Cosmological models with a change of the direction of energy transfer

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Let us consider one more type of interaction $Q$, whose sign (i.e., the direction of energy transfer) changes when the mode of decelerated expansion is replaced by the mode of accelerated expansion, and vice versa. The simplest interaction of this type is the one proportional to the deceleration parameter. An example of such interaction is \[Q=q(\alpha\dot\rho+\beta H\rho),\] where $\alpha$ and $\beta$ are dimensionless constants, and $\rho$ can be any of densities $\rho_{de}$, $\rho_{dm}$ or $\rho_{tot}$. In the following problems for simplicity we restrict ourselves to the decaying $\Lambda$ model, for which $\dot\rho_{de}=\dot\rho_\Lambda=-Q$ and $p_{de}=-\rho_{de}$. (after [1])

Problem 1

Construct general procedure to the Hubble parameter and the deceleratio parameter for the case \(Q=q(\alpha\dot\rho_{dm}+\beta H\rho_{dm})\).

Problem 2

Find deceleration parameter for the case $\alpha=0$ in the model considered in the previous problem.

Problem 3

Consider the model $Q=q(\alpha\dot\rho_\Lambda+3\beta H\rho_\Lambda)$. Obtain the Hubble parameter $H(a)$ and deceleration parameter for the case $\alpha=0$.