Dynamical Properties of a Cosmological Model with Diffusion

Abstract

The description of the dynamics of particles undergoing diffusion in general relativity has been an object of interest in the last years. Most recently a new cosmological model with diffusion has been studied in which the evolution of the particle system is described by a Fokker-Planck equation. This equation is then coupled to a modified system of Einstein equations, in order to satisfy the energy conservation condition. Continuing with this work, we study in the present paper a spatially homogeneous and isotropic spacetime model with diffusion velocity. We write the system of ordinary differential equations of this particular model and obtain the solutions for which the scale factor in the Robertson Walker metric is linear in time. We analyse the asymptotic behavior of the subclass of spatially flat solutions. The system representing the homogeneous and isotropic model with diffusion is rewritten using dynamical variables. For the subclass of spatially flat solutions we were able to determine all equilibrium points and analyse their local stability properties.

Appendix—Tools from Dynamical Systems

In this appendix we include some concepts and tools from the theory of dynamical systems that are used in the paper. The content of this appendix is mainly based on the books [6, 7].

We consider an autonomous dynamical system of the form

$$\begin{aligned} \dot{x} = f(x) , \end{aligned}$$

(35)

where $f \!: \mathbb {R}^n \longrightarrow \mathbb {R}^n$ is a ${\fancyscript{C}}^1$ vector field and $x: \mathbb {R}_0^+ \longrightarrow \mathbb {R}^n$ is a function, $x=x(t)$. In the particular case of a planar dynamical system ($n=2$) we introduce the following notation

$$\begin{aligned} x=(x,y)^\mathrm{T} , \quad f_1(x) = P(x,y) , \quad f_2(x) = Q(x,y), \end{aligned}$$

(36)

and rewrite Eq. (35) in the form

$$\begin{aligned} \dot{x} = P(x,y) , \qquad \dot{y} = Q(x,y) . \end{aligned}$$

(37)

Theorem 2

Consider the planar dynamical system (37) with $P(x,y)$ and $Q(x,y)$ being analytic functions of $(x,y)$ in some open subset $E\subset \mathbb {R}^2$ containing the origin. Assume that the Taylor expansions of $P$ and $Q$ about the origin begin with $m$th-degree terms $P^{(m)}(x,y)$ and $Q^{(m)}(x,y)$ with $m\ge 1$. Then any orbit of the planar dynamical system (37) that approaches the origin as $t\rightarrow +\infty$ either spirals toward the origin as $t\rightarrow +\infty$ or it tends toward the origin in a definite direction $\theta =\theta _0$ as $t\rightarrow +\infty$. If the function $g(x,y)={\textit{xQ}}^{(m)}(x,y) - {\textit{yP}}^{(m)}(x,y)$ is not identically null, then all directions of the approach $\theta =\theta _0$ satisfy the relation

$$\begin{aligned} \cos \theta _0 Q^{(m)}(\cos \theta _0, \sin \theta _0) - \sin \theta _0 P^{(m)}(\cos \theta _0, \sin \theta _0) = 0. \end{aligned}$$

(38)

Moreover, if one orbit of the system (37) spirals toward the origin as $t\rightarrow +\infty$ then all trajectories of (37) in a deleted neighborhood of the origin spiral toward the origin as $t\rightarrow +\infty$.

For details on Theorem 2, see Ref. [7], page 140, Th. 2.

Now, we recall the definition of a sector, with reference to the planar dynamical system (37), as well as the possible character of a sector. If the Taylor expansions of $P$ and $Q$ about the origin begin with $m$th-degree terms $P^{(m)}$ and $Q^{(m)}$ and if the function

$$ g(\theta ) = \cos \theta Q^{(m)}(\cos \theta , \sin \theta ) - \sin \theta P^{(m)}(\cos \theta , \sin \theta ) $$

is not identically null, from Theorem 2 it follows that there are at most $2(m+1)$ directions, obtained as the solutions of the equation $g(\theta ) = 0$, along which a orbit of the system (37) may approach the origin. Thus, the solution curves of the system (37) that approach the origin along these directions divide a neighborhood of the origin into a finite number of open regions called sectors. Three types of sectors can occur, namely either a hyperbolic, parabolic or an elliptic sector when it is topologically equivalent to the sector represented in Fig. 5, picture (a), (b) and (c), respectively, where the directions of the flow need not to be preserved. Moreover, the trajectories that lie on the boundary of a hyperbolic sector are called separatrixes.

Fig. 5

Sector in the neighborhood of the origin of hyperbolic type (a), parabolic type (b) and elliptic type (c)

Another important concept that is used in this paper is the Morse index of a fixed point of the dynamical system (37). We begin with the definition of the index of a Jordan curve. Let $f =(P,Q)^\mathrm{T}$ be a ${\fancyscript{C}}^1$ vector field on an open subset $E \!\subset \! \mathbb {R}^2$ and let $\fancyscript{C}$ be a Jordan curve contained in $E$, such that the system (37) has no fixed point on $\fancyscript{C}$. The index of $\fancyscript{C}$ relative to $f$ is the integer $i({\fancyscript{C}})$ computed as

$$ i({\fancyscript{C}}) = \frac{1}{2\pi } \oint _{\fancyscript{C}} \frac{P dQ - Q dP}{P^2 + Q^2} . $$

Let $x_0$ be an isolated fixed point of system (37) and assume that the Jordan curve $\fancyscript{C}$ contains $x_0$ and no other fixed points of (37) on its interior. The Morse index of $x_0$ with respect to $f$ is defined by

$$ i(x_0) = i({\fancyscript{C}}) . $$

The following result is very convenient for the evaluation of the Morse index of a fixed point. It is stated for a fixed point at the origin but it is also valid for an arbitrary fixed point $x_0$.

Theorem 3

Consider the planar dynamical system (37) with $P(x,y)$ and $Q(x,y)$ being analytic functions of $(x,y)$ in some open subset $E\subset \mathbb {R}^2$ containing the origin. If the origin is an isolated fixed point of the system (37) then the Morse index of the origin, say $i$, satisfies the relation

$$\begin{aligned} i = 1 + \frac{1}{2} \, (e-h) , \end{aligned}$$

(39)

where $e$ and $h$ indicate the number of elliptic and hyperbolic sectors, respectively, in a neighborhood of the origin.

For details on Theorem 3, see Ref. [7], page 305, Th. 7.

As a consequence of Theorem 3, it follows that the number $h$ of hyperbolic sectors and the number $e$ of elliptic sectors have the same parity.

Further concepts and properties that are used in the analysis developed in Sect. 4 are those related to the $\alpha$- and $\omega$- limit sets of and orbit. We come back to the dynamical system (35) and assume that, for each $x_0\in \mathbb {R}^n$, the system has a unique global solution $x\in {\fancyscript{C}}^1(\mathbb {R})$ such that $x(0)=x_0$. We say that an equilibrium point $x^*$ is an $\omega$-limit point of the solution $x(t)$ if there exists a sequence $t_n\rightarrow +\infty$ such that $\lim _{n\rightarrow +\infty } x(t_n) = x^*$. The set of all $\omega$-limit points of the solution $x(t)$ is called its $\omega$-limit set. Analogously, by considering a sequence $t_n\rightarrow -\infty$, such that $\lim _{n\rightarrow -\infty } x(t_n) = x^*$, we define the concepts of an $\alpha$-limit point and the $\alpha$-limit set of a solution $x(t)$. Since solutions with the same orbit have equal $\omega$- and $\alpha$-limit sets, we will refer to $\omega$- and $\alpha$-limit sets of an orbit $\gamma$, and we will denote them by $\omega (\gamma )$ and $\alpha (\gamma )$, respectively. The following two theorems state important results on the limit sets $\omega (\gamma )$ and $\alpha (\gamma )$, that are used in Sect. 4.

Theorem 4

(LaSalle Invariance Principle) Let $S\subset \mathbb {R}^n$ be a compact and positively invariant subset of the dynamical system (35), and $Z \! : S \longrightarrow \mathbb {R}$ a ${\fancyscript{C}}^1$ monotone function along the flow of the dynamical system. Let $\gamma$ be an orbit in $S$. Then

$$ \omega (\gamma ) \subseteq \Big \{ x \in S : \; Z'(x) = 0 \Big \} , $$

where $Z' = \varDelta Z \cdot f$.

For details on Theorem 4, see Ref. [6], page 103, Th. 4.11.

Theorem 5

(Monotonicity Principle) Let $S\subset \mathbb {R}^n$ be an invariant subset of the dynamical system (35) and $Z \! : S \longrightarrow \mathbb {R}$ a ${\fancyscript{C}}^1$ strictly monotonically decreasing function along the flow of the dynamical system. Let $a$ and $b$ be defined by $\;a = \inf \{Z(x) : \; x \in S \}$ and $b = \sup \{Z(x) : \; x \in S \}$. Let $\gamma$ be an orbit in $S$. Then

$$ \alpha (\gamma ) \subseteq \Big \{ s \in \partial S : \; \lim _{x\rightarrow s} Z(x) \not = a \Big \} , \quad \omega (\gamma ) \subseteq \Big \{ s \in \partial S : \; \lim _{x\rightarrow s} Z(x) \not = b \Big \} , $$

For details on Theorem 5, see Ref. [6], page 103, Th. 4.12. Finally, the following theorem states a crucial result about the limit sets $\omega (\gamma )$ and $\alpha (\gamma )$ in the particular case of a dynamical system in $\mathbb {R}^2$.

Theorem 6

(Generalized Poincaré-Bendixson) Consider the dynamical system (35) on $\mathbb {R}^2$, and suppose that the dynamical system has only a finite number of equilibrium points. Then, for any orbit $\gamma$ of the dynamical system, each one of the limit sets $\omega (\gamma )$ and $\alpha (\gamma )$ can only be one of the following: an equilibrium point; a periodic orbit  the union of equilibrium points and heteroclinic cycles.

For details on Theorem 6, see Ref. [7], page 101, Th. 4.10, or Ref. [6], page 245, Th. 2.