# Phenomenology of interacting models

### Problem 1

Find the effective state parameters $w_{(de)eff}$ and $w_{(dm)eff}$ that would allow one to treat the interacting dark components as non-interacting.

### Problem 2

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 3

Find the effective state parameters $w_{(de)eff}$ and $w_{(dm)eff}$ for the case of the warm dark matter ($w_{dm}\ne0$) and analyze the features of dynamics in this case.

### Problem 4

Show that the quintessence coupled to DM with certain sign of the coupling constant behaves like a phantom uncoupled model, but without negative kinetic energy.

Use the result of the previous problem. When $Q<0$ it is possible that $w_{(de)eff}<w_{de}$. This means that the coupled quintessence behaves like a phantom uncoupled model, but without negative kinetic energy.

### Problem 5

In order to compare dynamics of a model with observational results it is useful to analyze all dynamic variables as functions of redshift rather than time. Obtain the corresponding transformation for the system of interacting dark components.

### Problem 6

Show that energy exchange between dark components leads to time-dependent effective potential energy term in the first Friedman equation. (after [1]).

### Problem 7

Show that the system of interacting components can be treated as the uncoupled one due to introduction of the partial effective pressure of the dark components

    $\Pi_{de}\equiv\frac{Q}{3H},\quad \Pi_{dm}\equiv-\frac{Q}{3H}.$


### Problem 8

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

### Problem 9

Assume that the mass $m_{dm}$ of DM particles depends exponentially on the DE scalar field $m=m_*e^{-\lambda\varphi}$. Find the interaction term $Q$ in this case.

### Problem 10

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

### Problem 11

Make the transformation from the variables $(\rho_{de}, \rho_{dm})$ to $\left(r=\frac{\rho_{dm}}{\rho_{de}}, \rho = \rho_{dm} + \rho_{de}\right)$ for the system of interacting dark components.

### Problem 12

Generalize the result of previous problem to the case of warm dark matter.

### Problem 13

Calculate the derivatives $dr/dt$ and $dr/dH$ for the case of flat universe with the interaction $Q$.

### Problem 14

It was shown in the previous problem that $\dot r=r\left(\frac{\dot\rho_{dm}}{\rho_{dm}}-\frac{\dot\rho_{de}}{\rho_{de}}\right) = 3Hr \left(w_{de} +\frac{1+r}{\rho_{dm}}\frac Q{3H}\right)=(1+r)\left[3Hw_{de}\frac{r}{1+r}+\Gamma\right],\quad \Gamma\equiv\frac Q {\rho_{de}}.$ Exclude the interaction $Q$ and reformulate the equation in terms of $\rho_{de}$, $H$ and its derivatives.

### Problem 15

Generalize the result, obtained in the previous problem, for the case of non-flat Universe [2]

### Problem 16

Show that critical points in the system of equations obtained in problem \ref{IDE_12} exist only for the case of dark energy of the phantom type.

### Problem 17

Show that the result of previous problem holds also for warm dark matter.

### Problem 18

Show that existence of critical points in the system of equations obtained in problem \ref{IDE_12} requires a transfer from dark energy to dark matter.

### Problem 19

Show that the result of previous problem holds also for warm dark matter.

### Problem 20

Assume that the ratio of the interacting dark components equals $r\equiv\frac{\rho_{dm}}{\rho_{de}}\propto a^{-\xi}, \quad \xi\ge0.$ Analyze how the interaction $Q$ depends on $\xi$.

### Problem 21

Show that the choice $r\equiv\frac{\rho_{dm}}{\rho_{de}}\propto a^{-\xi}, \quad (\xi\ge0)$ guarantees existence of an early matter-dominated epoch.

### Problem 22

Find the interaction $Q$ for the Universe with interacting dark energy and dark matter, assuming that ratio of their densities takes the form

   $r\equiv\frac{\rho_{dm}}{\rho_{de}}=f(a),$ where $f(a)$ is an arbitrary differentiable function of the scale factor.


### Problem 23

Let $Q=\frac{\dot f(t)}{f(t}\rho_{dm}.$ Show that the sign of the deceleration parameter is defined by the ratio $\frac{\dot f}{fH}.$

### Problem 24

Show that in the model, considered in the previous problem, the transition from the accelerated expansion to the decelerated one can occur only due to time dependence of the interaction.