Difference between revisions of "Non-linear interaction in the dark sector"

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 \[\frac{\rho_1}{\rho_2}=\frac{3}{4A}e^{-4\gamma/3t}.\]

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 $t^*$.

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.

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.

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 \[\rho_{dm}=\rho_{dm0} a^{-3(1-\gamma)}\left[\frac{1+r_0a^{3(w+\gamma)}}{1+r_0}\right]^{-\frac{\gamma}{w+\gamma}},\] \[\rho_{de}=\rho_{de0} a^{-3(1+w)}\left[\frac{1+r_0a^{3(w+\gamma)}}{1+r_0}\right]^{-\frac{\gamma}{w+\gamma}},\] \[\rho_{dm0}=\frac{r_0}{1+r_0}\rho_0,\quad \rho_{de0}=\frac{1}{1+r_0}\rho_0.\]

Proceeding as in the previous problem, one finds \[r=r_0\frac w{(w=\gamma r_0)a^{-3w}-\gamma r_0},\] \[\rho=\rho_0a^{-3\left(1-\frac{w\gamma}{w-\gamma}\right)}\left[\frac{(w+\gamma r_0)a^{-3w}+r_0(w-\gamma)}{w(1+r_0)}\right]^{\frac{w}{w-\gamma}}.\]

For $w<0$ the solutions are \[r=\left(r_0-\frac\gamma{|w|}\right)a^{-3|w|}+\frac\gamma{|w|},\] \[\rho=\rho_0a^{-3\left(1-\frac{|w|}{|w|+\gamma}\right)}\left[\frac{|w|+\gamma (|w|r_0-\gamma)a^{-3|w|}}{|w|(1+r_0)}\right]^{\frac{|w|}{|w|+\gamma}}.\]

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 \[a_c=(\bar K/B)^{\frac n {3(1+\alpha)}}.\]

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 $c>a^{3(1+\alpha)/n}$.

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 $q=1/2$.