Non-linear interaction in the dark sector
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 $r(t)\equiv\rho_1(t)/\rho_2(t)$.
The equations of the system are: \[\dot\rho_1 + 3H\rho_1=\gamma\rho_1\rho_2,\] \[\dot\rho_2 + 3H\rho_2=-\gamma\rho_1\rho_2,\] \[3H^2=\rho_1+\rho_2.\] Substitute the relation $\rho_2=3H^2-\rho_1$ into the conservation equation for $\rho_1$, and take into account that \[H=\frac2{3t}\] for a matter-dominated Universe to obtain \[\dot\rho_1 + \frac2t\rho_1=\gamma\rho_1\left(\frac4{3t^2}-\rho-1\right).\] Its general solution reads \[\rho_1 = \frac4{t^2(4Ae^{4\gamma/3t}+3)},\quad \rho_1 = \frac4{t^2(\frac9{4A}e^{-4\gamma/3t}+3)},\] The ratio of these densities is \[\frac{\rho_1}{\rho_2}=\frac{3}{4A}e^{-4\gamma/3t}.\]
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 $t\to\infty$?
According to the result of the previous problem, for $t\to\infty$ the ratio $r$ tends exponentially quickly to the constant $3/(4A)$, with the individual densities evolving as $\rho_i\sim t^{-2}$. This suggests that even if there were several types of 'dust' in the Universe mutually interacting in this manner, we could approximate the evolution of their densities by treating them as non-interacting after a short initial period. In other words, we might write $\rho_i\sim C_it^{-2}$ for $t>t^*$, where the relative magnitudes of the $C_i$ are determined from a relatively short initial evolution up to time $t^*$.
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.
In this case the dynamical equations simplify to: \[\dot\rho_{dm} + 3H\rho_{dm}=\gamma\rho_{dm}\rho_\Lambda,\] \[\dot\rho_\Lambda = - \gamma\rho_{dm}\rho_\Lambda,\] \[3H^2=\rho_{dm}+\rho_\Lambda,\] \[2\dot H=-\rho_{dm}.\] Exclude $\rho_{dm}$ from the system \[\dot\rho_\Lambda = - \gamma\rho_{dm}\rho_\Lambda,\] \[2\dot H=-\rho_{dm},\] to find \[\rho_\Lambda=Ae^{2\gamma H}.\] One can see that $\rho_\Lambda$ never vanishes, so 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.
The general case for the considered model is described by the following equations: \begin{equation}\label{IDE_35_1} \dot\rho_{dm} + 3H\rho_{dm}=\gamma\rho_{dm}\rho_{de}, \end{equation} \begin{equation}\label{IDE_35_2} \dot\rho_{de} = - \gamma_{de} H \rho_{dm} - \gamma\rho_{dm}\rho_{de}, \end{equation} \begin{equation}\label{IDE_35_3} 3H^2=\rho_{dm}+\rho_{de}, \end{equation} Addition (\ref{IDE_35_1}) and (\ref{IDE_35_2}) yields \begin{equation} \label{IDE_35_4} \dot\rho_{dm} + \dot\rho_{de} = -\sqrt{3(\rho_{dm}+\rho_{de})}(\rho_{dm}+\gamma_{de}\rho_{de}) \end{equation} Subtraction of appropriate multiples of (\ref{IDE_35_1}) and (\ref{IDE_35_2}) yields \begin{equation} \label{IDE_35_5} \gamma_{de}\dot\rho_{dm}\rho_{de} -\gamma \gamma_{de}\rho_{dm}\rho_{de}^2 = \dot\rho_{de}\rho_{dm} + \gamma\rho_{dm}^2\rho_{de}. \end{equation} Combine (\ref{IDE_35_4}) and (\ref{IDE_35_5}) and integrate to get the following relation between the dark energy and dark matter densities: \[\rho_{de}=\rho_{dm}^{\gamma_{de}}e^{2\gamma\sqrt{\frac{\rho_{dm} + \rho_{de}}{3}}}\] Due to this rather awkward relation between the two densities, it is difficult to make further progress using purely analytical methods.
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 $(m,n,s)=(1,1,-2)$, $Q=3H\gamma\rho_{de}\rho_{dm}/\rho$.
In such a case equation for the ratio $r=\rho_{dm}\rho_{de}$ reduces to \[r'=r[w+\gamma],\] where $w=p_{de}\rho_{de}$, and the prime denotes the derivative with respect to $\ln a^3$. The solution of this equation is \[r=r_0a_*{3(w+\gamma)}.\] Substituting this solution into \[\rho'=-\left(1+\frac w{1+r}\right)\rho\] one finds the total energy density \[\rho=\rho_0 a^{-3(w+\gamma)}\left[\frac{1+r_0a^{3(w+\gamma)}}{1+r_0}\right]^{\frac{w}{w+\gamma}}.\] The densities of the components are \[\rho_{dm}=\rho_{dm0} a^{-3(1-\gamma)}\left[\frac{1+r_0a^{3(w+\gamma)}}{1+r_0}\right]^{-\frac{\gamma}{w+\gamma}},\] \[\rho_{de}=\rho_{de0} a^{-3(1+w)}\left[\frac{1+r_0a^{3(w+\gamma)}}{1+r_0}\right]^{-\frac{\gamma}{w+\gamma}},\] \[\rho_{dm0}=\frac{r_0}{1+r_0}\rho_0,\quad \rho_{de0}=\frac{1}{1+r_0}\rho_0.\]
Problem 7
Find analytical solution of non-linear interaction model for $(m,n,s)=(1,2,-2)$, $Q=3H\gamma\rho_{dm}^2/\rho$.
Proceeding as in the previous problem, one finds \[r=r_0\frac w{(w=\gamma r_0)a^{-3w}-\gamma r_0},\] \[\rho=\rho_0a^{-3\left(1-\frac{w\gamma}{w-\gamma}\right)}\left[\frac{(w+\gamma r_0)a^{-3w}+r_0(w-\gamma)}{w(1+r_0)}\right]^{\frac{w}{w-\gamma}}.\]
Problem 8
Find analytical solution of non-linear interaction model for $(m,n,s)=(1,0,-2)$, $Q=3H\gamma\rho_{de}^2/\rho$.
For $w<0$ the solutions are \[r=\left(r_0-\frac\gamma{|w|}\right)a^{-3|w|}+\frac\gamma{|w|},\] \[\rho=\rho_0a^{-3\left(1-\frac{|w|}{|w|+\gamma}\right)}\left[\frac{|w|+\gamma (|w|r_0-\gamma)a^{-3|w|}}{|w|(1+r_0)}\right]^{\frac{|w|}{|w|+\gamma}}.\]
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])
Integrate the conservation equation for polytropic component \[\dot\rho_{de} +3H\left(\rho_{de}+K\rho_{de}^{1\frac1n}\right)=-3\alpha H\rho_{de}\] to obtain \[\rho_{de}(a)=\left(\frac1{Ba^{3(1+\alpha)/n}-\bar K}\right)\] where $B$ is the integration constant and $\bar K\equiv K/(1+\alpha)$ . Note that to have a positive energy density for arbitrary value of $n$, it is required that $Ba^{3(1+\alpha)/n}>\bar K$. In the case $Ba^{3(1+\alpha)/n}=\bar K$ one has $\rho_{de}\to\infty$ and therefore the polytropic component has a finite-time singularity at \[a_c=(\bar K/B)^{\frac n {3(1+\alpha)}}.\]
Problem 10
Show that under certain conditions the interacting polytropic dark energy with $Q=3\alpha H\rho_{de}$ behaves as the phantom energy.
Substituting $Q=3\alpha H\rho_{de}$ into conservation equation for DE, one has \[\dot\rho_{de} +3H(1+\alpha+w_{de})\rho_{de}=0.\] Differentiate with respect to time the expression for $\rho_{de)}(a)$ obtained in the previous problem to get \[\dot\rho_{de}=-3BH(1+\alpha)a^{\frac{3(1+\alpha)}{n} } \rho_{de}^{1+1/n}.\] Substitute this equation into the conservation equation to obtain the effective EoS parameter of interacting polytropic model as \[w_{(de)eff}=-1-\frac{a^{\frac{3(1+\alpha)}{n}}}{c-a^{\frac{3(1+\alpha)}n}},\] where $c\equiv\bar K/B$. We see that the interacting polytropic model behaves as a phantom model, i.e. $w_{(de)eff}<-1$ when $c>a^{3(1+\alpha)/n}$.
Problem 11
Find deceleration parameter for the system considered in the problem.
Taking the time derivative of first Friedmann equation and using conservation equations for $\rho_{dm}$ and $\rho_{de}$ with $Q=3\alpha H\rho_{de}$ one obtains \[\frac{\dot H}{H^2}=-\frac32\left[1+\frac{c(1+\alpha)}{a^{3(1+\alpha)/n}-c}\Omega_{de}\right].\] Consequently, \[q=-1+\frac32\left[1+\frac{c(1+\alpha)}{a^{3(1+\alpha)/n}-c}\Omega_{de}\right].\] It is obvious that in the limiting case of matter-dominated Universe the obtained expression is reduced to well-known result $q=1/2$.
