Cosmological Consequences of a Parametrized Equation of State

: We explore the cosmic evolution of the accelerating universe in the framework of dynamical Chern–Simons modiﬁed gravity in an interacting scenario by taking the ﬂat homogeneous and isotropic model. For this purpose, we take some parametrizations of the equation of state parameter. This parametrization may be a Taylor series extension in the redshift, a Taylor series extension in the scale factor or any other general parametrization of ω . We analyze the interaction term which calculates the action of interaction between dark matter and dark energy. We explore various cosmological parameters such as deceleration parameter, squared speed of sound, Om-diagnostic and stateﬁnder via graphical behavior.


Introduction
It is believed that in present day cosmology, one of the most important discoveries is the acceleration of the cosmic expansion [1][2][3][4][5][6][7][8][9][10]. It is observed that the universe expands with repulsive force and is not slowing down under normal gravity. This unknown force, called dark energy (DE), and is responsible for current cosmic acceleration. In physical cosmology and astronomy, DE is a mysterious procedure of energy which is assumed to pervade all of space which tends to blast the extension of the universe. However, the nature of DE is still unknown which requires further attention [11] (for recent reviews on the so-called geometric DE, i.e., modified gravity theories to explain the late-time cosmic acceleration, see, for example [12][13][14][15][16][17]). In the standard Λ-cold dark matter (CDM) model of cosmology, the whole mass energy of the cosmos includes 4.9% of usual matter, 26.8% of DM and 68.3% of a mysterious form of energy recognized as dark energy. In astrophysics, DM is an unknown form of matter which appears only participating in gravitational interaction, but does not emit nor absorb light [18]. The nature of DM is still unknown, but its existence is proved by astrophysical observations [19]. The majority of DM is thought to be non-baryonic in nature [20].
On the other hand, different modified theories of gravity have been proposed in order to explain cosmic acceleration. The dynamical Chern-Simons modified gravity has been recently proposed [60] which is motivated from string theory and loop quantum gravity [61,62]. In this gravity, Jawad and Rani [63] investigated various cosmological parameters and planes for pilgrim DE models in that FRW universe. Jawad and Sohail [62] explored different cosmological planes as well as parameters for modified DE. Till now, various works have been done on the investigation of cosmic expansion scenario with different cosmological parameters [64][65][66][67][68][69][70][71]. In the present work, we use the constructed models in the frame work of dynamical Chern-Simons modified gravity and investigate the different cosmological parameters such as the deceleration parameter, squared speed of sound, state finder parameters and Om-diagnostic.
This paper is organized as follows: in the next section, we provide the basic cosmological scenario of dynamical Chern-Simons modified gravity and construct the field equations in for flat FRW spacetime. We take interaction scenario for constitutes DE and DM with the help of conservation equations. The holographic DE (HDE) density is used as DE model with Hubble horizon as IR cut-off. In Section 3, we provide the parametrization model of EoS parameter and construct the setup to discuss the cosmic evolution of the universe. Also, we analyze the interaction term for the corresponding parametrizations. In Section 4, we discuss the cosmological parameters such as deceleration, squared speed of sound, Om-diagnostic and statefinder. Last section comprises the results.

Dynamical Chern-Simons Modified Gravity
In this section, we describe the dynamical Chern-Simons modified gravity by the following action here R is the Ricci scalar, * R ρσµν R ρσµν is the topological invariant called the pontryagin term, l is the coupling constant, θ is the dynamical variable, S mat is the action of matter and V(θ) is the potential. Now in case of string theory, we take V(θ) = 0. The variation of Equation (1) corresponding to metric g µν and scalar field θ, respectively, give the following field equations where G µν is known as Einstein tensor and C µν appears as Cotton tensor which is defined as The energy-momentum tensor related to scalar field and matter are given bŷ hereT θ µν shows the scalar field contribution and T µν represents the matter contribution while p and ρ indicate the pressure and energy density respectively. Also, u µ = (1, 0, 0, 0) is the four velocity. In case of flat FRW universe, first Friedmann equation for dynamical Chern-Simons modified gravity becomes where H =˙a a represents the Hubble parameter, a is the scale factor and dot indicates the derivative with respect to cosmic time, ρ = ρ m + ρ d is the effective density and 8πG = 1. We assume p m = 0 then for ordinary matter, the conservation equations are given aṡ For FRW universe the pontryagin term vanishes, so the scalar field in Equation (3) reduces to the following form By taking θ = θ(t), we get the following equation The solution of this equation forθ isθ = ba −3 where b is an integration constant. Using this solution in Equation (6), we have Taking into account the equation of continuity Equation (8), Equation (12) takes the form Equation (12) can be re-written as where E(z) = H H 0 is a normalized Hubble parameter, z is the redshift function which is defined as 1 + z = a 0 a . Interaction is an idea of two way action that occur when two or more objects have effect on each other. The continuity equations for energy densities are defined as Here, prime denotes the derivative with respect to the redshift and Q is the interaction term which calculates the action of interaction between the DM and DE. Basically, Q tells about the rate of energy exchange between DM and DE. When Q > 0, it means that energy is being converted from DE to DM. For Q < 0, the energy is being converted from DM to DE [72]. In the preceding prospectus of DE, HDE is one of the sensational attempts to analyze the nature of DE in the frame of quantum gravity. The HDE is based on holographic principle which states that all information relevant to a physical system inside a spatial region can be observed on its boundary instead of its volume. The relation of ultra-violet UV (Λ) and infra-red IR (L) is introduced by Cohen et al. [73] which plays a key role in the construction of HDE model [74]. The relation is about the energy of vacuum of a system with particular size whose maximal quantity should not be greater than the black hole mass of the similar size. This can be indicated as L 3 ρ d ≤ LM 2 p , here M 2 p = (8πG) −1 and L represents the reduced Planck mass and IR cutoff respectively [75]. From the above inequalities, the HDE density takes the following form where c is the constant parameter of the dimensionless HDE and describes the expansion of universe and it lies in the interval 0 < c 2 < 1 and L is taken as Hubble horizon. By inserting Equations (15)- (17) in (14) we get the following expression after some calculation we get the following result

Parametrizations of Equation of State Parameter
Parametrization is a process of choosing different parameters and is used for the comparison of two datasets. In cosmological context, the EoS parameter is the relation between energy density and pressure and it helps to classify the accelerated and decelerated phases of the universe. At ω = 0, this parameter corresponds to non-relativistic matter and involves the radiation era 0 < ω < 1 3 for the accelerated phase of the universe. At ω < −1, ω = −1 and −1 < ω < − 1 3 it represents the phantom, cosmological constant and quintessence eras respectively. A parametrized formation of ω is assumed for deviating DE. We construct two different models; one with a constant EoS parameter and other with a dark fluid in the existence of DM [1,76]. By using a function of redshift the variation of EoS parameter can be estimated and many parametrizations have been suggested so far. We use the following parametrizations At present, ω 0 is the value of EoS parameter, ω 1 is the parameter of the model that is determined by using the observational data [76] and q is the deceleration parameter. By inserting Equation (20) in (19), we have where A is a constant of integration. Similarly by inserting Equation (21) in (19), we get the following result Now, we analyze the interaction term Q for the chosen parametrization of EoS parameter. Using Equations (14) and (17), we obtain the energy density of DM in the following form Taking the derivative with respect to z of Equation (24) along with ρ m in the continuity equation related to DM, we have Using Equations (14)- (16), we get the following result where r is the coincidence parameter which is defined as r = ρ m ρ d with the help of Equations (17) and (24). Comparing the above equations, it yields At present time, the above equation becomes It is significant to express that the value of Q-term predicts the rate at which the universe expands and coincidence parameter decreases. Using the positivity condition of the Q-term at present time, Equation (28) takes the following form The normalized Hubble parameter in terms of coincidence parameter is obtained by the ratio of Equations (17) and (24) such that where is a constant quantity. This parameter shows the singular behavior at r(z) = r c .
At present time, r c = 1+r 0 . For the coincidence parameter, we can consider a CPL-type parametrization form [42] We can notice that above parametrization becomes singular at z = −1 and it has a linear behavior and bounded nature for low and high value of redshift respectively. Taking into account the above parametrization, we get the value of redshift z s , such that z s = − . For the singular behavior, we have the condition −1 < z s < 0. After some manipulation, we obtain which yields (− o z s ) (1+z s ) ≥ 0 at present time. Substituting these results in Equation (31), it can be written as Moreover, we define the function θ(z) := (1+z) (z−z s ) and substitute Equation (32) in (26), we get the following result Using above result the expression (27) for the Q-term can be written as For ω 1d the expression for the Q-term takes the following form Similarly for ω 2d , the Q-term is reduced in the following relation In Figure 1

Cosmological Parameters
In this section, we construct some cosmological parameters such as the deceleration parameter, stability analysis, statefinder and Om-diagnostic corresponding to parametrizations of EoS parameter in the presence of dynamical Chern-Simons modified gravity.

Deceleration Parameter
The deceleration parameter can be described as follows This parameter characterizes the accelerated as well as decelerated phases of the universe. For q ∈ [−1, 0), it shows the accelerated phase of the universe and q ≥ 0 exhibits the decelerated phase of the universe. The time derivative of Hubble parameter gives the following relation in terms of redshift functionḢ Inserting Equation (38) into (37), we have The deceleration parameter for ω 1d can be evaluated by using Equations (22) and (39) such that The plot of this equation in shown in Figure 3 (left) versus z for three different values of ω 0 . The particular values of other constants are same as in above case. For z > 0, the deceleration parameter transits towards the range for accelerated phase. For present and future epochs, this parameter represents the accelerated phase of the evolving universe. Substituting the Equation (23) into (39), the expression of deceleration parameter for ω 2d takes the following form For ω 2d , we plot the deceleration parameter q 2 as shown in Figure 3 (right) for same parametric values. In this scenario, the deceleration parameter exhibits accelerated phase of the universe since it remains between −1 and 0 for all values of (ω 0 , ω 1 ).

Stability Analysis
The squared speed of sound can be described as follows v 2 s = dp dρ . (42) The squared speed of sound parameter is used to discuss the stability of model. That is, v 2 s < 0 leads to the unstable behavior of the model while v 2 s ≥ 0 corresponds to the stable behavior. Inserting Equations (12), (13) and (22) in (42), we obtain the squared speed of sound for ω 1d as follows Taking into account Equations (12), (13) and (23) in (42), the relation of squared speed of sound for ω 2d takes the following form The plot of v 2 s1 is expressed in Figure 4 (left). It can be seen that the trajectories of squared speed of sound show positive behavior for a positive range of z (except some values) which gives the stability of the model. However, for a small range of positive values of z, z = 0 and z < 0, the model expresses unstable behavior. In Figure 4 (right), the graph of squared speed of sound versus redshift parameter is given. The trajectories for ω 0 = −0.8, −0.9 give the positive behavior for all values of z while for ω 0 = −1, the squared speed of sound represents negative behavior for all values. This shows the stable behavior in first case while in latter case, the model is unstable.

Statefinder Parameters
The statefinder parameters (r, s) are two new cosmological parameters introduced by Sahni [78] which are defined for flat universe model as which help in differentiating the DE models. That is, for (r, s) = (1, 0) then it shows the ΛCDM limit, (r, s) = (1, 1) represents the CDM limit. Also, s > 0 and r < 1 represent the DE regions such that phantom and quintessence and r > 1, s < 0 give the Chaplygin gas behavior. We can obtain statefinder parameters (r, s) for ω 1d by using Equations (22) and (38) in (45), such that Inserting Equations (23) and (38) in (45), the statefinder parameters for ω 2d take the following form In Figure 5 (left), the graph of s 1 displayed against r 1 for three different values of ω 0 . We can observe that the (r, s) parameters corresponds to Chaplygin gas behavior for the underlying scenario. However, the trajectory for ω 0 = −1 does not yield any result for some region which is related to r > 1, s > 0. The model constitutes the ΛCDM limit (r, s) = (1, 0) for the trajectories of ω 0 = −0.8, −0.9. In the right side plot, we draw s 2 versus r 2 for ω 2d which gives the r < 1 and s > 0 for the trajectory ω 0 = −0.9, ω 1 = −0.5. This shows the DE eras, phantom and quintessence. The remaining two trajectories do not give any fruitful results.

Om-Diagnostic
The Om-diagnostic is another tool to differentiate different phases of the universe. The positive trajectory of Om-diagnostic represents the DE era like phantom while quintessence era is obtained from negative behavior. This parameter is given by The Om-diagnostic for ω 1d and ω 2d can be obtained by substitution of Equations (22) and (23) in above relation, such that In Figure 6, we draw the Om-diagnostic versus z for ω 1d in left plot and for ω 2d in the right plot. It can be observed that the trajectories of Om-diagnostic for both cases represent the negative slopes at a past epoch which implies the quintessence era while give positive slopes at future epoch which constitutes the phantom era of the universe.

Conclusions
In the framework of dynamical Chern-Simons modified gravity, we have assumed the flat FRW spacetime and discussed different DE models by using a collection of observations at low redshift. We have taken the parametrizations of EoS parameter to explore the cosmic evolution of accelerating universe in interacting scenario. The parametrization may be a Taylor series extension in the redshift, a Taylor series extension in the scale factor or any other parametrization of ω. We have evaluated the different cosmological parameters, such as the deceleration parameter, squared speed of sound, Om-diagnostic and statefinder parameters. The deceleration parameter is a cosmological parameter which helps to classify the accelerated as well as decelerated phases of the universe. The squared speed of sound is another cosmological parameter which is used to check the stability of the models. The statefinder parameters differentiate various DE models, their behavior and cosmological evolution at present time. The Om-diagnostic is used to differentiate the phantom and quintessence behavior. The trajectories of the constructed models have been plotted with different constant parametric values.
The interaction term represented the positive behavior and is observationally verified that the interaction term must not change its sign during cosmic evolution. The deceleration parameter indicated the consistent result while squared speed of sound expressed some stable solutions. The statefinder parameters for the first parametrization represented Chaplygin gas model behavior and met the ΛCDM limit for specific choice of parameters and DE era is obtained for second choice of parametrization. The Om diagnostic parameter indicated the phantom and quintessence eras of the universe.
Author Contributions: A.J., S.R. and S.S. completed the manuscript, while K.B. and R.J. done its proof reading.
Funding: This research received no external funding.