# Simple linear models

### Problem 1

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

### Problem 2

Obtain the equation for the evolution of the DE energy density for $Q=\delta(a) H\rho_{dm}$.

### Problem 3

Find $\rho_{dm}$ and $\rho_{de}$ in the case $Q=\delta H\rho_{dm}$, $\delta=const$, $w_{de}=const$.

### Problem 4

As was shown above, 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 of previous problem ($Q=\delta H\rho_{dm}$, $\delta=const$, $w_{de}=const$) the relative change of particles mass per Hubble time equals to the interaction constant.

### Problem 5

Find $\rho_{dm}$ and $\rho_{de}$ in the case $Q=\delta H\rho_{de}$, $\delta=const$, $w_{de}=const$.

### Problem 6

Find $\rho_{dm}$ and $\rho_{de}$ in the case $Q=\delta(a) H\rho_{de}$, $\delta(a)=\beta_0a^\xi$, $w_{de}=const$.
(after [1])

### Problem 7

Let's look at a more general linear model for the expansion of a Universe that contains two interacting fluids with the equations of state $p_1 = (\gamma_1-1)\rho_1,$ $p_2 = (\gamma_2-1)\rho_2,$ and energy exchange $\dot\rho_1+3H\gamma_1\rho_1 = -\beta H\rho_1 + \alpha H\rho_2,$ $\dot\rho_2+3H\gamma_2\rho_2 = \beta H\rho_1 - \alpha H\rho_2.$ Here $\alpha$ and $\beta$ are constants describing the energy exchanges between the two fluids. Obtain the equation for $H(t)$ and find its solutions (After [2], [3]).

### Problem 8

Show that the energy balance equations (modified conservation equations) for $Q\propto H$ do not depend on H.

### Problem 9

The Hubble parameter is present in the first Friedmann equation quadratically. This gives rise to a useful symmetry within a class of FLRW models. Because of this quadratic dependence, Friedmann's equation remains invariant under a transformation $H\to-H$ for the spatially flat case. This means it describes both expanding and contracting solutions. The transformation $H\to-H$ can be seen as a consequence of the change $a\to1/a$ of the scale factor of the FLRW metric. If, instead of just the first Friedmann equation, we want to make the whole system of Universe-describing equations invariant relative to this transformation, we must expand the set of values that undergo symmetry transformations. Then, when we refer to a duality transformation, we have in mind the following set of transformations $H\to\bar H=-H,\quad \rho\to\bar\rho=\rho,\quad p\to\bar p=-2\rho-p,\quad \gamma\equiv\frac{\rho+p}{\rho}\to\bar\gamma\equiv\frac{\bar\rho+\bar p}{\bar\rho}=-\gamma.$

Generalize the duality transformation to the case of interacting components.(after [4].)