# Interactions in the Dark Sector

Dark energy is the main component of the Universe's energy budget, thus it is necessary to consider the possibility of its interaction with other components of the Universe, in particular with the component second in importance - the dark matter. Additional interest to that possibility is related with the fact that within its framework it is possible to solve the so-called "coincidence problem": the coincidence by the order of magnitude (at present) of dark energy and dark matter densities (0.7 and 0.3 respectively). As the nature of those two components is yet unknown, we cannot describe interaction between them starting from the first principles and we are forced to turn to phenomenology. In the base of the phenomenology one can put the conservation equation $\dot\rho_i+3H(\rho_i+p_i)=0.$ In the case of interaction between the components it is necessary to introduce the interaction (a source) into the right-hand side of the equation. It is natural to assume that interaction is proportional to the energy density multiplied by a constant of inverse time dimension. For that constant it is natural to choose the Hubble constant.

## General Analysis

### Problem 1

Construct a model of the Universe containing only interacting dark energy and dark matter, while their total energy density is conserved.

### Problem 2

Within the framework of problem, find the effective state parameters $w^{(\varphi)}_{eff}$ and $w^{(m)}_{eff}$ that would allow one to treat the components as non-interacting.

### Problem 3

Using the effective state parameters obtained in the previous problem, analyze dynamics of dark matter and dark energy depending on sign of the rate of energy density exchange in the dark sector.

### Problem 4

Show that the coupling quintessence behaves like a phantom uncoupled model, but without any negative kinetic energy.

### Problem 5

Show that within the framework of the model of the Universe described in problem the Klein-Gordon equation for the scalar field takes the form: $$\ddot{\varphi}+3H\dot{\varphi}+\frac{dV}{d\varphi}=-\Gamma ,\; \Gamma \equiv \frac{Q}{\dot{\varphi}}$$ Here $Q$ is the constant of interaction between dark energy and dark matter.

### Problem 6

The dynamics of present Universe is assumed to be dominated by dark energy and dark matter. In frame of the model of interacting dark energy and dark matter (see problem) obtain equations for the densities $\rho_{DE}$ and $\rho_{DM}$ under assumption that $Q=-3H\Pi$, where the quantity $\Pi$ can be considered as the effective pressure.

### Problem 7

For the scalar field $\varphi$ in the form of quintessence use the variables $x^2=\frac{\kappa^2\dot\varphi^2}{6H^2},\ y^2=\frac{\kappa^2V}{3H^2}$ to express the total equation of state parameter $w_{tot}\equiv p_{tot}/\rho_{tot}$, quintessence state parameter $w_\varphi\equiv p_\varphi/\rho_\varphi$, deceleration parameter $q$ and determine the allowed variation range of the parameters. The Universe is assumed to be flat.

## Linear Models

### Problem 8

Show that the energy balance equation (modified conservation equations) for $Q\propto H$ or $Q\propto\dot\varphi$ are independent from $H$ when expressed in the variables $x(N), y(N)$ where $N=\log a$ thus phase space of such coupling model is two-dimensional $(x,y)$ space.

### Problem 9

Use the variables $(x,y)$, introduced in the problem in order to obtain the system of equations describing the quintessence in the potential $V(\varphi)=V_0\exp(-k\lambda\varphi),$ where $k^2\equiv8\pi G$, $\lambda$ is a dimensionless constant and $V_0>0$, in the case of flat Universe and under assumption $Q=\alpha H\rho_{DM}$

### Problem 10

Find the scale factor dependence for the dark matter density assuming that the constant of interaction between the dark matter and the dark energy equals $Q=\delta(a)H\rho_{DM}$.

### Problem 11

Find the scale factor dependence for the dark matter and energy densities assuming that $Q=\delta H\rho_{DE}$ $(\delta=const)$.

### Problem 12

Let the dark energy state equation be $p_{DE}=w\rho_{DE}$, where $w=const$. Within the framework of problem, find the dependence of the dark energy density on scale factor, assuming that $\rho_{DM}=\rho_{DM0}a^{-3+\delta}$, where $\delta$ characterizes the deviation of the dark matter density's evolution from the standard one (in the absence of interaction).

### Problem 13

Let the densities' ratio in model of interacting dark energy and dark matter has the form $\frac{\rho_{DM}}{\rho_{DE}}\propto Aa^{-\xi}.$ Determine the interaction $Q$ between the components.

### Problem 14

Determine the statefinder $\{r,s\}$ (see Chapter Dark Energy) in the model of interacting dark energy and dark matter with the interaction intensity $Q=-3\alpha H$.

### Problem 15

Consider a flat Universe filled with dark energy in the form of the Chaplygin gas ($p_{ch}=-A/\rho_{ch}$) and dark matter. Let the components interact with each other with intensity $Q=3\Gamma H\rho_{ch}$ ($\Gamma>0$). Show that for large $a$ ($a\to\infty$) holds $w_{ch}\equiv p_{ch}/\rho_{ch}<-1$, i.e. in such a model the Chaplygin gas behaves as the fantom energy.

### Problem 16

Interaction between dark matter and dark energy leads to non--conservation of matter, or equivalently, to scale dependence for the mass of particles that constitute the dark matter. Show that, within the framework of the model considered in problem, the relative change of particles mass per Hubble time equals the interaction constant.

### Problem 17

Consider a model of flat homogeneous and isotropic Universe filled with matter (baryon and dark), radiation and a negative pressure component (dark energy in the form of quintessence). Assuming that the baryon matter and radiation conserve separately and the dark components interact with each other, describe the dynamics of such a system.

### Problem 18

Assume that the dark matter particles' mass $m_{DM}$ depends on a scalar field $\varphi$. Construct the model of interacting dark energy and dark matter in this case.

### Problem 19

Find the equation of motion for the scalar field interacting with dark matter if its particles' mass depends on the scalar field.

### Problem 20

Find the interaction $Q$ for the Universe, with interacting dark energy and dark matter, assuming that their densities' ratio takes the form $\rho_m/\rho_{DE}=f(a)$, where $f(a)$ is an arbitrary differentiable function of the scale factor.

### Problem 21

Using the results of previous problem, find the quantity $E^2\equiv H^2/H_0^2$, which is necessary to test the cosmological models and to find restrictions on the cosmological parameters. Assume that $f(a)=f_0 a^\xi$, where $\xi$ is constant.

### Problem 22

In the Universe described in problems 1 and 2, calculate the distance module corresponding to the redshift's range $0.014\leq z\leq 1.6$ and find the quantity $E=H/H_0$.

### Problem 23

Consider a model of the Universe with interacting components, in which the scale factor dependence for one of them takes the form $\rho _{1} (a)=C_{1} a^{\alpha } +C_{2} a^{\beta }$, where $C_1$, $C_2$, $\alpha$ and $\beta$ are constants. Find the interaction $Q$.

### Problem 24

In the model of the Universe considered above assume that $\rho_1=\rho_m$, $\rho_2=\rho_{DE}$ and $Q=\gamma H\rho_1$. Find the range of possible values of the interaction constant $\gamma$.

### Problem 25

Show that in the model considered in problem 61 there is no coincidence problem.

### Problem 26

In the model of problem 61 find the scale factor's value $a_{eq}$ at the time when the dark matter density was equal to that of dark energy, and the scale factor's value $a_{ac}$ at the time when the Universe started to accelerate. Find the ranges of the state parameter $w_{DE}$ corresponding to the cases $a_{ac}<a_{eq}$ and $a_{ac}>a_{eq}$.

## Non-linear Models

The interactions studied so far are linear in the sense that the interaction term in the individual energy balance equations is proportional to either dark matter density or to dark energy density or to a linear combination of both densities. Also from a physical point of view an interaction proportional to the product of dark components seems preferred: an interaction between two components should depend on the product of the abundances of the individual components, as, e.g., in chemical reactions. Moreover, such type of interaction looks much more attractive in comparison with the observations than the linear one. Below we investigate the dynamics for a simple two-component model with a number of non-linear interactions (F.Arevalo, A.Bacalhau and W. Zimdahl, arXiv: 1112.5095)

### Problem 27

Let interaction term $Q$ is a non-linear function of the energy densities of the components and/or the total energy density. Motivated by the structure $\rho_{DM}=\frac{r}{1+r}\rho,\ \rho_{DE}=\frac{1}{1+r}\rho,$ $\rho\equiv\rho_{DM}+\rho_{DE},\ r\equiv\frac{\rho_{DM}}{\rho_{DE}},$ consider the ansatz $Q=3H\gamma\rho^m r^n(1+r)^s,$ where $\gamma$ is a positive coupling constant. Show that
1) for $s=-m$ interaction term is proportional to a power of products of the densities of the components;
2) for $(m,n,s)=(1,1,-1)$ and $(m,n,s)=(1,0,-1)$ the linear case is reproduced.

### Problem 28

Find analytical solution of non-linear interaction model covered by the ansatz of previous problem for $(m,n,s)=(1,1,-2)$, $$Q=3H\gamma\rho_{DE}\rho_{DM}/\rho$$.

### Problem 29

Find analytical solution of non-linear interaction model for $(m,n,s)=(1,2,-2)$, $$Q=3H\gamma\rho_{DM}^2/\rho$$.

### Problem 30

Find analytical solution of non-linear interaction model for $(m,n,s)=(1,0,-2)$, $$Q=3H\gamma\rho_{DE}^2/\rho$$.

## The Chaplygin Gas

Any fundamental science shows an obvious tendency to decrease number of the fundamental substances. The well-known examples are transition from chemical elements to nucleons and electrons, and then from baryons and mesons - to quarks. Cosmology manifests an intention to develop a unified model for dark energy and dark matter. The observed transition from the matter domination to that of dark energy makes it attractive to introduce a dynamical substance which would mimic properties of matter in early Universe and possess the negative pressure to provide the accelerated expansion in the present epoch. The Chaplygin gas represents the simplest substance with the required properties. Its equation of state is postulated to be: $p=-\frac{A}{\rho}, \ A>0.$

### Problem 31

Find the scale factor dependence for the density of the Chaplygin gas.

### Problem 32

Show that in the early Universe the Chaplygin gas behaves as matter with zero pressure, and at later times---as the cosmological constant.

### Problem 33

Find the range of the energy density $\rho_{ch}$ corresponding to the accelerated expansion of the Universe filled with dark energy in the form of Chaplygin gas with pressure $p=-A/\rho_{ch}$ and non--relativistic matter with density $\rho_m$.

### Problem 34

Show that the sound speed in the Chaplygin gas in the late Universe is close to the light speed.

### Problem 35

Show that the sound speed in the Chaplygin gas behaves as $c_s\propto t^2$ in the matter--dominated epoch.

### Problem 36

Show that the cosmological solution corresponding to the Chaplygin gas can be obtained in the quintessence model.

### Problem 37

Find the dependence of density on the scale factor in the generalized Chaplygin gas model with state equation $p=-\frac{A}{\rho^\alpha},\ (A>0,\ \alpha>0.)$

### Problem 38

In the generalized Chaplygin gas model (see previous problem) find the state equation parameter $w$.

### Problem 39

Determine the sound speed in the generalized Chaplygin gas model (see problem). Can it exceed the speed of light?

### Problem 40

Let us present the energy density in the generalized Chaplygin gas model of problem in the form of the sum $\rho_{ch}^{(gen)} = \rho_{DM}+\rho_{DE}$, where $\rho_{DM}$ is the component with the properties of non-relativistic mater ($p_{DM}=0$) and $\rho_{DE}$ is the component with the properties of dark energy described by the state equation $p_{DE}=w_{DE}\rho_{DE}$. Find the restriction on the model's parameter $w_{DE}$.

### Problem 41

Find the dependence of density on scale factor for the Chaplygin gas model with the state equation $p = \left( {\gamma - 1} \right)\rho - \frac{A}{\rho ^\alpha };\quad 0 \le \alpha \le 1$ (the so-called modified Chaplygin gas).

### Problem 42

Construct the effective potential for the Chaplygin gas considering it as a scalar field. Do the same for the generalized Chaplygin gas (see problem) and the modified Chaplygin gas of the previous problem.

### Problem 43

Show that for the Chaplygin gas model the line $w=-1$ cannot be crossed.

### Problem 44

Show that for the generalized Chaplygin gas model the line $w=-1$ cannot be crossed.

## Universe as the Dynamical System

The Universe described by the Friedman equations can be treated as an autonomous dynamical system. Its behavior is determined by the system of differential equations of the form: $\dot{\vec{x}}=\vec{f}(\vec{x}).$ To study dynamics of the system it is of crucial importance to find the so-called critical points $\vec{x}^*$ defined by the following condition: $\vec{f}(\vec{x}^*)=0.$ In order to study the stability of the critical points, we expand around them: $\vec{x}=\vec{x}^*+\vec{u},\ \dot{\vec{u}}=\vec{f}'(\vec{x}^*)\vec{u}+\vec{g}(\vec{x}).$ Here $\vec{g}(\vec{x})/||x||\to0$ as $\vec{x}\to\vec{x}^*$, and $f'_{ij}(\vec{x}^*)\hat M(\vec{x}^*)=\frac{\partial f_i}{\partial x_j}(\vec{x}^*)$ is constant non-singular stability matrix, whose eigenvalues encode the behavior of the dynamical system near the critical point. The eigenvalues are just the roots of the equation $\det|\hat M -\lambda \hat I|=0$ We yet limit ourself to two-dimensional case. If the solutions are non-degenerate and real, they describe a stable node for $\lambda_{\pm}<0$ an unstable node for $\lambda_{\pm}<0$ and a saddle if $\lambda_{+}$ and $\lambda_{-}$ have different signs. For complex eigenvalues $\lambda_{\pm}=\alpha\pm i\beta$, it is the sign of $\alpha$ that determines the character of the critical point. For $\alpha=0$ the critical point is a center, for $\alpha<0$ it is a stable focus and for $\alpha>0$ it is an unstable focus.

Inspired by Yi Zhang, Hui Li, arXiv:1003.2788

### Problem 45

Assume that there are two components in the Universe: background matter and the dark energy. Obtain equations of motion for relative densities of both components.

### Problem 46

Find fixed points for the dynamical system considered in the previous problem and analyze their stability.

### Problem 47

Find critical points for the model of interacting dark components with $Q=-3H\Pi$.

### Problem 48

Show that for the model considered in the previous problem, independently from the specific interaction, the existence of the critical points $r_c$ and $\rho_c$ requires a transfer from dark energy to dark matter.

### Problem 49

Find eigenvalues of the stability matrix for the model of Universe considered in the problems 1 and 2.

### Problem 50

Find eigenvalues of the stability matrix under assumption that $Q=3H\gamma\rho^mr^n(1+r)^s$ (see problem).

### Problem 51

Classify critical points for the model of interacting dark components considered in the previous problem.