Generic autonomous system approach to interacting dark energy models Parth
Shah1, Gauranga C. Samanta2, Kazuharu Bamba3, R. Myrzakulov4 1Department of
Mathematics, VIT AP University, Amaravati 522237, India 2P.G. Department of
Mathematics, Fakir Mohan University, Balasore, Odisha 756019, India 3Division
of Human Support System, Faculty of Symbiotic Systems Science, Fukushima
University, Fukushima 960-1296, Japan 4Eurasian National University, Nur-
Sultan 010008, Kazakhstan

###### Abstract

We explore an autonomous system analysis of dark energy models with
interactions between dark energy and cold dark matter in a general systematic
approach to cosmological fluids. We investigate two types of models such as
local and non-local ones. In particular, a local form of interaction is
directly proportional to only the energy density, while a non-local
interaction is directly proportional to the energy density as well as the
Hubble parameter. As a consequence, it is explicitly demonstrated that in both
cases there exist the stability points in terms of cosmological parameters.
This work aims at obtaining acceleration and stability using interaction
models without modifying the matter or geometric component of the Universe.

Keywords: General relativity $\bullet$ Dynamical System analysis $\bullet$
Interaction models
Mathematics Subject Classification Codes: 83C05; 83C15; 83F05.

##  I Introduction

There are large number of cosmological evidence in the form of observation
which proves that our universe is presently experiencing an accelerated
expansion. In the standard Einstein gravity, the late time cosmological
acceleration is explained by introducing an exotic energy component with huge
negative pressure, known as dark energy. In this quest, the cosmological
constant $\Lambda$ was introduced as the simplest candidate of dark energy.
However, it has some difficult theoretical issues such as fine tuning and
cosmic coincidence problem. It is important to know the exact nature of dark
energy. Since then various dynamical dark energy models have also been
explored in several articles [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15,
16, 17, 19, 20, 21, 22, 23]. Moreover to obtain late time cosmic acceleration
a large scale modification of gravity has been used. At present late time
cosmic acceleration is considered to be an established phenomenon but its
cause is still not clearly understood. There are numerous models in Einstein
gravity (with cosmological constant) and modified gravity theories which
explain the phenomenon of accelerated expansion [24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34].

Even though $\Lambda CDM$ model is consistent with current observations, yet
coincidence problem does not have a valid argument. Interaction of dark energy
with dark matter is one other approach that is useful to address the mentioned
problem. The interacting dark energy models have been recently proposed by
several authors [35, 36, 37, 38]. The interaction between dark energy and dark
matter may enhance the nature of dark matter, and also affect structure
formation. The investigation of phase space analysis is the one of the most
effective test for dark energy models [39, 40, 41, 42]. If the dark energy
models have $\frac{\Omega_{DE}}{\Omega_{DM}}$ of the order $1$ and an
accelerated scaling attractor solution, then the coincidence problem can be
resolved. The non-interacting quintessence [43, 44] and quintom models [46,
47, 48] show late time accelerated attractors, and possess $\Omega_{DM}$ as 1,
therefore, they do not provide an adequate solution for coincidence problem.
In the literature, two forms of linear interactions have been studied namely
local and non-local. Local forms of interactions are the ones which are
directly proportional to energy density whereas non- local forms are directly
proportional to Hubble parameter H and energy density $\rho$. In this work we
consider one example of both local form which is proportional to energy
density and non local form of interactions proportional to both Hubble
parameter H and energy density. Many local forms interaction models have been
discussed in references [49, 50, 51, 52]. There is also an approach to discuss
the interacting term without the assumption of a specific form of interacting
term which is discussed in [53]. To study these interaction models we have
incorporated the technique of autonomous systems. This technique gives us the
qualitative behaviour of the systems without solving the equations
analytically. The critical points are the zeros of the systems for the
interacting dynamical model. These points can be plotted in different ways to
visualize the role of interaction models. This work is organised in the
following manner: In section II we establish the interacting quintessence
cosmological framework and construct an autonomous dynamical system which is
worthy for phase space investigation. We then discuss phase space analysis and
find stationary points and their stability for interacting cosmological
models. Conclusion and discussions are presented in section III.

##  II Interacting Dark Energy and Dark Matter

Consider that our universe is well described by a flat
Friedmann–Lemaitre–Robertson–Walker (FLRW) metric

$ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2})$ (1)

with the matter distribution obeying the perfect fluid distribution with the
energy-momentum tensor

$T_{\mu\nu}=(p+\rho)u_{\mu}u_{\nu}+pg_{\mu\nu}$ (2)

where $u_{\mu}$ is the four velocity vector of the perfect fluid. $\rho$ and
$p$ are the energy density and the thermodynamic pressure of the perfect
fluid. The explicit form of the Einstein’s field equations (assuming c = 1)
would be

$G_{\mu\nu}=8\pi GT_{\mu\nu}$ (3)

The two independent Friedmann’s equations derived are

$H^{2}=\frac{8\pi G}{3}(\rho_{m}+\rho_{r}+\rho_{d})$ (4)
$2\dot{H}+3H^{2}=-8\pi G(p_{m}+p_{r}+p_{d}),$ (5)

where $H$ = $\dot{a}/a$ the Hubble parameter. Dot represents the derivative
with respect to cosmic time t. $\rho_{m}$ and $\rho_{d}$ are the energy
densities of dark matter and dark energy. $p_{m}$ and $p_{d}$ are the
thermodynamic pressure of Dark Matter and Dark Energy. We assume dark matter
as a pressureless dust i.e. $p_{m}=0$. The dark energy satisfies the equation
of state $p_{d}=\omega_{d}\rho_{d}$, where $\omega_{d}$ is the equation of
state for dark energy. We consider the universe to be filled with dark matter,
radiation and dark energy. We consider the interaction only between dark
energy and dark matter components and write the conservation equations in the
following coupled form:

$\dot{\rho}_{m}+3H\rho_{m}=Q,$ (6)

$\dot{\rho}_{d}+3H(1+\omega_{d})\rho=-Q$ (7)

$\dot{\rho}_{r}+4H\rho_{r}=0,$ (8)

Here, $Q$ is the rate of energy density exchange between dark matter and dark
energy and its positive value denotes the energy transfer from dark energy to
dark matter while negative value denotes the transfer from dark matter to dark
energy. $Q=0$ would mean that there is no energy transfer between these two
quantities and it would be the case similar to non interacting model.

$Q$ is assumed to be positive for the validity of the second law of
thermodynamics. If we see the continuity equations (6) and (7), the
interaction between dark energy and dark matter must be a function of the
energy densities multiplied by a quantity having units of the inverse of time
which without the loss of generality can be chosen as the Hubble parameter.
Thus interaction between dark energy and dark matter could be expressed in the
most general form as $Q=Q(H,\rho_{m},\rho_{d})$. In various literature, the
nature of Dark Energy has been studied considering different type of
interactions [54, 55]. We consider for simplicity that the interaction is in
linear combinations of the dark sector densities as [49, 56, 57, 58].

$Q=3\lambda_{m}H\rho_{m}+3\lambda_{d}H\rho_{d}$ (9)

where $\lambda_{m}$ and $\lambda_{d}$ are dimensionless constants. As from
observational point of view the interaction should be sub-dominant today [37],
so, $\lambda_{m}$ and $\lambda_{d}$ are very small. The factor ‘3’ in the
above expression for interaction is motivated purely from mathematical ground.
This general form of interaction has been studied recently by several authors
[49, 56, 57, 58] and the particular cases $\lambda_{m}=\lambda_{d}$, in
cite46, and $\lambda_{d}=0$, in [60]. Inserting (9) in the energy conservation
equations (6) and (7) we have

$\dot{\rho}_{m}+3H\left(1-\lambda_{m}-\frac{\lambda_{d}}{u}\right)\rho_{m}=0,$
(10)

$\dot{\rho}_{d}+3H\left(1+\omega_{d}+\lambda_{d}+\lambda_{m}u\right)\rho_{d}=0$
(11)

Define various density parameters as:

$\Omega_{m}=\frac{\rho_{m}}{3H^{2}},~{}~{}\Omega_{r}=\frac{\rho_{r}}{3H^{2}},~{}~{}\Omega_{d}=\frac{\rho_{d}}{3H^{2}}$
(12)

##  III Generic autonomous system approach

We analyze this model by considering cosmological constant as the governing
force of acceleration of the universe. Hence $\omega_{d}=-1$. The technique of
dynamical systems is used by introducing certain dimensionless parameters to
study the stability of interaction model. The dimensionless parameters
introduced are:

$x=\frac{\rho_{m}}{3H^{2}},~{}~{}y=\frac{\rho_{r}}{3H^{2}}$ (13)

So we have,

$\Omega_{d}=1-x-y$ (14)

We divide the above problem into three cases:

Case 1: $\lambda_{d}=0$

When $\lambda_{d}=0$, we have $Q=3\lambda_{m}H\rho_{m}$. Assuming this
interaction term will lead us to following set of autonomous differential
equations:

${}x^{\prime}=x\left(-3(1-\lambda_{m})+3x+4y\right)$ (15)

${}y^{\prime}=y\left(3x+4y-4\right)$ (16)

We can solve these two equations. Let $x^{\prime}=\frac{dx}{dz}$. Then we can
one times integrate Eqs.(15)-(16). As result we obtain

${}\frac{x}{y}=e^{[C_{1}+(1+3\lambda_{m})z]}$ (17)

or

${}x=ye^{[C_{1}+(1+3\lambda_{m})z]}$ (18)

where $C_{1}=const$. In figure 1, we have presented the 3-dimensional portrait
of (18) in terms of coordinates $x,y,z$.

Figure 1: 3D plot of (18) for $\lambda_{m}=1$

We analyze this system using dynamical system analysis. O(0,0), A(0,1) and
B($1-\lambda_{m}$,0) are the critical points of this system. Physically (0,0)
would mean that the universe is completely filled with dark energy and (0,1)
would mean that universe is completely filled with radiation.

By evaluating Jacobian at the above mentioned critical points and finding its
eigenvalues we get:

Point | $\omega_{eff}$ | Eigenvalues | Stability
---|---|---|---
(0, 0) | $-1$ |  -4, $-3(1-\lambda_{m})$ | Stable
(0, 1) | $\frac{1}{3}$ |  4, $1+3\lambda_{m}$ | Unstable
($1-\lambda_{m}$, 0) | $-\lambda_{m}$ |  $-1-3\lambda_{m}$, $3(1-\lambda_{m})$) | Saddle Point
Table 1: Stability analysis for $\lambda_{d}=0$

Phase space portrait of the dynamical system (15)-(16) is plotted now for
$\lambda_{m}=0.1$. Clearly origin is the stable point and ($1-\lambda_{m}$, 0)
i.e. $(0.9,0)$ is the saddle point in this system.

Figure 2: Phase space portrait of (15) and (16)

We can also study the relative energy density of dark matter, radiation and
dark energy together with effective equation of state parameter in $\Lambda
CDM$ model.

Figure 3: Relative energy density for $\lambda_{d}=0$

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. $\omega_{eff}$ starts with $\frac{1}{3}$ 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 $\omega_{\Lambda}$ which is
-1.

Case 2: $\lambda_{d}=\lambda_{m}=\lambda$ When
$\lambda_{d}=\lambda_{m}=\lambda$, we have $Q=3\lambda H(\rho_{m}+\rho_{d})$.
For this interaction term we arrive at following set of autonomous
differential equations:

${}x^{\prime}=-3x+3\lambda x+3\lambda(1-x-y)+3x^{2}+4xy$ (19)

${}y^{\prime}=y\left(3x+4y-4\right)$ (20)

The critical points of this system are A $(0,1)$, B
$\left(\frac{1}{2}\left(1-\sqrt{1-4\lambda}\right),0\right)$ and C
$\left(\frac{1}{2}\left(1+\sqrt{1-4\lambda}\right),0\right)$. We here note
that the critical points are real only when $\lambda\leq 1/4$. Eigenvalues of
point A are $\frac{1}{2}\left(5-3\sqrt{1-4\lambda}\right)$ and
$\frac{1}{2}\left(5+3\sqrt{1-4\lambda}\right)$, eigenvalues of point B are
$-4+\frac{3}{2}\left(1-\sqrt{1-4\lambda}\right)$ and
$-3+3\left(1-\sqrt{1-4\lambda}\right)$ and eigenvalues of point C are
$-4+\frac{3}{2}\left(1+\sqrt{1-4\lambda}\right)$ and
$-3+3\left(1+\sqrt{1-4\lambda}\right)$. The brief analysis of these critical
points is as follows:

Point | $\omega_{eff}$ | Stability
---|---|---
A | $\frac{1}{3}$ | Unstable
B | $-\frac{1}{2}\left(1+\sqrt{1-4\lambda}\right)$ | Stable
C | $-\frac{1}{2}\left(1-\sqrt{1-4\lambda}\right)$ | Saddle Point
Table 2: Stability analysis for $\lambda_{d}=\lambda_{m}$

Phase space portrait of the dynamical system (19)-(20) is plotted now for the
value $\lambda=0.1$. Clearly point B i.e. $(0.1127,0)$ is the stable point in
this system and point C i.e. $(0.8873,0)$ would be saddle point of this
system.

Figure 4: Phase space portrait of (19) and (20)

Case 3: $\lambda_{m}=0$ When $\lambda_{m}=0$, we have
$Q=3\lambda_{d}H\rho_{d}$. For this interaction term we arrive at following
set of autonomous differential equations:

${}x^{\prime}=-3x+3\lambda_{d}(1-x-y)+3x^{2}+4xy$ (21)

${}y^{\prime}=y\left(3x+4y-4\right)$ (22)

The critical points of this system are O $(\lambda_{d},0)$, A $(0,1)$ and B
$(1,0)$. The brief analysis of these critical points is as follows:

Point | $\omega_{eff}$ | Eigenvalues | Stability
---|---|---|---
($\lambda_{d}$, 0) | $-1+\lambda_{d}$ | $3\lambda_{d}-4,3\lambda_{d}-3$ | Stable
(0, 1) | $\frac{1}{3}$ |  1, 4 - 3$\lambda_{d}$ | Unstable
(1, 0) | $0$ |  -1, 3 - 3$\lambda_{d}$ | Saddle Point
Table 3: Stability analysis for $\lambda_{m}=0$

Phase space portrait of the dynamical system (21)-(22) is plotted now for the
value $\lambda=0.1$. Clearly point O: (0.1,0) is the stable point in this
system and point B: (1,0) would be saddle point of this system.

Figure 5: Phase space portrait of (21) and (22)

As the next form of interaction let us consider $Q=\alpha\dot{\rho}_{m}$. This
interaction term will lead to the following set of dynamical system equations.

${}x^{\prime}=\frac{-3x}{1-\alpha}+3x^{2}+4xy$ (23)

${}y^{\prime}=y\left(3x+4y-4\right)$ (24)

We can solve these two equations. Let $x^{\prime}=\frac{dx}{dz}$. Then we can
one times integrate Eqs.(21)-(22). As result we obtain

${}\frac{x}{y}=e^{[C_{2}-(\frac{1-4\alpha}{1-\alpha})z]}$ (25)

or

${}x=ye^{[C_{2}-(\frac{1-4\alpha}{1-\alpha})z]}$ (26)

where $C_{2}=const$. In Figure 6, we have presented the 3-dimensional portrait
of (26) in terms of coordinates $x,y,z$.

Figure 6: 3D plot of (26) for $\alpha=0.5$

We analyze this system using dynamical system analysis. The critical point of
the system are O: $(0,0)$, A: $(0,1)$ and B: $(\frac{1}{\alpha-1},0)$. The
brief analysis of these critical points is as follows:

Point | $\omega_{eff}$ | Eigenvalues | Stability
---|---|---|---
$(0,0)$ | $-1$ | $-4,\frac{3}{\alpha-1}$ |  Stable for $\alpha<1$
$(0,1)$ | $\frac{1}{3}$ | $4,\frac{4\alpha-1}{\alpha-1}$ | Unstable
$(\frac{1}{\alpha-1},0)$ | $\frac{2-\alpha}{\alpha-1}$ | $\frac{3}{\alpha-1},\frac{1-4\alpha}{\alpha-1}$ |  Stable for $\alpha>1$
Table 4: Stability analysis for $Q=\alpha\dot{\rho}_{m}$

Phase space portrait of the dynamical system (23)-(24) is plotted now for the
value $\alpha=0.5$. Clearly point O: (0,0) is the stable point in this system
and point A: (0,1) would be unstable point of this system.

Figure 7: Phase space portrait of (23) and (24) for $\alpha=0.5$

We can also study the relative energy density of dark matter, radiation and
dark energy together with effective equation of state parameter in $\Lambda
CDM$ model.

Figure 8: Relative energy density for $Q=\alpha\dot{\rho}_{m}$

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. However matter starts to
diminish soon after that and dark energy starts to dominate and hence causing
the accelerated expansion of the universe which we currently observe.
$\omega_{eff}$ starts with $\frac{1}{3}$ as universe was completely radiation
dominated then during the matter formation era, it reduced and further when
dark energy starts to dominates, it goes negative and ultimately reach the
value of $\omega_{\Lambda}$ which is -1.

Phase space portrait of the dynamical system (23)-(24) is plotted now for the
value $\alpha=2$. Clearly point B: (1,0) is the stable point in this system
and point A: (0,1) would be unstable point of this system.

Figure 9: Phase space portrait of (23) and (24) for $\alpha=2$

Hence as obtained from the mathematical expression point O is stable when
$\alpha<1$ and point B is stable when $\alpha>1$. Whereas point A is always
unstable.

##  IV Discussion and Conclusion

In this work we discuss the interaction between dark matter and dark energy in
the general theory of gravity using the approach of dynamical system of
analysis. The universe is assumed to be described by the flat FLRW metric. The
coupled form of the conservation equations are mentioned with $Q$ as the
interaction term. Interaction term indicates the rate of energy transfer
between dark matter and dark energy. Positive $Q$ indicates that the energy is
transferred from dark energy to dark matter and negative $Q$ indicates that
the energy is transferred from dark matter to dark energy. $Q=0$ indicates the
model similar to the no-interaction model. We study this with autonomous
system analysis which helps us to obtain cosmological parameters for the
interaction model.

We have discussed two types of models namely local and non-local. An
interaction is said to be of local form if it is directly proportional to
energy density only and a non-local interaction is directly proportional to
energy density as well as Hubble parameter $H$. The non-local interaction
which is studied in this work is
$Q=3\lambda_{m}H\rho_{m}+3\lambda_{d}H\rho_{d}$. This is one of the most
general interaction model studied in the literature. Further for this
interaction we have studied three different cases. Case 1 discusses
$\lambda_{d}=0$, $\lambda_{d}=\lambda_{m}$ is studied in case 2 and case 3
discusses $\lambda_{M}=0$. In each of these cases, we first obtain the set of
autonomous differential equations. From this, we get the critical points of
the system. The eigenvalues of these critical points gives a conclusive
evidence about the stability of the critical point and $\omega_{eff}$ is also
evaluated in each of the case to study the acceleration at that critical
point. We also obtain a 2D phase space portrait to show the stability. The $x$
coordinate of the critical point describes the energy density of matter part
of the universe. Similarly $y$ coordinate describes the energy density of
radiation part of the universe. It must be noted that the point $(0,0)$
describes the universe completely dominated by dark energy. We also study the
relative energy density of the universe at various times. For a non-local
interaction form, we obtain a universe which begins with complete radiation
dominance and later when the matter formation begins the radiation part
decreases. At one stage the universe is completely matter dominated and other
components are zero. At the end stage, the universe is completely dark energy
dominated and other two components are zero which leads to the late time
acceleration of the universe. We note here that although the completely
radiation dominated universe and completely dark energy dominated match with
the observational evidence however a completely matter dominated universe does
not support any observational evidence. Later the local interaction form
$Q=\alpha\dot{\rho}_{m}$. is discussed. All the calculations done for local
form of interaction are also done for the non-local form of interaction. Here
different stable points are achieved for different values of $\alpha$. In the
non-local form of interaction when we evaluate the relative energy density, we
observe that beginning of the universe is with the radiation dominated
universe and at the end it is completely dark energy dominated universe which
is similar to what was observed in local form also. But here we saw that
universe is never completely dominated by the matter part. This plot makes
this form of interaction very similar to the observational evidence.

From all the calculations and discussions, we could conclude that for the
linear interaction models acceleration and stability phase could be observed
without modifying the geometry part of the Einstein’s field equations. In this
work, we obtain stable points which corresponds to a particular physical state
of the Universe and these points are the attractors for our model.
Subsequently, we observed that a physical system with some initial conditions,
will always try to attend the stable state which is evident from the plots
(Fig. 2, 4, 5, 7 and 9).

In future, other interactions models could be studied. There are several non-
linear form of interactions which might also help in theoretical studies of
the cosmological models. Moreover, these interactions could also be studied in
the modified gravity which can help in further understanding of local, non-
local, linear and non-linear interactions. In particular, there are studies on
non-local gravity under Integal Kernel Theories of Gravity (IKGs) which take
into account integral kernels of differential operators, such as the inverse
d’Alembert operator [61] and [62]. This approach of IKGs emerge in searching
for unitary and renormalizability of quantum gravity models. Non-local gravity
models are being considered as a natural mechanism to address dark energy
dynamics and dark matter issue in the large scale structure [63] and [64]. In
future a comparative study can be done with various interaction models and
non-local gravity. We also plan to investigate the method of dominant balances
to see whether such cosmic singularities can occur in this framework like [65]
and [66].

## Acknowledgments

The work of PS is supported by Department of Science and Technology (DST),
Govt. of India via INSPIRE Fellowship (Ref. No. IF160358) and VIT AP
University. The work of KB has partially been supported by the JSPS KAKENHI
Grant Number JP21K03547. The work of RM is supported by the Grant AP09261147
of the MES Of Kazakhstan. We acknowledge the reviewers for giving their
valuable comments and suggestions to improve the manuscript.

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