^{1}

^{2}

^{3,4}

^{1}

^{1}

^{2}

^{3}

^{4}

^{3}.

We make steps in a new direction by considering fluids with EoS of more general form

Experimental data interpretation claims that we have accelerated expansion for our universe. However this phenomenon can be understood as a theoretical model based consequence. In general relativity concepts of dark energy and dark matter were introduced by hand and it seems that they deal with the problem at intermediate level, because the considered number of models and articles is going to be behind reasonable limit. However, still the questions concerning the nature of dark energy and dark matter, about possible interactions and so forth, are open. Dark energy thought to be responsible to accelerate expansion. On theoretical and phenomenological levels scalar fields were considered as thought that scalar field can be a base of dark energy. One of them is a tachyonic scalar field. Concerning some fundamental problems, dynamical models of dark energy were proposed and considered from different corners. However, it is not the unique approach and the geometrical part of gravitational action was modified.

A set of observational data reveal, from the following picture of our universe which is called modern era in theoretical cosmology, that an expansion of our universe is accelerated [

gives the energy density and pressure as

A quintessence field [

Dark energy models based on idea of fluid are not less popular and are well studied. Fluids in cosmology are convenient, because, as practice teaches us, we can, for instance, different modifications in geometrical part of action encode in fluid part of field equations, giving illusion that in nature fluids with general form of EoS could be considered like to Chaplygin gas and its generalizations [

where

In this paper we would like to propose a modification in the interaction term

Due to the lack of information about dark energy and dark matter, usually the interaction terms are assumed to be proportional to the energy density, scale factor, Hubble parameter, and their derivative. In [

Before to main formulation of our problem we would like to pay our attention to the question of interaction in cosmology between fluid components. Usually, three forms of

where

where

This new type of interaction, where deceleration parameter

In stellar astrophysics, the polytropic gas model can explain the equation of state of degenerate white dwarfs, neutron stars, and also the equation of state of main sequence stars [

As we know the viscous cosmology is an important theory to describe the evolution of the universe. It means that the presence of viscosity in the fluid introduces many interesting pictures in the dynamics of homogeneous cosmological models, which is used to study the evolution of universe.

We consider that the composed models of a fluid consist of barotropic fluid

(1) viscous modified Chaplygin gas

(2) and viscous polytropic gas

where

There are several theoretical models to describe dark energy. Among them the model based on Chaplygin gas EoS and its extensions is interesting because of the possibility of dynamical analysis and solving some famous problems in cosmological constant model. Therefore, in order to construct a real model of our universe, we consider the modified Chaplygin (or polytropic) gas-like dark energy including viscosity and time-dependent interaction between components. Above points are strong theoretical motivation to consider a toy model of our universe which needs observational data for confirmation or rejection.

This paper is organized as follows. In the next section we will introduce the equations which govern our model. Then, we give numerical results corresponding to both models. In the discussion section we summarize our results. In Appendices

The field equations that govern our model of consideration are

By using the following FRW metric for a flat universe

field equations can be reduced to the following Friedmann equations:

where

Energy conservation

In order to introduce an interaction between DE and DM, we should mathematically split (

For the barotropic fluid with

where the index

In the above equation, index PG refers to polytropic gases which serves as dark energy. Cosmological parameters of our interest are EoS parameters of each component

and deceleration parameter

where

This model is based on differential equation (

Plots of Figure

Behavior of Hubble parameter

The second plot of Figure

In the next plot of Figure

Finally the last plot of Figure

Plots of Figure

Behavior of EoS parameter

From the second plot we find that increasing

In the third plot we can find variation of

Finally we find that viscous coefficient decreases value of

Observational data needs to have

Plots of Figure

Behavior of deceleration parameter

This model is based on differential equation (

Plots of Figure

Behavior of Hubble parameter

In the first plot of Figure

The second plot of Figure

In the next plot of Figure

Finally the last plot of Figure

It seems that the value of the viscosity in the interval

Behavior of EoS parameter

From the second plot we find that increasing

In the third plot we can find variation of

Finally we find that viscous coefficient decreases value of

Comparing with observational data suggests that

Plots of Figure

Behavior of deceleration parameter

In Appendices

We considered two different models of viscous interacting cosmology with modified interaction term so it is depending on Hubble parameter and discussed numerically cosmological parameters of the models. In the first model we consider viscous modified Chaplygin gas which interacts with barotropic fluid. We obtained effect of interaction and viscous parameters on the cosmological quantities. We found that these parameters increase Hubble expansion parameter. If we neglect interaction parameters and viscosity, then evolution of Hubble parameter is faster than the case of interacting viscous cosmology. In the noninteracting case the Hubble parameter yields to constant after sudden reduction at initial stage. Also we studied equation of state parameters and found that interaction parameters and viscosity decrease value of EoS parameters. This situation is similar to deceleration parameter. In the noninteracting case, EoS and deceleration parameters yield to −1 as expected. We then studied effect of these parameters on total density and pressure. We found that both interaction parameters and viscosity increase value of total density but decrease value of total pressure. At the initial stage the total density suddenly decreased and yielded to a constant for noninteracting case, but it is increasing the function of time in presence of interaction term. We show that this model may agree with some observational data which say that

In the second model we consider viscous polytropic gas which interacts with barotropic fluid. Just before, we obtained effect of interaction and viscous parameters on the cosmological quantities. We found that interaction parameters decrease but viscosity increases Hubble expansion parameter. Behavior of interaction term in Hubble expansion parameter of this model is the opposite of previous model. If we neglect interaction parameters and viscosity, then evolution of Hubble parameter is faster than the case of interacting viscous cosmology. In the noninteracting or nonviscous cases the Hubble parameter yields to approximately a constant after sudden reduction at initial stage. Also we studied equation of state parameters and found that interaction parameters increase and viscosity decreases value of EoS parameters. EoS parameter yields to −1 for the noninteracting case and yields to 0 for nonviscous case. The effect of interaction parameters on the deceleration parameter is similar to the EoS parameter but the deceleration parameter yields to approximately 0.5 for the nonviscous cosmology. Finally we studied effect of this parameter on total density and pressure. We found that interaction parameters decrease but viscosity increases value of total density. On the other hand interaction parameters increase total pressure but viscosity decreases one. This model also may agree with some observational data even more than the first model. In both models, the phantom regime is obtained by adding interaction and we have

For the future work it is interesting to consider the effects of varying viscosity [

In this appendix we study equation of state parameter corresponding to viscous modified Chaplygin gas, total density, and pressure of the model numerically. Plots of Figure

Behavior of EoS parameter of viscous Chaplygin

Behavior of

Behavior of

In this appendix we study equation of state parameter corresponding to viscous polytropic gas, total density, and pressure of the model numerically. Plots of Figure

Behavior of EoS parameter of viscous polytropic

Behavior of

Behavior of

The authors declare that there is no conflict of interests regarding the publication of this paper.