FRW Cosmology with the Extended Chaplygin Gas
We propose extended Chaplygin gas equation of state for which it recovers barotropic fluid with quadratic equation of state. We use numerical method to investigate the behavior of some cosmological parameters such as scale factor, Hubble expansion parameter, energy density, and deceleration parameter. We also discuss the resulting effective equation of state parameter. Using density perturbations we investigate the stability of the theory.
Accelerated expansion of universe may be described by dark energy which has positive energy and adequate negative pressure [1, 2]. There are several theories to describe the dark energy such as quintessence . Another candidate is Einstein’s cosmological constant which has two crucial problems so called fine tuning and coincidence . There are also other interesting models to describe the dark energy such as -essence model  and tachyonic model . An interesting model to describe dark energy is Chaplygin gas [7, 8] that are based on Chaplygin equation (CG) of state , which are not good consistent with observational data . Therefore, an extension of CG model is proposed [11–13], which is called generalized Chaplygin gas (GCG). It is also possible to study viscosity in GCG [14–19]. However, observational data ruled out such a proposal. Then, GCG was extended to the modified Chaplygin gas (MCG) . Recently, viscous MCG is also suggested and studied [21, 22]. A further extension of CG model is called modified cosmic Chaplygin gas (MCCG) which was recently proposed [23–25]. Also, various Chaplygin gas models were studied from the holography point of view [26–28].
The MCG equation of state (EoS) has two parts: the first term gives an ordinary fluid obeying a linear barotropic EoS, and the second term relates pressure to some power of the inverse of energy density. So here we are essentially dealing with a two-fluid model. However, it is possible to consider barotropic fluid with quadratic EoS or even with higher orders EoS [29, 30]. Therefore, it is interesting to extend MCG EoS which recovers at least barotropic fluid with quadratic EoS.
MCG is described by the following EoS: Now we would like to introduce the extended Chaplygin gas EoS: which reduces to MCG EoS for () and can recover barotropic fluid with quadratic EoS by setting . Also higher may recover higher order barotropic fluid which is indeed our motivation to suggest extended Chaplygin gas. We hope this model will be consistent with observational data compared to previous models.
This paper is organized as follows. In Section 2 we review FRW cosmology and give some useful equations to study cosmological parameters. In Section 3 we introduce our model and numerically analyze some cosmological parameters. In Section 4 we study the deceleration parameter and compare our results with some observational data. In Section 5 we investigate the stability of our model and study density perturbations and speed of sound in the same context. Finally in Section 6 we summarize our results and give a conclusion.
The spatially flat Friedmann-Robertson-Walker (FRW) universe is described by the following metric: where . Also, represents time-dependent scale factor. The energy-momentum tensor for a perfect fluid is given by where is the energy density and is the isotropic pressure. Also, and () with . The independent field equations for the metric (3) and the energy-momentum tensor (4) are given by where dot denotes derivative with respect to the cosmic time , and we take . It is also assumed that the total matter and energy are conserved with the following conservation equation:
3. Extended Chaplygin Gas EoS
Modified Chaplygin gas was introduced with the following equation of state: where , , and are positive constants. In this model, one gets a constant negative pressure at low energy density and high pressure at high energy density. Choosing one gets generalized Chaplygin gas EoS, and together recovers the original Chaplygin gas EoS. Moreover, the first term on the right hand side of (8) gives an ordinary fluid obeying a barotropic EoS, while there are other barotropic fluids with EoS being quadratic and higher orders . Since modified Chaplygin gas can only recover linear form of barotropic EoS, here we would like to extend this model so that resulting EoS also can recover EoS of barotropic fluids with higher orders. In that case we propose the following EoS: which is called the extended Chaplygin gas EoS. Now, recovers ordinary MCG with . In order to obtain the scale factor-dependence of energy density we should use EoS given by (9) in conservation equation (7). The term is somewhat similar to a cosmological constant (if was set equal to minus 1 it would be exactly like a lambda term). We know that the cosmological constant presents a fine tuning problem, so our model is superior. In the following special cases we only consider the last term of expansion in (9) and find special solution.
Special case of reduces to the modified Chaplygin gas EoS with the following density : where is an integration constant, and as we mentioned above, the last term of expansion considered. Therefore, we can obtain Hubble parameter as the following: This case is completely discussed in .
Here, we assume that the last term of expression in EoS (9) is dominant. In that case one can express energy density in terms of scale factor as where is an integration constant.
We can use numerical method to solve the Friedmann equations and obtain the time dependence of the scale factor in plots of Figures 1 and 2 for various parameters. In Figure 1 we fix and and see that the variation of has opposite effect at the early and late time. In the early universe, increasing increases the value of the scale factor (Figure 1(a)). But at the late times, increasing decreases the value of the scale factor (Figure 1(b)). In Figure 2(a) we fix and and vary . We find that, in the early universe, the value of is not important, and it is reasonable because density is high at the initial stage and the second term of EoS becomes negligible. But, at the late time, increasing increases the value of the scale factor. Another interesting case is when we set and fix to investigate time-dependent scale factor by variation of (see Figure 2(b)). Under this assumption we get from (9) that as increases (), the first term on the right hand side reduces to zero while the second term decreases for a chosen energy density. Special case of yields to which is illustrated by blue line of Figure 2(b).
In order to compare this state with observational data we study Hubble expansion parameter in terms of redshift in Figure 3 for selected values of parameters , , , and constant . We can see that current value of the Hubble expansion parameter obtained as corresponding to which is near several observational data . Also, we can use observational data given by [35, 36] to compare at different redshifts. We can see from Figure 3 that the later value of has higher value than results presented by [35, 36]. In order to have agreement with these data we can choose smallest value of the constant . It is clear that the dashed line of Figure 3 is near the results obtained by SJVKS10 or best fitted values of [35, 36]. However, we can obtain exact agreement by choosing appropriate small value of .
Similar to the previous case, we focus on the last summation term of (9). In that case, one can express energy density in terms of the scale factor as where is an integration constant. Therefore, we can obtain Hubble parameter as the following: This is indeed dual of the first case (Section 3.1) by and .
In that case the EoS (9) reduces to the following expression: We assume that and use (7) to obtain the following integral: It gives us the following energy density: where is root of the following equation: In this case the Hubble parameter in terms of redshift plotted in the Figure 4, which shows that this case is far from observations in any time.
In that case the EoS (9) reduces to the following expression: We assume that and use (7) to find the following integral: It gives us the following energy density: where is root of the following equation: Resulting Hubble parameter of this case is similar to the previous case (), therefore we check the next case.
3.6. Arbitrary and
In the previous subsections we discussed some particular cases of and . Now, we would like to consider general case and give numerical analysis of the cosmological parameters such as scale factor, dark energy density, and Hubble expansion parameter with an arbitrary choice of and . Before doing this, we obtain an expression for the energy density corresponding to which is extension of the previous subsections. Repeating procedure of the Sections 3.4 and 3.5 gives us the following expressions: where is root of the following equation: We can investigate using the expression where being the Hubble parameter today, in the form, where and are constants related to the constants , , and . Since there is only one fluid and the spatial section is flat, . Using Friedmann’s equation, it is possible to obtain and compare it with the observational data. We perform it in Figure 5. We can see that choosing and are the best fit in agreement with observational data [35, 36]. Therefore, always we can choose appropriate values of and to have a model in agreement with observational data better than CDM model.
However, some disagreement of these cases with observational data of may be because of choosing . Other choice of should investigate numerically as the following.
In order to find real solutions which will be interesting from observational point of view, first of all, we combine (5), (6), and (9) to obtain the following second order differential equation: where we assume for simplicity and reducing free parameters. Numerically, we can solve (15) and obtain behavior of scale factor against . In Figure 6 we fix , , and and vary to find that increasing decreases value of the scale factor. Figure 6(a) shows long term behavior of the scale factor, while Figure 6(b) shows variation of the scale factor at the early universe. Also, Figure 7 shows that decreases by increasing which is in agreement with the late time behavior of the Figure 1.
Equation (28) can also be solved numerically to obtain the behavior of the dark energy density. Figure 8 shows that increasing decreases value of energy density. As expected, energy density obtained here is a decreasing function of time which yields an infinitesimal constant at the late times.
By using (5) we can rewrite (6) as follows: Now, we use (5) and (9) in (29) to study behavior of Hubble expansion parameter via the following equation: Numerical analysis of this equation is illustrated in Figure 9 and shows that increasing decreases value of the Hubble expansion parameter.
4. Deceleration Parameter
In the previous section we gave a numerical analysis the behavior of the scale factor, energy density, and Hubble expansion parameter. Since it is not clear to see the analytical behavior of these parameters from (27), (28), and (30), we would like to look at another parameter to get more information about the dynamics.
An important parameter in cosmology, from theoretical and observational point of views, is called the deceleration parameter which is given by Using (5), (9), and (31) one can obtain In Figure 10 we draw deceleration parameter in terms of for various values of . We can see that increasing increases the value of . The green line of Figure 10 corresponds to modified Chaplygin gas. At the early universe with high density the deceleration parameter may be reduced to the following expression: In the case of and we recover result of CDM model where . On the other hand, late time behavior (low density limit) of deceleration parameter may be described by Again, special case of and give .
At the late stage of evolution one can obtain an effective EoS parameter as follows: as then so we asymptotically get from extended Chaplygin gas as well as MCG, which corresponds to an empty universe with cosmological constant. In Figure 11 we draw the effective EoS parameter and find that at the late stage, increasing decreases the value of and yields it −1. On the other hand, at the early universe with high density we have positive effective EoS. It is interesting to note that always remains greater than −1, thus avoiding the undesirable feature of big rip similar to the previous cases of Chaplygin gas EoS. Also, we can see that evolution of with higher is faster than the case with lower .
5. Density Perturbation
In this section we give density perturbation analysis of our model. Already, density perturbation of a universe dominated by Chaplygin gas was studied by . Now, we use their results to write the following perturbation equation corresponding to flat FRW universe which expands with acceleration. The perturbation equation of density is given by where is a density fluctuation, is the wave-number of the Fourier mode of the perturbation, , and is sound speed. This is an important parameter to investigate stability of the theory. Extended Chaplygin gas with real positive sound speed is stable therefore we should seek regions in which the squared sound speed will be positive to have stability. In the following subsections we solve (36) numerically for some values of .
In the case of one can obtain where is an integration constant. Then, the behavior of is illustrated in Figure 12, which shows evolution of perturbation for various values of . We can see that at the initial time there is no difference between various values of . After that, increasing increases the value of . Analytical study of a similar case with can be found in . Reference  suggests to be proportional to combination of hypergeometric and exponential function of time which is decreasing function of time at initial stage. Therefore, our results are in agreement with . However, there is also a numerical analysis with which suggest is increasing function of scale factor , so this case is not relevant to our study. The case corresponding to MCG was ruled out by  by using observational constraints. Therefore, we should consider other cases with higher to investigate the validity of our model.
In Figure 13 we fix and study variation of squared sound speed for various values of . We can see that this model is completely stable for .
In the case of we can obtain the following dark energy density: where is an integration constant. Therefore, (36) can be solved numerically which is illustrated in Figure 14. This shows time evolution of for various values of . We find that small values of yield to positive which are decreasing function of time. Larger values of yield to as periodic function, which are damped at the late time.
Also, in the Figure 15 we can see the behavior of squared sound speed for some values of . Significantly, we can see a periodic behavior of and find that our model is completely stable.
Interesting point in this case is that one can get a stable universe for some nonzero values of unlike the MCG models that were studied in . It gives us good motivation to continue our work to construct a valuable model of the universe.
In that case we can obtain the following time-dependent density: where and is an integration constant. Also, we assumed that the last term of expansion in (9) is dominant.
Figure 16 shows that grows periodically at the initial time and then behaves as damping periodic function of time. We study evolution of only for .
Also, Figure 17 shows a variation of squared sound speed with time which is positive for all values of and tells that our model is completely stable with . Also, there are critical times where the sound speed vanishes and again grows to high value. It means that small value of should choose to avoid causality.
Before end of this section, it may be useful to present sound speed in terms of scale factor. In that case we recall special cases of and with , which is discussed in Sections 3.4 and 3.5. In that case, Figure 18 shows that the squared sound speed is positive for both cases. In both cases of and the sound speed yields a constant for the large scale factor. It is clear that increasing increases sound speed. We can fix parameter to have well defined sound speed.
In this paper, an extended model of Chaplygin gas (ECG) as a model of dark energy is proposed which recovers barotropic fluid with quadratic EoS. Scale-factor dependence energy density is obtained for special cases. In the general case, we obtained the evolution of scale factor, Hubble expansion parameter, and time-dependent dark energy density and found the effect of in cosmological parameters. For instance we found that evolution of scale factor corresponding to (linear barotropic fluid) is faster than the case with (quadratic barotropic fluid). We also found that Hubble expansion parameter and dark energy density are decreasing with .
Then, we investigated deceleration parameter and discussed initial time and late time behavior of it. We have shown that is verified for low densities (late time). Then, we discussed about effective EoS parameter and confirmed that is valid also in our model.
We analyzed and compared our results with observational data. We found that, by choosing appropriate values of constant parameters, our model has more agreement with observational data than CDM.
Finally, we studied density perturbations and investigated the stability of our model under assumption that the first term of (9) was dominant. We focused on the special case of (MCG), , and . We found that the cases of and are completely stable by choosing appropriate values of parameters.
We found that adding higher order terms which recovers second and higher order barotropic EoS also may solve the problem of MCG which was ruled out . Therefore, we concluded that ECG may be a more appropriate model than MCG and GCG and has agreement with the observational data. This paper is one of the first steps to introduce extended Chaplygin gas model and there are many things to investigate in future works such as construction of holographic version of this model.
It is not clear that if the model is to produce a sufficiently long matter dominated period, followed by a dark energy dominated epoch at the correct redshift, if the values of the extended Chaplygin gas density and pressure are sufficiently small to pass current experimental tests. For example, if the pressures and densities need to be fairly large in order to produce the appropriate eras, the extended Chaplygin gas presence may be detectable via, for example, galactic dynamics where it condenses with the regular matter (like dark matter); it will be part of our future work. An important point is that our solution did not recover the standard cosmological model in the past, since the extended Chaplygin gas does not behave as a dust component. In order to have a model which behaves as a dust we should consider varying in (9), so at the initial time and gives . This is also left for future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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