġρ + Λ = 0. (11) By using equations ( 8) and ( 9), we obtain Further for the construction of the homogeneous and isotropic model, we assume the dependence of ρ and p as where ξ is the equation of state (EoS) parameter, not necessarily a constant.If ξ becomes zero, then we will get the pressure less model with p = 0 and if ξ acquires a non-negative value ξ = 1 3 , then model becomes the radiation dominated model.As suggested by Mishra et al., Pande et al (2000), Berman Berman (1991) and Rahman Abdel-Rahman (1990), we consider the dependence of G and Λ as For any physically relevant model, there exist two important observational quantities defined by Hubble's parameter H(t) and deceleration parameter q(t).While the deceleration parameter q(t) quantifies how the expansion rate varies over time, H(t) indicates the time dependence of the universe's expansion, or how quickly the cosmos is expanding.For further investigation, we may define the Hubble and deceleration parameter as and where q ≡ q(t) > 0 represents the decelerating phase of universe while q<0 denotes accelerated expansion phase.The results from Type Ia supernova suggested that the universe shows the transitional phase change from early deceleration to late time cosmic acceleration phase.As stated above, this describes that the DP must change its values.Numerous authors including our own group derived the Einstein field equation's solution by applying the time-varying Hubble parameter, which yields a CDP.Akarsu and Dereli have presented a cosmological model with a time-varying linear deceleration parameter and took the advantage of it in attaining an accelerated solution.In the next section of this communication, we are ready to present the detailed investigation as above.
3 Model with deceleration parameter

As a linear time-varying function
In this model, we will find the solution of Einstein field equation using the assumption where l is a constant which has a dimension of time inverse, β is a dimensionless constant and t 0 ≥ 0. If l = 0, LVDP law reduced to Berman's law of constant DP.On the basis of LVDP, one can generalize the cosmological solution.From the observations, it is clear that if q < −1, then with acceptable assumptions under LVDP, the universe face rapid expansion.From equation ( 17), we obtain where the value of integrating constant is assumed to be zero.Integrating equation ( 18), we obtain and will get Now using equations ( 10) and ( 13) along with scale factor given in equation ( 19), we will get following set of equations Ġ where α ′ is the proportionality constant derived from equation ( 14).Now after integrating the equations ( 22), ( 23) and ( 24), we will get the values of G, ρ, Λ in terms of cosmic time t as follows where A, B and C are constants of proportionality.We propose that during the expansion of the universe, the density ρ and gravitational constant G decreases with the increase in cosmic time, i.e., ρ < 0 and Ġ < 0 during the evolution of the universe.Thus equations ( 26) and (( 27)) implies that 1 + ξ > 0 and 1 + α Further after few more computational steps from equations ( 12), ( 13), ( 25), ( 26) and ( 27), we may write the expression for pressure p as where

As the function of time varying Hubble parameter H
In this model we assume DP as the function of time varying Hubble parameter defined by where H is Hubble parameter and α is a arbitrary constant.Now from equation ( 30) we may obtain the expression for H as where the value of integrating constant is assumed to be zero.Further integrating equation ( 31) with initial condition a(t) = 0 at t = 0, we get Following the steps as done in the previous section, one may obtain the expressions for pressure (p), energy density (ρ), and cosmological constant (Λ) as where

Expression for jerk parameter
In cosmology, the different approach to understanding the cosmic acceleration is to examine the dimensionless jerk parameter (j), which is obtained from the scale factor a(t).The expression for the jerk parameter is given by Since this parameter includes the third derivative of scale factor a(t), it helps us calculate the universe's expansion rate more precisely.From the literature survey, it can be concluded that all the statements made regarding the universe's experience can be added together to make the claim that the value of the jerk parameter of the universe is 1 Dunajski and Gibbons (2008).For the construction of the cosmological model, we find the expression for j(t) in the above sections as follows • When q = −1 + 2α(H−1)

The Physical and geometrical properties in different cases of DP
The physical and mathematical traits of both models are covered in subsections 5.1 and 5.2 of this section.Within the framework of our investigation of FLRW cosmological models with time-varying DP, the examination of these features offers insightful information regarding the dynamics and evolution of the cosmos as a whole.

In case of varying Hubble parameter
The Lambda Cold Dark Matter (ΛCDM) model is the newest embodiment of our understanding of the beginning of the universe.It is referred to as a refinement of the big bang theory by positing that most of the physical substances in the universe consist of material called dark matter.Now equation ( 1) after putting the value of a(t) with c = 1 takes the form Now to discuss the physical and geometrical nature of this model, we wish to refer the equations ( 30), ( 31), ( 33), ( 35) and (38).For better understanding of the solutions, we have taken the pictorial representation of parameters in Fig 8,9,10,11,12,13 & 14.

Results and discussion
From the pictorial representation of the above cited figures, following conclusions are made: • Figure 1 depicts the variation of Hubble parameter (H) with respect to cosmic time t for different choices of l, β and t 0 as cited in the figure.The graph of H diverges at the beginning and also at the end of the universe.One can observe that it remains positive throughout the expansion of universe.• Figure 2 infers that q = β − 1 > 0 provided β > 1 at t = t 0 , q ⩽ 0 for t ⩾ β−1 l + t 0 , q ⩽ −1 for t − t 0 ⩾ β l and q = −β − 1 at t = t 0 + 2β l .Such behaviour conclude that the model begin with decelerating expansion phase, enters into accelerated expansion phase at t = β−1 l + t 0 , enters into super exponential state at t = t 0 + β l and ends at t = t 0 + 2β l .From the above discussion we can say that the Universe was decelerating in the early stage and accelerating at present moment.
• The alteration in energy density parameter (ρ) with respect to cosmic time (t) is shown in Figure 3 and 4, for different choices of l, β and t 0 as cited in the figure assuming two values of ξ = 0, 1 3 .We noticed that ρ > 0 and ρ < 0, i.e., the energy density is positive decreasing function of time.
• From Figure 5 and 6, it can be observed that the pressure p is negative throughout the expansion of the universe for different choices of l, β and t 0 as cited in the figure assuming two values of ξ = 0, 1 3 .The negative pressure ensures the accelerated evolution of the universe.Also p → 0 as t → ∞.
• Figure 7 depicts the increasing value of the jerk parameter in the late-time universe expansion, which means the model shows the deflection from the ΛCDM model.• Figure 8 describes the variation of DP w.r.t.cosmic time t.As t → ∞, the graph of q tends to −1 which is the case same in the flat ΛCDM model.• The graph of H shows the steep decline behaviour and approaches to negligible value after crossing early inflationary phase of universe in Figure 9 for various values of α. • The pressure (p) in Figure 10 and 11, changing its values from non-negative to present negative ones.It may be concluded that, the negative pressure is responsible for the universe?spresent-day expansion.• For numerous choices of α as cited in the Figure 12 and Figure 13, the graph of energy density ρ decreases with the increase in time and remains positive throughout the evolution.• Since from equation (38), it is noticed that j → 1 as t → ∞.Similar behaviour can be observed from the Figure 14.The value of jerk parameter remains positive throughout the cosmic evolution.Also, the flat ΛCDM model have constant jerk parameter j = 1.Chiba and Nakamura (1998); Visser (2004) Thus, from the above one can predict that the model with DP as a function of time varying Hubble parameter reaches to flat ΛCDM model in late-time cosmic evolution.

Concluding remarks
Although we have presented the details of the obtained results in Section 6 under "Results and discussion" for the study entitled "Dynamics of the Universe with Variable Parameters that Govern the Gravitational Interactions".In this study, the study of cosmological models with variable DP has revealed important insights into the behaviour of the Hubble constant, energy density, and expansion dynamics of the universe.The formulated model in this study begins with a decelerating expansion phase, reflecting the slowing down of the universe's expansion over time.However, an interesting transition occurs where the expansion of the universe shifts into an accelerating phase.This transition is associated with the presence of dark energy (a hypothetical form of energy) that causes the expansion rate to increase.The Hubble constant that characterized the expansion rate shows an interesting behaviour in this model.It diverges at the beginning and end of the universe, indicating significant changes in the expansion dynamics during these phases of the universe.Furthermore, the energy density in this model is described as a positive, increasing function of time.This behaviour indicates that the universe evolves from a state of high energy density, such as in the early inflationary phase, to a lower energy density as it expands.The graph of the Hubble constant indicates a steep decline, reflecting the decreasing expansion rate during the decelerating phase.The Hubble constant also approaches a negligible value, which implies that the expansion rate gradually slows down after crossing the early inflationary phase.