Phase space structure of models with interaction

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The evolution of a Universe filled with interacting components can be effectively analyzed in terms of dynamical systems theory. Let us consider the following coupled differential equations for two variables \begin{equation} \label{IDE_s6_1} \begin{array}{l} \dot x=f(x,y,t),\\ \dot y=g(x,y,t). \end{array} \end{equation} We will be interested in the so-called autonomous systems, for which the functions $f$ and $g$ do not contain explicit time-dependent terms. A point $(x_c,y_c)$ is said to be a fixed (a.k.a. critical) point of the autonomous system if \[f(x_c,y_c)=g(x_c,y_c)=0.\] A critical point $(x_c,y_c)$ is called an attractor when it satisfies the condition \(\left(x(t),y(t)\right)\to(x_c,y_c)\) for $t\to\infty$. Let's look at the behavior of the dynamical system (\ref{IDE_s6_1}) near the critical point. For this purpose, let us consider small perturbations around the critical point \[x=x_c+\delta x,\quad y=y_c+\delta y.\] Substituting it into (\ref{IDE_s6_1}) leads to the first-order differential equations: \[\frac{d}{dN}\left(\begin{array}{c}\delta x\\ \delta y\end{array}\right) = \hat M \left(\begin{array}{c}\delta x\\ \delta y\end{array}\right).\] Taking into account the specifics of the problem that we are solving, we made the change \[\frac{d}{dt}\to\frac{d}{dN},\] where $N=\ln a$. The matrix $\hat M$ is given by \[\hat M = \left( \begin{array}{lr} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\ {} & {}\\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{array} \right) \] The general solution for the linear perturbations reads \[\delta x=C_1e^{\lambda_1 N} + C_2e^{\lambda_2 N},\] \[\delta y=C_3e^{\lambda_1 N} + C_4e^{\lambda_2 N},\] The stability around the fixed points depends on the nature of the eigenvalues.

Let us treat the interacting dark components as a dynamical system described by the equations \[\rho'_{de}+3(1+w_{de})\rho_{de}=-Q\] \[\rho'_{dm}+3(1+w_{dm})\rho_{dm}=Q\] Here, the prime denotes the derivative with respect to $N=\ln a$. Note that although the interaction can significantly change the cosmological evolution, the system is still autonomous. We consider the following specific interaction forms, which were already analyzed above: \[Q_1=3\gamma_{dm}\rho_{dm},\quad Q_1=3\gamma_{de}\rho_{de},\quad Q_1=3\gamma_{tot}\rho_{tot}\]

Problem 1

Find effective EoS parameters $w_{(dm)eff}$ and $w_{(de)eff}$ for the interactions $Q_1$, $Q_2$ and $Q_3$.

Problem 2

Find the critical points of equation for ratio $r=\rho_{dm}/\rho_{de}$ if $Q=3\alpha H(\rho_{dm}+\rho_{de})$, where the phenomenological parameter $\alpha$ is a dimensionless, positive constant, $w_{dm}=0$, $w_{de}=const$.

Problem 3

Show, that the remarkable property of the model, considered in the previous problem, is that for the interaction parameter $\alpha$, consistent with the current observations $\alpha<2.3\times10^{-3}$ the ratio $r$ tends to a stationary but unstable value at early times, $r_s^+$, and to a stationary and stable value, $r_s^-$ (an attractor), at late times. Consequently, as the Universe expands, $r(a)$ smoothly evolves from $r_s^+$ to the attractor solution $r_s^-$.

Problem 4

Transform the system of equations \[\rho'_{de}+3(1+w_{de})\rho_{de}=-Q,\] \[\rho'_{dm}+3(1+w_{dm})\rho_{dm}=Q,\] into the one for the fractional density energies.(after [1])

Problem 5

Analyze the critical points of the autonomous system, obtained in the previous problem \[\Omega'_{dm}=3f_j \Omega_{dm}\Omega_{de},\] \[\Omega'_{de}=-3f_j \Omega_{dm}\Omega_{de},\] by imposing the conditions $\Omega'_{dm}=\Omega'_{de}=0$ and $\Omega_{dm}+\Omega_{de}=1$ (flatness of Universe).

Problem 6

Construct the stability matrix for the dynamical system considered in the problem and determine its eigenvalues.

Problem 7

Using result of the previous problem, determine eigenvalues of the stability matrix for the following cases: i) $\Omega_{dm} = 1$, $\Omega_{de} = 0$, $f_j \ne 0$; ii) $\Omega_{dm} = 0$, $\Omega_{de} = 1$, $f_j \ne 0$; iii) $f_j = 0$.

Problem 8

Obtain position and type of the critical points obtained in the previous problem for the case of cosmological constant interacting with dark matter as $Q=3\gamma_{dm}\rho_{dm}$.

Problem 9

Construct the stability matrix for the following dynamical system \begin{align} \nonumber \rho' & = - \left(1+\frac{w_{de}}{1+r}\right)\rho,\\ \nonumber r' & = r \left[w_{de} - \frac{(1+r)^2}{r\rho}\Pi\right], \end{align} and determine its eigenvalues. (After [2])