# Physical mechanism of energy exchange

### Problem 1

Models with interaction between DM and the DE field can be realized if we make just an obvious assumption: the mass of the cold DM particles is a function of the DE field. Let the dark matter particles will be collisionless and nonrelativisic. Hence, the pressure of this fluid and its energy density are $$p_{dm}=0$$ and $$\rho_{dm}=nm$$ respectively, where $m$ is the rest mass and $n$ is the number density of DM particles. We define $m=\lambda\varphi$ where $\varphi$ is a scalar field and $\lambda$ is a dimensionless constant. Show how such assumption affects the scalar field dynamics (after [1]). [2]

### Problem 2

Show that the DM on a inhomogeneous vacuum background can be treated as as interacting DE and DM. (after [3].)

### Problem 3

Obtain general equations of motion for DE interacting with DM (after [4]).