Qualitative behavior of the cosmological models in various matter fields
Abstract
In this current work we try to understand the late time accelerated expansion of the universe using scalar field with potential . It studies the consequences of three fluid mixtures in the universe using dynamical system analysis. These three fluids consist of dark matter, perfect fluid and scalar field with exponential potential and power law potential. The use of this technique has allowed to understand the behavior of various models of the universe. term has let us understand the acceleration phase in these models. This work also tries to determine the domination of the particular fluid at certain stages of the universe along with their stability analysis.
Keywords: Dynamical system analysis, Scalar field, General relativity.
1 Introduction
From the recent observational data, it is assumed that, our universe is spatially flat and suffered two acceleration phases. The first acceleration occurred at the time of the big bang is called an inflationary phase, prior to the radiation dominated stage. Since the discovery of accelerated expansion in 1998 [1, 2], there have been several studies for search of candidates capable for this. First and foremost among them being positive cosmological constant. Even though this model fits with astronomical observational data, but there have been issues with particle physics prediction i. e. cosmological constant problem [3, 4] and coincidence problem [5]. Another way handling this problem is by letting cosmological constant to be dynamical. This can be done by introducing some scalar field which can help understanding the late time acceleration of the universe. Any entity which at the cosmic distance provide an accelerated expansion of the late times is termed as dark energy. We have two basic approaches for explaining dark energy. The first way is to “modify matters” i.e. term in Einstein’s field equation which would contain matter with negative pressure. The second way is to “Modify geometry” in which the geometry of spacetime is changed which leads to change in term [6]. Here we are modifying matter by adding terms with negative pressure, purpose of which is fulfilled by scalar field potential which is one of the components of our fluid. Dark energy equation of state parameter between and is termed as quintessence [7, 8]. Scalar fields are very important in cosmology not just because they handle dark energy, but they also replicate the inflationary era, dark matter models and other cosmological features [9]. One of the major problems in theories of gravity is the difficulty in finding the analytic or numerical solution due to highly non linear terms and hence comparison with observations cannot be carried out easily. So it is important that some other techniques are efficiently used for solving such equations. One such method is the Dynamical System Analysis. Application of dynamical systems analysis to cosmology has been deeply discussed in these books [10] and [11]. Apart from these, there are several other books on dynamical systems and differential equations like [12, 13, 14, 15, 16, 17] which provide some fundamental understanding of this using certain computational techniques. It is advantageous as it provides a simple method for finding the numerical solution and also it helps us understand the qualitative behaviour of the system[20, 24, 26, 25, 18, 19, 27, 21, 22, 23]. The most important concept in dynamical systems is of critical points of the system of first order ODE. They are the zeros of the vector field . Phase space points are those that satisfy the condition . The stability conditions are obtained by calculating the Jacobian matrix at critical points and finding their eigenvalues. After the identification of critical points and eigenvalues we can identify the flow near any of these points by linearising the system in a neighborhood of the point. The fundamental idea is to analyze if the trajectories in the neighborhood of the point are either attracted or repelled or what we call the study of stability properties near that particular critical point.
In the present work we try to analyze the stability of the universe which is assumed to be composed of various kinds of fluids. We begin our hypothesis by try to understand the universe by considering completely dark matter filled universe, Subsequently, we take universe completely filled with perfect fluid (with equation of state ). Then we consider certain non interacting mixtures of various fluids in which we start with dark matter and perfect fluid. Then, we introduce dark energy and consider the universe filled with a mixture of dark matter, perfect fluid and dark energy wherein we consider their various kinds and try to understand the stability of the universe. Lastly, we consider a mixture of dark matter, perfect fluid and scalar field with potential as a candidate for dark energy, where we consider two kinds of potential as exponential and power law. In each of these cases, we try to analyze the domination era, stability conditions, as well as the acceleration phase.
2 Mathematical Structure
The action for general theory of relativity with scalar field could be written as
(1) 
where is Ricci Scalar, is determinant of metric, is matter Lagrangian , = diag and is scale factor.
Let us consider the matter part of the universe divided into three parts, such as Dark Matter (DM), Dark Energy (DE) and other matter.
The Dark Matter (DM) is assumed to be described by dust and whose energy momentum tensor can be defined as
(2) 
where is the four velocity vector.
Dark Energy (DE) is described by a scalar field (quintessence or phantom) rolling down a potential . The energy momentum tensor for scalar field is defined as
(3) 
where . The energy density and the isotropic pressure of the field are
(4) 
and the equation of state parameter can be written as
(5) 
Here, the other matter is assumed to be described by perfect fluid (PF) having energy momentum tensor
(6) 
and a linear equation of state
(7) 
for .
CaseI: Dark Matter , it represents stiff matter and radiation fluid model respectively.
Throughout this work we assumed
In this case, we have considered the universe filled with Dark Matter only. The explicit form of the field equations are:
(8) 
(9) 
(10) 
(11) 
(12) 
and deceleration parameter
(13) 
This positive deceleration parameter indicates that, our universe is experiencing a deceleration phase. However, type Ia supernovae observational data suggest that our universe is undergoing an accelerated expansion phase. Hence, we may conclude that our present universe cannot be filled with dark matter completely.
CaseII: Perfect Fluid ()
In this case, we have considered the universe to be completely filled with perfect fluid. The explicit form of the field equations are:
(14) 
(15) 
(16) 
So here we can conclude that for and similarly we cannot have accelerated expansion of the universe, when the universe is completely filled with perfect fluid.
(19) 
which is sometimes called the Raychaudhari equation. Note that here acceleration and deceleration of the universe depends on the sign of . Hence from equation (19) we can say that the universe is decelerating if , while it is accelerating if . If the linear equation of state holds, then the condition can be transferred to equation of state parameter with for decelerating universe and for accelerating universe.
CaseIII:Mixture of dark matter and perfect fluid
In this case, we consider the universe to be filled with dark matter (with density and pressure = 0) and perfect fluid (with density and pressure ). We also consider here that there is no interaction of dark matter with perfect fluid. The explicit form of field equations are
(20) 
(21) 
(22) 
(23) 
Now we define density parameters as:
(24) 
For introducing dynamical systems, for our further analysis, we define
(25) 
This comes because of the fact that sum of total density parameters of the entire universe is 1 i.e. for our case, . All assumptions taken above will lead us to following system of autonomous equation.
(26) 
where prime denotes derivative w.r.t. and the variable x is function of time parameter . There are only two critical points for this dynamical system, and . The first one is unstable (), while the second one is stable (). The evolution of the physical universe starts from the completely radiation dominated universe and ends in a matter dominated universe. Corresponding equation of state for this universe would be , which starts from the value of and then drops to zero as matter start to dominate.
Critical Point  Stability 
Unstable  
Stable 
CaseIV: Mixture of matter, perfect fluid () and Dark Energy.
Let us consider the universe filled with perfect fluid (with density and pressure ), nonrelativistic dark matter(with density and pressure = 0) and Dark Energy (with density and pressure ). We assume that there is no interaction between dark energy and dark matter. The explicit form of field equations are:
(27) 
(28) 
(29) 
(30) 
(31) 
Define various density parameters as:
(32) 
For this case, introduce following dimensionless variables:
(33) 
So we have,
(34) 
Assuming that dark matter, perfect fluid and dark energy do not interact with each other will lead us to following set of autonomous differential equations:
(35) 
(36) 
O(0,0), A(0,1) and B(1,0) are the critical points of this system. Physically (0,0) would mean that the universe is completely filled with dark energy, (1,0) would mean that universe is completely filled with dark matter and (0,1) would mean that universe is completely filled with perfect fluid.
The Jacobian Matrix for this set of autonomous equations would be:
By evaluating Jacobian at the above mentioned critical points and finding its eigenvalues we get:
Point  Eigenvalues  Stability  
(0, 0)  3,  Stable  
(0, 1)  3,  Unstable  
(1, 0)  ,  Saddle Point 
Now in this case, if we consider radiation as a particular case of perfect fluid(), then we will have:
Point  Eigenvalues  Stability  
(0, 0)  3,  Stable  
(0, 1)  , 1  Unstable  
(1, 0)  ,  Saddle Point 
Subcase I:
Further if we consider cosmological constant as a type of dark energy , we will have:
Point  Eigenvalues  Stability  
(0, 0)  3,4  Stable  
(0, 1)  1,4  Unstable  
(1, 0)  3,1  Saddle Point 
These results are same as discussed by [28].
By evaluating differential equations(35) and (36) by taking above assumption, we get the following set of autonomous non linear differential equations
(37) 
(38) 
Phase space portrait of the dynamical system 3738 is plotted now. Clearly origin is the stable point in this system.
We can also study the relative energy density of dark matter, radiation and dark energy together with effective equation of state parameter in model.
In above figure we see that during initial stage, we have universe completely filled with radiation, which then slowly reduces and matter starts to form, hence increasing its relative energy density. In the later stage dark energy starts to dominate and hence causing the accelerated expansion of the universe which we currently observe. starts with as universe was completely radiation dominated then during the matter formation era, it goes down to zero and further when dark energy starts to dominates, it goes negative and ultimately reach the value of which is 1.
Subcase II:
Now we assume stiff fluid as a form of a perfect fluid and we still stick to cosmological constant. Here we get a different set of autonomous differential equations which are as follows:
(39) 
(40) 
Here, the corresponding set of critical points and stability conditions remains same, but as the system of differential equations is different, so we get a different set of eigenvalues, which is as follows:
Point  Eigenvalues  Stability  
(0, 0)  3,6  Stable  
(0, 1)  3,6  Unstable  
(1, 0)  3,3  Saddle Point 
For the dynamical set of equations 39 and 40, again origin is the critical point here and we have a phase space portrait as:
Here also we can study the relative energy density of dark matter, stiff fluid and dark energy together with effective equation of state parameter in model.
In above figure we see that during initial stage, we have universe completely filled with stiff fluid, which then slowly reduces and matter starts to form, hence increasing its relative energy density. So is behaving same as energy density of stiff fluid, till dark energy comes into the picture. In the later stage when dark energy starts to dominate and causing the accelerated expansion of the universe which we currently observe. starts with as universe was completely radiation dominated then during the matter formation era, it goes down to zero and further when dark energy starts to dominates, it goes negative and ultimately reach the value of which is 1.
Subcase III:
As there was no significant difference in this phase portrait, so we now try to analyze this system by taking . As discussed previously, the stability conditions are not dependent on values of equation of state parameter . But stability comes faster and prominent with . Here it can be seen that when there was cosmological constant alongside radiation and dark matter, the universe was moving from radiation domination towards dark matter domination which is the saddle point and then ultimately moving towards the stable point (0,0). But here, whatever may be your beginning, the universe quickly moves towards the stable point(0,0) and hence making the saddle point (1,0) weak. Conversely if we take , but negative, then we observe the universe to be moving from radiation domination to dark matter domination and then to dark energy domination quite slowly, which makes the saddle point (1,0) stronger and the stable point (0,0) weaker.
So from this caseIV, we may conclude one important fact that the critical points are not dependent quantities for the mixture of dark matter, perfect fluid and dark energy. We get a different set of eigenvalues for different kinds of mixture, which leads to some difference in stability properties which have been discussed briefly.
Case V Mixture of dark matter, Perfect fluid and scalar field as a form of dark energy.
Let us consider the universe filled with perfect fluid (with density and pressure ), nonrelativistic dark matter(with density and pressure = 0)and scalar field (as a form of dark energy). We assume that there is no interaction of any component with any other component. The explicit form of field equations are:
(41) 
(42) 
(43) 
(44) 
(45) 
Now we introduce dimensionless quantities as which are normally referred as Expansion Normalized variables (EN) [18]:
(46) 
with these variables and Friedmann constraint we have,
(47) 
where relative energy densities are and .
This effectively gives us
(48) 
with our realistic assumption of we have .
We also define here
(49) 
We here note that these EN variables fail to be a system of an autonomous equation as still depends on scalar field . We can study in general, with exponential potential, where is just a parameter and system of equations becomes autonomous. In general, the equation for variable would be given by:
(50) 
So we will have,
(51) 
where,
Subcase I: Exponential Potential
We take exponential potential i.e., , in which , hence is just a parameter, which would mean that y 0 and our phase space becomes upper half of sphere.
Dynamical systems for cosmological scalar fields with an exponential potential was studied long before the discovery of cosmic acceleration[19]. All assumptions taken above will lead us to following system of autonomous equation.
(52)  
(53)  
(54) 
System of equations 52, 53, 54 remains invariant under the transformation y y and z z. Now the acceleration equation gives
(55) 
We define the effective Equation of state parameter of the universe as which can be written in terms of EN variables as
(56) 
where is any point in our phase space and when is , it describes a universe undergoing accelerated phase of expansion.
The Jacobian of the set of autonomous differential equations would be:
Real critical points include
Point A i.e. behaves as a saddle point as eigenvalues take both the sign. Also in this region is , which lie between [0,1]. So there cannot be any accelerated region for this point. In this case all the values i.e. are zero, hence this type of universe would be perfect fluid dominated.
Point i.e. is unstable for and saddle point for . Also in this region is 1. As here the value of is 1, this model of universe is completely scalar field dominated. Although we cannot have accelerated phase for this model.
Point i.e. is unstable for and saddle point for . Also in this region is 1. Like in the previous case, this is called completely scalar field dominated and we can’t have accelerated phase for this model.
Point i.e. is not feasible as the condition of is not getting satisfied here. So no further evaluation regarding this critical point is to be done.
Point i.e. , else it becomes unstable. here becomes , which states that for , this model gives us the accelerated phase. This model is scalar field dominated. behaves as a saddle point if
Point i.e.
Detailed description of points with its eigenvalues and stability is as follows.
Critical Point  Eigenvalues  Stability 
  Saddle Point  
  Unstable for  
Saddle for  
  Unstable for  
Saddle for  
Saddle Point if ,  
else Unstable  

Critical Point  Acceleration Phase  Existence  
No  all and all  
and  1  No  all and all 

Subcase II: Power Law Potential
Here we will consider power law type potential. Explicitly we will consider:
(57) 
where is a dimensionless parameter and M is a positive constant with units of mass. For this potential, we have
(58) 
or
(59) 
As , corresponds to exponential potential studied in subcase I, in which becomes a constant, we will exclude that from here. We can now here define a new variable as
(60) 
when we get again , but when , we have , meaning that this new variable is only bounded to . In this new variable 60, the dynamical system becomes:
(61)  
(62)  
(63)  
(64) 
Jacobian matrix for this system of equation looks like:
By substituting critical points in this Jacobian Matrix, we can study detailed description of point with its eigenvalues and stability as given below:
Critical Point  Eigenvalues  Hyperbolicity  Stability 
Non Hyperbolic  Saddle Point  
0,  Non Hyperbolic  Saddle Point  
0, 3, 1,  Non Hyperbolic  Stable Point for  
0, 3, 1,  Non Hyperbolic  Saddle Point  

Point O i.e. is entire axis which is critical. Since these are not isolated critical points we expect that, at least one eigenvalues of the Jacobian vanishes. This is evident from the table above. From the opposite signs of other eigenvalues, its is clear that this critical point behave as a saddle point. Also it is a nonhyperbolic as one of the eigenvalue vanishes.
Point i.e. are scalar field kinetic dominated solution. So here , which means that it would be a stifffluid dominated universe. Similar to previous case, here also we have opposite signs of the eigenvalues, apart from one vanishing eigenvalue. So again it would be a non hyperbolic and saddle kind of critical point.
Point i.e. is a scalar field dominated point. Here , which means that it would be a dominated by potential energy of scalar field. Here, apart from one vanishing eigenvalue, all other eigenvalues are negative which leads to stability in the neighborhood of this point. Nonhyperbolicity still holds here due to one vanishing eigenvalue.
Point i.e. is also a scalar field dominated point. Here , which means that it would be a dominated by potential energy of scalar field. Here, apart from one vanishing eigenvalue, other eigenvalues have opposite signs which leads to its being a saddle point. Nonhyperbolicity still holds here due to one vanishing eigenvalue, which is evident from the table above.
3 Conclusion and Further work
In this work, we characterize cosmological evolution by considering a mixture of various fluids. The work was precisely distributed into five cases of fluid mixtures. We began with considering an entirely dark matter filled universe and evaluated its field equations and scale factor, from which we arrived at its deceleration parameter, which comes out to be a positive constant, . This states that the universe is currently experiencing a deceleration phase, which contradicts the observational fact of expanding universe. From this, it can be concluded that the universe cannot be completely dark matter filled. Secondly, we consider completely perfect fluid dominated universe with equation of state as , with . Similar calculations as in the previous case and from Raychaudhari equation, it can be found that for the accelerated expansion of the universe, must be less that . So it leads us to the conclusion that for an accelerated expansion of the universe, it must not be completely perfect fluid filled type. Later, as a third case we consider the mixture of both dark matter and perfect fluid, with both not interacting with each other and get the explicit field equations. Then considering radiation as a kind of perfect fluid and by using dynamical system analysis, we arrive at two critical points and conclude that such a universe will start with radiation domination which is an unstable state and slowly it reaches to the matter dominated state, which is stable. Although still we cannot arrive at the accelerated expansion of the universe as is still nonnegative.
In the fourth case, we introduce the dark energy for the first time. We consider the noninteracting mixture of dark matter, perfect fluid and dark energy. With its field equations, we arrive at a set of two autonomous differential equations whose critical points leads us to stability analysis. As a subcase, radiation was once considered as a kind of perfect fluid and cosmological constant as a kind of dark energy and in the next case, we stick to cosmological constant but consider stiff fluid as a kind of perfect fluid. It has been concluded that in all kinds of such cases the completely dark energy dominated universe is stable and it also leads us to accelerated expansion of the universe. Energy density parameter has also been looked at, whose behaviour is plotted against to understand the domination of the particular quantity in a different era of the universe. Point , the stable critical point of this system which corresponds to dark energy dominated universe was looking for various kinds of dark energies such as the cosmological constant, and (but negative). We conclude that more stronger the dark energy (low value of ) more stronger the stable critical point . It was also evident that as , the critical point was so strong that there was no role for saddle point in that case.
In the fifth case, we consider universe filled with a non interacting mixture of dark matter, perfect fluid and scalar field with potential as a form of dark energy. Exponential potential has been considered in the first subcase. Here we arrive at the set of three autonomous differential equations from which critical points are obtained and Jacobian matrix has been constructed. Based on the eigenvalue stability analysis has been done, as discussed in previous cases. Apart from that acceleration phase has been studied based on the fact that should be less than  to achieve it. Subsequently, we consider the power law potential, wherein we obtain set of four autonomous differential equations, in which the fourth equation is dependent on potential parameter and similar calculations were done. We arrive at the critical point which is a completely scalar field dominated as a stable point in this case.
The use of the dynamical system technique has allowed to understand some behavior of various models of the universe. has helped us understand the acceleration phase in this model. Coordinate values help us determine the domination of the particular fluid in our universe. Then, using the critical point and evaluating Jacobian at those points, let us arrive at the eigenvalues. Signs of those eigenvalues help us understand the stability of particular points. In several cases we have reached at certain condition for this stability as well as acceleration phase.
In this work, we considered Einstein’s gravity and all calculations have been done based on that. We could further carry out this work by modified gravity i.e. gravity and look into certain conditions for stability analysis.
4 Acknowledgement
The author Parth Shah is extremely thankful to Department of Science and Technology (DST), Govt. of India, for providing INSPIRE Fellowship (Ref. No. IF160358) for carrying out his research work.
References
 [1] A. G. Riess et al., Astron. J. 116(1998) 1009.
 [2] S. Perlmutter et al., Astrophys. J. 517 (1999) 565.
 [3] S. Weinberg, Rev. Mod. Phys. 61 (1989) 1.
 [4] J. Martin, C. R. Phys. 13 (2012) 566.
 [5] I. Zlatev, L. M. Wang and P. J. Steinhardt, Phys. Rev. Lett. 82 (1999) 896.
 [6] Luca Amendola and Shinji Tsujikawa, Dark Energy: Theory and Observations, Cambride Publication (2010)
 [7] E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15 (2006) 1753.
 [8] S. Tsujikawa, Classical Quantum Gravity 30 (2013) 214003.
 [9] L. Arturo et.al. , Journal of Physics: Conference Series 761 (2016) 012076.
 [10] Wainright and Ellis, Dynamical Systems in Cosmology, (1997).
 [11] A. A. Coley, Dynamical Systems and Cosmology, (2003).
 [12] Arrowsmith D. K., Place C. A. An Introduction to Dynamical Systems, Cambridge University Press (1990).
 [13] Hirsch, Devaney and Smale, Differential Equations, Dynamical Systems and Introduction to Chaos, (1974).
 [14] Lefschetz S., Differential Equations. Geometric Theory, Interscience, New York (1957).
 [15] Lynch Stephen, Dynamical Systems with applications using Mathematica (2007).
 [16] Perko Lawrence, Differential equations and Dynamical systems (2001).
 [17] Wiggins Stephen, Introduction to Applied Nonlinear Dynamical Systems and Chaos (2003).
 [18] J. Wainwright, G. F. R. Ellis, Dynamical Systems in Cosmology, Cambridge University Press (2005).
 [19] E. J. Copeland et. al.,Phys. Rev. D 57 (1998) 4686.
 [20] N. Roy, arXiv: 1511.07978[grqc], 2015.
 [21] K. Bamba, D. Momeni and M. Al Ajmi, arXiv: 1711.10475[grqc] (2017).
 [22] S. D. Odintsov and V. K. Oikonomou, Phys.Rev. D 96 (2017) 104049.
 [23] S. D. Odintsov, V. K. Oikonomou and Petr V. Tretyakov, Phys.Rev. D 96 (2017) 044022.
 [24] M. Hohmann, L. Jarv and U. Ualikhanova, Phys.Rev. D 96 (2017) 043508.
 [25] A. S. Bhatia and S. Sur, Int. J. Mod. Phys. D 26 (2017) 1750149.
 [26] S. Carneiro and H Borges, Gen. Rel. Grav. 50 (2018) 1.
 [27] S. Santos Da Costa et al., Class.Quant.Grav. 35 (2018) 075013.
 [28] N. Tamanini, Dynamics of cosmological scalar fields, Phys. Rev. D 89 (2014) 083521