Does the diffusion dark matter-dark energy interaction model solve cosmological puzzles?

We study dynamics of cosmological models with diffusion effects modeling dark matter and dark energy interactions. We show the simple model with diffusion between the cosmological constant sector and dark matter, where the canonical scaling law of dark matter $({\rho }_{dm,0}{a}^{-3}(t))$ is modified by an additive $\epsilon (t)=\gamma t{a}^{-3}(t)$ to the form ${\rho }_{dm}={\rho }_{dm,0}{a}^{-3}(t)+\epsilon (t)$. We reduced this model to the autonomous dynamical system and investigate it using dynamical system methods. This system possesses a two-dimensional invariant submanifold on which the dark matter-dark energy (DM-DE) interaction can be analyzed on the phase plane. The state variables are density parameter for matter (dark and visible) and parameter $\delta$ characterizing the rate of growth of energy transfer between the dark sectors. A corresponding dynamical system belongs to a general class of jungle type of cosmologies represented by coupled cosmological models in a Lotka-Volterra framework. We demonstrate that the de Sitter solution is a global attractor for all trajectories in the phase space and there are two repellers: the Einstein-de Sitter universe and the de Sitter universe state dominating by the diffusion effects. We distinguish in the phase space trajectories, which become in good agreement with the data. They should intersect a rectangle with sides of ${\mathrm{\Omega }}_{m,0}\in [0.2724,0.3624]$, $\delta \in [0.0000,0.0364]$ at the 95% CL. Our model could solve some of the puzzles of the $\mathrm{\Lambda }\mathrm{CDM}$ model, such as the coincidence and fine-tuning problems. In the context of the coincidence problem, our model can explain the present ratio of ${\rho }_m$ to ${\rho }_{de}$, which is equal $0.4576_{-0.0831}^{+0.1109}$ at a $2\sigma$ confidence level.

Figure 1

Phase portrait of dynamical system $x^{\prime }=x(\delta +3x-3)$, ${\delta }^{\prime }=\delta (-\delta +\frac{3}{2}x)$, where $x={\mathrm{\Omega }}_m=\frac{{\rho }_m}{3H^2}$, $\delta =\frac{\gamma a^{-3}}{H{\rho }_m}$ and $\prime \equiv \frac{d}{d\ln a}$ on the Poincaré sphere coordinates are $X=\frac{x}{\sqrt{x^2+y^2+z^2}}$, $\mathrm{\Delta }=\frac{\delta }{\sqrt{x^2+y^2+z^2}}$. Critical point (1) represents the de Sitter universe—a global attractor for all physical trajectories. Critical point (2) represents the scaling universe. Critical point (3) represents the Einstein-de Sitter universe. Critical points (4) and (5) represent the static universe. Critical point (6) represents the de Sitter universe. The gray region represents the domain of the present value of $X$ and $\mathrm{\Delta }$, which is distinguished by astronomical data. Let us note trajectories lie in the domain with $\mathrm{\Delta }<0$ represent the contracting model but there is no symmetry with respect to the $\mathrm{\Delta }$-axis. At critical point (6), energy density of baryonic matter is negligible as well as density of dark matter and only effects of the relativistic diffusion are important.

Figure 2

The evolution of dark matter energy density for trajectories of type II (for the best fitted values of model parameter together with confidence level at 95%). Dark matter ${\rho }_{dm}$ is expressed in $[100\times \mathrm{km}/(\mathrm{s}\mathrm{Mpc}){]}^2$. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 3

The evolution of dark matter energy density for trajectories of type II (for the best fitted values of model parameter together with confidence level at 95% for the present epoch). Dark matter ${\rho }_{dm}$ is expressed in $[100\times \mathrm{km}/(\mathrm{s}\mathrm{Mpc}){]}^2$. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$. The value of the age of the Universe for the best fit with errors are presented by the dashed lines.

Figure 4

Diagram of scale factor as $a$ function of cosmological time $t$ for trajectories of type II (for the best fitted values of model parameter together with confidence level at 95%). For the present epoch $T$ $a(T)=1$. A universe is starting from the initial singularity toward a de Sitter universe. This type of behavior is favored by the observational data. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 5

Evolution of dimensionless parameter $\delta$ of cosmological time $t$ for trajectories of type II (for the best fitted values of model parameter together with confidence level at 95%). Note that as trajectory in the phase space achieved the state of the pericentrum located in the saddle point, this state is corresponding on the diagram the maximum. Note that the existence of a maximum value of $\delta$ parameter $(\frac{{\rho }_{m,0}}{\gamma }+t_{\max }-t_0){\rho }_m(t_{\max })=2H(t_{\max })$. For the late time $\delta (t)$ function is decreasing function of $t$ and $\rho (\infty )=0$. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 6

Dependence of Hubble function of trajectories of type II (for the best fitted values of model parameter together with confidence level at 95%). For late times $H(t)$ goes to constant values (${\mathrm{deS}}_{+}$). The Hubble function $H(t)$ is expressed in $[100\times \mathrm{km}/(\mathrm{s}\mathrm{Mpc})]$. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 7

The evolution of dark energy density for the best fitted values of model parameter for trajectories of type II. Dark energy ${\rho }_{de}$ is expressed in $[100\times \mathrm{km}/(\mathrm{s}\mathrm{Mpc}){]}^2$. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 8

Diagram of relation ${\mathrm{\Omega }}_m/{\mathrm{\Omega }}_{de}$ for trajectories of type II (for the best fitted values of model parameter together with confidence level at 95%). We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 9

Diagram of relation ${\mathrm{\Omega }}_m/{\mathrm{\Omega }}_{de}$ for trajectories of type II for the present epoch. Note that at the present epoch ${\rho }_{m,0}\propto {\rho }_{de,0}$ (therefore coincidence problem is solved). We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$. The value of the age of the Universe for the best fit with errors are presented by the dashed lines.

Figure 10

The relation of $H(t)$ for typical trajectory of type I. The $H(t)$ function is expressed in $[100\times \mathrm{km}/(\mathrm{s}\mathrm{Mpc})]$. Note that $H(0)$ is finite therefore it is not a singularity. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 11

Diagram of $a(t)$ for typical trajectory of type I. Note that $a(0)$ is finite therefore it is not a singularity. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 12

Diagram of ${\rho }_m(t)$ for typical trajectory of type I. Note that ${\rho }_m(0)$ is equal zero therefore it is not a singularity. Matter ${\rho }_m$ is expressed in $[100\times \mathrm{km}/(\mathrm{s}\mathrm{Mpc}){]}^2$. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 13

Diagram of ${\rho }_{de}(t)$ for typical trajectory of type I. Note that ${\rho }_{de}(0)$ is finite therefore it is not a singularity. Dark energy ${\rho }_{de}$ is expressed in $[100\times \mathrm{km}/(\mathrm{s}\mathrm{Mpc}){]}^2$. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 14

Diagram of ${\mathrm{\Omega }}_m(t)/{\mathrm{\Omega }}_{de}(t)$ for typical trajectory of type I. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 15

Diagram of $\delta (t)$ for typical trajectory of type I. We choose $(\mathrm{s}\mathrm{Mpc})/(100\times \mathrm{km})$ as a unit of the cosmological time $t$.

Figure 16

Diagram of stability of critical point (3), depending on $w$ and $\tilde{w}$. In the gray domains there is the focus type of critical point and the boundaries of this domains is given by the lines $w=\frac{1}{3}(1+6\tilde{w}+\sqrt{-1+9\tilde{w}(2+3\tilde{w})})$ and $w=\frac{1}{3}(1+6\tilde{w}-\sqrt{-1+9\tilde{w}(2+3\tilde{w})})$. In the blue regions there are the saddle type of critical point and is limited by lines $w=-1/3$ and $\tilde{w}=-1/3$. In the white top and bottom regions there are the stable and unstable nodes, respectively.

Figure 17

Phase portrait of the dynamical system (36)–(37). Note that trajectories for the $\mathrm{\Delta }<0$ represent solutions with the negative value of $H$. From the cosmological point of view trajectories representing expanding models with $\mathrm{\Delta }>0$ are physical. Critical points (1) represents the de Sitter universe without diffusion effect. Critical point (2) represents the scaling universe. Critical point (3) represents the Einstein-de Sitter universe. Critical points (4) and (5) represent the static universe. Critical point (6) and (8) represent the de Sitter universe with diffusion effect.