# Dynamics of the universe with disformal coupling between the dark sectors

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## Abstract

We use a dynamical analysis to study the evolution of the universe at late time for the model in which the interaction between dark energy and dark matter is inspired by a disformal transformation. We extend the analysis in the existing literature by assuming that the disformal coefficient depends both on the scalar field and its kinetic terms. We find that the dependence of the disformal coefficient on the kinetic term of scalar field leads to two classes of the scaling fixed points that can describe the acceleration of the universe at late time. The first class exists only for the case where the disformal coefficient depends on the kinetic terms. The fixed points in this class are saddle points unless the slope of the conformal coefficient is sufficiently large. The second class can be viewed as the generalization of the fixed points studied in the literature. According to the stability analysis of these fixed points, we find that the stable fixed point can take two different physically relevant values for the same value of the parameters of the model. These different values of the fixed points can be reached for different initial conditions for the equation of state parameter of dark energy. We also discuss the situations in which this feature disappears.

## 1 Introduction

The observed cosmic acceleration at late time is one of the most important mysteries in the universe [1, 2, 3]. This phenomenon may be explained by introducing an unknown form of energy to govern the dynamics of the late-time universe [4, 5]. For the simplest case, this unknown form of energy, dubbed dark energy, is supposed to be in the form of evolving scalar fields. In general, viable dark energy models should have a mechanism to alleviate the coincidence problem, which is the problem why the energy density of dark energy and matter are comparable in magnitude at present although they evolve independently throughout the whole evolution of the universe [6, 7, 8]. A possible assumption for alleviating the coincidence problem is based on the introduction of an interaction between dark energy and dark matter. Various phenomenological forms of the interaction between dark energy and dark matter have been proposed and investigated in the literature [9, 10, 11, 12]. Interestingly, it has recently been shown that models of dark energy in which an interaction between dark energy and dark matter is assumed can satisfy the bound on the Hubble parameter at redshift 2.34 from BOSS data, while the \(\Lambda \)CDM model predicts too large Hubble parameter at this redshift [13].

*C*depend only on scalar field \(\phi \), while the disformal coefficient

*D*can depend both on \(\phi \) and its kinetic term

*X*. In the case where

*D*depends only on the scalar field, the above transformation provides relations among some pieces of Lagrangian in the Horndeski theory, but cannot generate a piece of the GLPV Lagrangian from the Horndeski theory [22]. However, it has been shown that if

*C*also depends on

*X*, the application of the transformation to GLPV action can generate terms that do not belong to the GLPV theory [20] and therefore these terms might be a cause of the Ostrogradski instabilities in the theories. Nevertheless, according to the discussion in [23], the Ostrogradski instabilities can be eliminated by hidden constraints in some cases.

The cosmological consequences of the interaction between dark energy and dark matter due to the disformal transformation, called disformal coupling between dark energy and dark matter, has been studied in various aspects for the case where the conformal and disformal coefficients depend on the field \(\phi \) only. It has been shown in [24] that the disformal coupling between dark energy and dark matter leads to a new stable fixed point compared with the case of conformal coupling, and the cosmological parameters at this fixed point can satisfy the observational bounds. The metric singularity of the new fixed point found in [24] presents the phantom behavior in the Jordan frame [25]. The influences of the disformal coupling on the observational quantities such as the CMB and matter power spectra have been investigated in [26, 27, 28]. In this work, we study the disformal coupling between dark energy and dark matter for more general case where the disformal coefficients depends on both \(\phi \) and its kinetic terms *X*. The physical motivation for such disformal coupling is related to the frame transformation among the general scalar–tensor theories presented above. Our aims are to study how the kinetic-dependent disformal coupling influences the evolution of the universe at late time by finding and analyzing the physically relevant fixed points of the model, rather than search for all possible fixed points of the model. We will show in the following sections that there are features arising only for the case where the disformal coefficient depends on both \(\phi \) and *X*.

In Sect. 2, the evolution equations for dark energy and dark matter with disformal coupling are presented in the covariant form. The autonomous equations for this model of dark energy are computed in Sect. 3, and the evolution of the late-time universe is studies using the dynamical analysis in Sect. 4. The conclusions are given in Sect. 5.

## 2 Disformal coupling between dark energy and dark matter

*Q*in terms of unbarred quantities, we write Eq. (8) as

*Q*as

## 3 Dynamical equations

### 3.1 Evolution equations for the FLRW universe

*Q*in Eq. (20) becomes

### 3.2 Autonomous equations

*M*and \(M_v\) are the constant parameters with dimension of mass. Here, we extend the analysis in the literature by supposing that the disformal coefficient

*D*also depends on the kinetic term

*X*through \(X^{\lambda _3}\) which is the simplest extension. Using the following dimensionless variables:

*N*. From the above equation, we see that

*t*is the cosmic time.

## 4 Dynamical analysis

Here, we concentrate on dynamics of the universe at late time, so that we ignore the contribution from radiation density in the autonomous equations. Moreover, we are mainly interested in the physical fixed points that correspond to the acceleration of the universe at late time.

We note that the relation in Eq. (43) is derived by supposing \(x_{1f}\ne 0\). In the case \(x_{1f}= 0\), Eq. (42) gives \(x_{2f}= 1\), i.e., this fixed point is the potential dominated solution. We will consider this case in detail in the next section.

### 4.1 Potential dominated solution

*X*, i.e., \(\lambda _3 = 0\). This result can easily be understood by noting that the field \(\phi \) is nearly constant in time within the potential dominated regime, so that the ratio of disformal coefficient to conformal coefficient nearly vanish if the disformal coefficient depends on kinetic term of the scalar field. Performing the usual stability analysis, one can show that the fixed point for the potential dominated solution is stable for \(\lambda _3 \ge 0\).

### 4.2 Scaling and field dominated solutions

We now consider the case where both \(x_{1f}\) and \(x_{2f}\) do not vanish. In our consideration, \(x_{1f}^2 + x_{2f}\le 1\), i.e., \(\Omega _c \ge 0\) at the fixed points, so that these fixed points correspond to scaling and field dominated solutions.

#### 4.2.1 Conformal scaling solutions

In the case \(\lambda _4 < 0\), the universe evolves towards the stable fixed point \((w_{d},\Omega _{d}) = (w_{df},1)\) if \(\lambda _1 < - 6 w_{df}/ \sqrt{3\gamma _{f}}\) and \(\lambda _3\) as well as \(\lambda _2\) are suitably chosen, e.g. \(\lambda _3 \sim \lambda _2 \sim \mathcal{O}(1)\). Similarly, for the case \(\lambda _4 > 0\), the universe will evolve towards the stable fixed point \((w_{d},\Omega _{d}) = (w_{df},1)\) if \(\lambda _1 > 6 w_{df}/ \sqrt{3\gamma _{f}}\). Here, \(w_{df}\) can be specified by \(\lambda _4\) through Eq. (50). However, if \(\lambda _4\) and \(\lambda _1\) satisfy Eq. (51) and \(\lambda _3\) as well as \(\lambda _2\) are suitably chosen, the universe will reach the stable fixed point \((w_{d}, \Omega _{d}) = (w_{df}, \Omega _{df})\) at late time, where \(w_{df}\) and \(\Omega _{df}\) are related to \(\lambda _1\) and \(\lambda _4\) through Eq. (51). We note that if we set \(\lambda _1, \lambda _2, \lambda _3\) and \(\lambda _4\) such that Eq. (46) is satisfied, the first eigenvalue in Eqs. (52) and (53) will be zero. Consequently, one can show that these fixed points are saddle points, and therefore the universe will finally evolve towards the disformal scaling solutions discussed in the next section. Hence, in this section, we will consider only the cases where the relation in Eq. (46) is not satisfied.

#### 4.2.2 Disformal scaling solutions

Class | \(x_{1f}\) | \(x_{2f}\) | \(x_{3f}\) |
---|---|---|---|

I | \(\displaystyle {\pm \frac{\sqrt{(1 + w_{df})\Omega _{df}}}{\sqrt{2}}}\) | \(\displaystyle {\frac{1}{2} \left( 1 - w_{df}\right) \Omega _{df}}\) | \(x_{3f}{}_1\) |

II | \(\displaystyle {\pm \frac{\sqrt{(1 + w_{df})\Omega _{df}}}{\sqrt{2}}}\) | \(\displaystyle {\frac{1}{2} \left( 1 - w_{df}\right) \Omega _{df}}\) | \(x_{3f}{}_2\) |

*i*and

*j*run from 1 to 3 and \(E_i\) is the RHS of Eqs. (33)–(35), respectively. Evaluating this matrix at the fixed points, we get

\(x_{1f}\) | \(x_{2f}\) | \(r_0\) | |
---|---|---|---|

Class I | |||

\(\mathrm{I}^+\) | \(\displaystyle {\frac{\sqrt{(1 + w_{df})\Omega _{df}}}{\sqrt{2}}}\) | \(\displaystyle {\frac{1}{2} \left( 1 - w_{df}\right) \Omega _{df}}\) | \(r_{01}\) |

\(\mathrm{I}^-\) | \(\displaystyle {- \frac{\sqrt{(1 + w_{df})\Omega _{df}}}{\sqrt{2}}}\) | \(\displaystyle {\frac{1}{2} \left( 1 - w_{df}\right) \Omega _{df}}\) | \(r_{01}\) |

Class II | |||

\(\mathrm{II}^+\) | \(\displaystyle {\frac{\sqrt{(1 + w_{df})\Omega _{df}}}{\sqrt{2}}}\) | \(\displaystyle {\frac{1}{2} \left( 1 - w_{df}\right) \Omega _{df}}\) | \(r_{0+}\) |

\(\mathrm{II}^-\) | \(\displaystyle {- \frac{\sqrt{(1 + w_{df})\Omega _{df}}}{\sqrt{2}}}\) | \(\displaystyle {\frac{1}{2} \left( 1 - w_{df}\right) \Omega _{df}}\) | \(r_{0-}\) |

Let us first consider the case \(\lambda _3 = 0\). It is clear that this case is not allowed for the fixed points in class I. The numerical investigation shows that the fixed point \(\mathrm{II}^-\) is stable for a wide range of \(\lambda _1\) when \(-1< w_{df}< 0\) and \(0< \Omega _{df}< 1\). However, it follows from Fig. 1 that the parameters region in which the fixed point \(\mathrm{II}^+\) is saddle increases area with increasing \(\lambda _1\). The fixed point \(\mathrm{II}^+\) can become a saddle point within the region \(w_{df}\in (-0.99, -0.97)\) and \(\Omega _{df}\in [0.7,1)\) until \(\lambda _1 \gg 1\) and \(\lambda _3 \gg 1\).

We now turn to the case \(\lambda _3 > 1\). It follows from Fig. 2 that, for the fixed points in class *II*, the area of the saddle region in the \(w_{df}\)–\(\Omega _{df}\) plane increases when \(\lambda _3\) or \(\lambda _1\) increases. Similar to the case \(\lambda _3 = 0\), the fixed points in class II can become a saddle points within the region \(w_{df}\in (-0.99, -0.97)\) and \(\Omega _{df}\in [0.7,1)\) unless \(\lambda _1 \gg 1\).

In the cases where \(\lambda _3 \ne 0\) and \(-1< w_{df}< 1\), Eq. (63) suggests that \(w_{df}\) takes a single value for a given real value of \(r_0\) and \(\lambda _3\). Using a similar analysis to the case where \(\lambda _3 = 0\), we conclude that the fixed points \((w_{df}, \Omega _{df})\) belonging to class I take single physically relevant value for a given value of \(\lambda _1, \lambda _3, \lambda _4\) and \(r_0\).

It follows from Fig. 4 that, for the fixed point \(\mathrm{II}^+\), the solution \(w_{df}^s\) for \(E_{w_{df}} = 0\) shifts to the lower value when \(\lambda _1\) increases, and shifts to the larger value when \(\lambda _3\) increases. According to our numerical investigation, we also find that \(w_{df}^s\) gets closer to 0 as \(\lambda _3\) gets larger, but \(w_{df}^s\) will not exist when \(\lambda _1 \gtrsim 30\). This means that the fixed point \(\mathrm{II}^+\) can take only one physically relevant value for a given value of the parameters when \(\lambda _1 \gtrsim 30\). From Fig. 4, we see that, for the fixed point \(\mathrm{II}^-\), the solution \(w_{df}^s\) shifts towards 0 when \(\lambda _3\) increases. From the detail of the numerical analysis for fixed point \(\mathrm{II}^-\), we find that \(\Omega _{df}\) associated with \(w_{df}^s\) becomes larger than unity when \(\lambda _1 \gtrsim 1\). Nevertheless, the value of \(\Omega _{df}\) can be reduce by enhancing the value of \(\lambda _3\), e.g., \(\Omega _{df}< 1\) for \(\lambda _1 = \lambda _3 = 5\). For both \(\mathrm{II}^+\) and \(\mathrm{II}^-\) fixed points, the solution \(w_{df}^s\) does not exist if \(\Omega _{df}^* \gtrsim 0.9\). Based on the above analysis, we conclude that the fixed point associated with \(w_{df}^s\) will not exist if \(\lambda _1\) is sufficiently larger than unity or the values of \(r_0\) and \(\lambda _4\) correspond to \(\Omega _{df}^* \gtrsim 0.9\).

We now study the situation in which the different fixed points with the same value of the parameters of the model can be reached. In order to perform, we solve Eqs. (33)–(36) numerically by setting the present value of \(\Omega _{c}, \Omega _b, \Omega _r, \Omega _d\) and \(w_d\) to be \(0.27, 0.03, 10^{-4}, 0.7 - 10^{-4}\) and \(-0.99\), respectively. For illustration, we plot in Fig. 5 the evolution of \(\Omega _{c}\) and \(w_{d}\) for the case where \(\lambda _1 = \lambda _3 = 1\) and \(r_0, \lambda _4\) are computed from \((w_{df},\Omega _{df}) = (w_{df}^*,\Omega _{df}^*) = (-0.99,0.9)\). From the figure, we see that if the initial conditions are chosen such that the initial value of \(w_{d}\) is significantly larger than \(w_{df}^*\), the universe will evolve towards the fixed point associated to the solution \(w_{df}^s\) in Fig. 4. Unfortunately, the present value of \(w_{d}\) may lie outside the observational bounds if the universe evolves towards this fixed point. Hence, the existence of the solution \(w_{df}^s\) seems to be a problem, which can be avoided by setting \(\lambda _1\) to be sufficiently larger than unity or setting the value of \(r_0\) and \(\lambda _4\) to be matched with \(\Omega _{df}^* \gtrsim 0.9\). We stress that these conclusions are based on the situation where \(\lambda _3 > 0\) and \(\lambda _1 > 0\).

## 5 Conclusions

In this work, we study the dynamics of the universe at late time for the model in which dark energy directly interacts with dark matter through disformal coupling. When the disformal coefficient depends on the kinetic terms of scalar field, there exist two classes of fixed points which can describe the acceleration of the universe in addition to that found in literature. The fixed points in the first class are saddle points unless \(\lambda _1\) is sufficiently larger than unity, and it exists only for the case where the disformal coefficient depends on the kinetic terms of scalar field. The fixed points in the second class can be stable within the parameter ranges that correspond to the accelerating universe.

In the case where the disformal coefficient depends only on the scalar field, the fixed points in the second class become the fixed points that found in the literature. Interestingly, in the case where the disformal coefficient also depends on the kinetic terms of scalar field, the stable fixed points in the second class can take different physically relevant values for the same value of the parameters of the model. For the case where \(\lambda _1 \sim \lambda _3 \sim 1\) and the values of \(r_0\) and \(\lambda _4\) are set such that the fixed point can occur at \(0.9 \gtrsim \Omega _{df}\gtrsim 0.7\) and \(w_{df}\sim -0.99\), the universe will evolve towards the fixed point \((w_{d}, \Omega _{d}) = (w_{df},\Omega _{df})\) if the initial value of \(w_{d}\) is close to \(w_{df}\). Nevertheless, if the initial value of \(w_{d}\) is sufficiently larger than \(w_{df}\), the universe will evolve towards another value of the fixed point at which the present value of \(w_{d}\) may not be in agreement with the observational bounds. The existence of two different values of the fixed point for the same value of the parameters can be avoided if \(\lambda _1\) is sufficiently larger than unity, or the values of \(r_0\) and \(\lambda _4\) are set from the fixed point \(w_{df}\sim -0.99\) and \(\Omega _{df}\gtrsim 0.9\).

## Notes

### Acknowledgements

KK is supported by Thailand Research Fund (TRF) through Grant RSA5780053.

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