Non-linear interaction in the dark sector

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The interaction studied so far are linear in the sense that the interaction term in the individual energy balance equations is proportional either to dark matter density or to dark energy density or to a linear combination of both densities. Also from a physical point of view an interaction proportional to the product of dark components seems preferred: an interaction between two components should depend on the product of the abundances of the individual components, as, e.g., in chemical reactions. Moreover, such type of interaction looks more preferable then the linear one when compared with the observations. Below we investigate the dynamics for a simple two-component model with a number of non-linear interactions.

In problems 32-34 let a spatially flat FLRW Universe contain perfect fluids with densities $\rho_1$ and $\rho_2$. Consider a nonlinear interaction of the form $Q=\gamma\rho_1\rho_2$. (after [1])

Problem 1

Consider a model in which both fluids are dust. Find $r(t)\equiv\rho_1(t)/\rho_2(t)$.

Problem 2

Consider a Universe with more than two CDM components interacting with each other. What is the asymptotic behavior of the individual densities of the components in the limit $t\to\infty$?

Problem 3

Consider a Universe containing a cold dark matter and a dark energy, in which the dark energy behaves like a cosmological constant. Show that in such model dark energy is a perpetual component of the Universe.

Problem 4

Consider a two-component Universe with the interaction $Q=\gamma\rho_1\rho_2$. Let one component is CDM ($\rho_1=\rho_{dm}$, $w_1=0$), and the second is the dark energy with arbitrary state equation ($\rho_2=\rho_{de}$, $w_2=\gamma_{de}-1$). (The case considered in the previous problem corresponds to $\gamma_{de}=0$.) Find the relation between the dark energy and dark matter densities.

Problem 5

Let interaction term $Q$ be a non-linear function of the energy densities of the components and/or the total energy density. Motivated by the structure \[\rho_{dm} = \frac r{1+r}\rho, \quad \rho_{de} = \frac r{1+r}\rho,\] \[\rho\equiv\rho_{dm}+\rho_{de},\quad r\equiv\frac{\rho_{dm}}{\rho_{de}}\] consider ansatz \[Q=3H\gamma\rho^m r^n(1+r)^s.\]

where $\gamma$ is a positive coupling constant. Show that for $s=-m$ interaction term is proportional to a power of products of the densities of the components. For $(m,n,s)=(1,1,-1)$ and $(m,n,s)=(1,0,-1)$ reproduce the linear case.(After [2])

Problem 6

Find analytical solution of non-linear interaction model covered by the ansatz of previous problem for $(m,n,s)=(1,1,-2)$, $Q=3H\gamma\rho_{de}\rho_{dm}/\rho$.

Problem 7

Find analytical solution of non-linear interaction model for $(m,n,s)=(1,2,-2)$, $Q=3H\gamma\rho_{dm}^2/\rho$.

Problem 8

Find analytical solution of non-linear interaction model for $(m,n,s)=(1,0,-2)$, $Q=3H\gamma\rho_{de}^2/\rho$.

Problem 9

Consider a flat Universe filled by CDM and DE with a polytropic equation of state \[p_{de}=K\rho_{de}^{1\frac1n}\] where $K$ and $n$ are the polytropic constant and polytropic index, respectively. Find dependence of DE density on the scale factor under assumption that the interaction between the dark components is $Q=3\alpha H\rho_{de}$. (after [3])

Problem 10

Show that under certain conditions the interacting polytropic dark energy with $Q=3\alpha H\rho_{de}$ behaves as the phantom energy.

Problem 11

Find deceleration parameter for the system considered in the problem.