Physical mechanism of energy exchange

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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]).