Friedmann Cosmology with Interaction Between Dark Energy and Multi-Phase Matter

Abstract

We now consider a more realistic cosmological model with a number of different matter species (as, in particular, in the $\Lambda$CDM model); for the sake of generality, we assume the number, $N>1$, of matter phases arbitrary.

We now consider a more realistic cosmological model with a number of different matter species (as, in particular, in the $\Lambda$CDM model); for the sake of generality, we assume the number, $N>1$, of matter phases arbitrary. Each of the phases is described by its density, $\rho _{n}=\rho _{n}(t)$, and pressure, $p_{n}=p_{n}(t)$, related by the partial equation of state:

$$\begin{aligned} p_{n}=w_{n}\rho _{n},\quad w_{n}>-1,\quad n=1,2,\ldots ,N\; . \end{aligned}$$

(4.1)

Two species with the same equation of state might still differ by other physical properties, so we do not assume $w_m\not =w_{n}$ for $m\not =n$. In the presence of DE, the total density and pressure thus become:

$$\begin{aligned} \rho _{tot}=\rho _{vac}+\rho =\rho _{vac}+\sum \limits _{n=1}^{N}\,\rho _{n}\; ;\nonumber \qquad \qquad \\ p_{tot}=p_{vac}+p=p_{vac}+\sum \limits _{n=1}^{N}\,p_{n}=-\rho _{vac}+\sum \limits _{n=1}^{N}\,w_{n}\rho _{n}\,. \end{aligned}$$

(4.2)

Cosmological evolution is again described by the Friedmann equations (2.4) but with the total density and pressure (4.2). The first Friedmann equation defines the time dependence of the scale factor (or the co-moving volume $V(t)=a^3(t)$) by the formula (2.8). The second one, the equation of energy conservation in the form (2.7), $d\rho _{tot}/dV=-(\rho _{tot}+p_{tot})/V$, turns to

$$ \frac{d}{dV}\left( \rho _{vac}+\sum \limits _{n=1}^{N}\,\rho _{n}\right) = -\frac{1}{V}\sum \limits _{n=1}^{N}\,(1+w_{n})\rho _{n}\; $$

(compare with the corresponding single phase Eq. (3.4)), or

$$\begin{aligned} \frac{d\rho _{vac}}{dV}+\sum \limits _{n=1}^{N}\,\left[ \frac{d\rho _{n}}{dV} +\frac{(1+w_{n})\rho _{n}}{V}\right] = 0\; . \end{aligned}$$

(4.3)

In the usual approach each specie is assumed to be conserved,

$$\begin{aligned} \frac{d\rho _{n}}{dV} +\frac{(1+w_{n})\rho _{n}}{V} = 0,\qquad n=1,2,\ldots ,N\; . \end{aligned}$$

(4.4)

Under this condition Eq. (4.3) requires that the DE density is constant, and the whole cosmological solution becomes thus

$$\begin{aligned} \rho _{vac}=\text{ const },\qquad \rho _{n}=C_n/V^{(1+w_{n})},\quad C_n>0,\quad n=1,2,\ldots ,N\; . \end{aligned}$$

(4.5)

There are enough grounds for considering matter species in our universe not interacting with each other. We retain this standard assumption, but, as everywhere in this book, do not forbid any of them to interact with DE. In case when $N_i\ge 1$ species interact with heavy vacuum, $N_i$ equations describing this interaction should be added to the Eq. (4.3), to determine the evolution of all the relevant densities. We discuss two cases: (a) when a single matter phase interacts with DE, $N_i=1$, and (b) when several phases are interacting, $1< N_i\le N$.

4.1 Single Matter Phase Interacting with Dark Energy

Let the matter component interacting with DE have the number $n=1$, so it is described by $\rho _1$; all other matter phases are conserved. The densities of the latter are as in Eq. (4.5),

$$\begin{aligned} \rho _{n}=C_n/V^{(1+w_{n})},\quad C_n>0,\quad n=2,\ldots ,N\; , \end{aligned}$$

(4.6)

and the conservation equation (4.3) reduces to

$$\begin{aligned} \frac{d(\rho _{vac}+\rho _1)}{dV}=-\frac{(1+w_1)\rho _1}{V}\; . \end{aligned}$$

(4.7)

To find $\rho _1(t)$ and $\rho _{vac}(t)$, we need an equation specifying the interaction between the two. It is natural to take it in the same general form (3.4), that is,

$$\begin{aligned} \frac{d\rho _{vac}}{dV}= \frac{ F(\rho _{vac},\rho _1)}{V}\; , \end{aligned}$$

(4.8)

where $F(\rho _{vac},\rho _1)$ is some interaction function. Combining the last two equations gives the governing system

$$\begin{aligned} \frac{d\rho _1}{dV}= -\frac{(1+w_1)\rho _1+ F(\rho _{vac},\rho _1)}{V},\qquad \qquad \frac{d\rho _{vac}}{dV}= \frac{ F(\rho _{vac},\rho _1)}{V}\; , \end{aligned}$$

(4.9)

which is, naturally, nothing else as the system (3.5) controlling the cosmology of a single matter phase interacting with DE (up to the notations $\rho ,\;w$ replaced with $\rho _1,\;w_1$). Therefore all the general features and all exact solutions found and discussed in Chap. 3, including non-singular cosmologies of Sect. 3.4, remain valid for $\rho _1(t)$ and $\rho _{vac}(t)$, with all other densities given by the usual expressions (4.6).

Remarkably, a toy non-singular cosmological solution of Sect. 3.3.1, with the initial jump in the DE density (see Fig. 3.1) and the linear interaction law (3.35),

$$ F(\rho _{vac},\rho _1)=-s\rho _1+\theta (\rho _{vac}-\rho _{\infty })\; , $$

acquires a physical meaning due to the presence of other matter species. Namely, the initial jump in the DE density from $\rho _{\infty }$ to $\rho _{0}$ can be explained by a phase transition between the matter phases (4.6) (otherwise not interacting with DE) and the heavy vacuum that keeps the total energy conserved. The corresponding solution not violating, unlike the solution (3.42), energy conservation, is:

$$\begin{aligned}&\rho _{vac}=\rho _{\infty },\;\; \rho _1=0,\;\;\rho _{n}=C_n/V^{(1+w_{n})},\; C_n>0,\; n=2,\ldots ,N,\quad \text{ for }\; 0<V<V_{*}\;; \nonumber \\&\rho _{vac}=\rho _\infty +{|s|}\rho _{*}\left[ Q_2\left( \frac{V_{*}}{V}\right) ^{|\mu _{2}|}+Q_1\left( \frac{V_{*}}{V}\right) ^{|\mu _{1}|}\right] ,\;\rho _1=\rho _{*}\left[ \left( \frac{V_{*}}{V}\right) ^{|\mu _{2}|}-\left( \frac{V_{*}}{V}\right) ^{|\mu _{1}|}\right] \; ,\nonumber \\&\rho _{n}=C_n^{'}/V^{(1+w_{n})},\; C_n^{'}>0,\; n=2,\ldots ,N,\quad \text{ for }\quad V_{*}<V<+\infty \; ;\\&\Delta \rho _{tot}\Bigl |_{V=V_{*}}=\left( \Delta \rho _{vac}+\Delta \rho \right) \Bigl |_{V=V_{*}}= \rho _{0}-\rho _{\infty }+\sum \limits _{n=2}^N\, \frac{C_n^{'}- C_n}{V_{*}^{1+w_{n}}}=0\nonumber \; . \end{aligned}$$

(4.10)

The constants $\mu _{1,2}$ and $Q_{1,2}$ are defined in the Eqs. (3.38) and (3.40), respectively, the interaction parameter s is in the physical range (3.12), and $\rho _{0}=\rho _{vac}({V_{*}+0})$. Note that some of the matter phases can turn to DE ($C_n^{'}<C_n$) at the jump, some might gain from DE ($C_n^{'}>C_n$), and some might stay unchanged ($C_n^{'}=C_n$), provided that the last equality holds. That is, since $\Delta \rho _{vac}=\rho _{0}-\rho _{\infty }$ is positive, $\Delta \rho =-\Delta \rho _{vac}$ must be negative. Note also that the (partial) phase trajectory of this solution in the plane $\{\rho _{vac},\;\rho _1\}$ remains as shown in Fig. 3.1.

The universe (4.10) starts ($t,V\rightarrow +0$) with $\rho _1=0$, $\rho _{vac}=\rho _{\infty }$, and all other matter phases singular, undergoes an instant matter–DE phase transition raising $\rho _{vac}$ to $\rho _{0}$ at some moment $t=t_*\;(V_{*}=V(t_*))$, and drives finally ($t,V\rightarrow \infty$) to the initial de Sitter universe with $\rho _{vac}=\rho _{\infty }$ and $\rho =0$.

4.2 Any Number of Matter Phases Interacting with Dark Energy

Let now more than one matter phases interact with DE, so that the number of the interacting species is $N_i,\;1<N_i\le N$. The densities of all other, non-interacting, matter species are again given by the standard expressions

$$\begin{aligned} \rho _{n}=C_n/V^{(1+w_{n})},\quad C_n>0,\quad n=N_i+1,\ldots ,N\; , \end{aligned}$$

(4.11)

and N should be replaced with $N_i$ in the energy conservation equation (4.3). Keeping the assumption that the matter species do not interact with each other, we take the following law of their interaction with DE ($F_{n}$ is an arbitrary function):

$$ \frac{d\rho _{n}}{dV}+\frac{(1+w_{n})\rho _{n}}{V}= -\frac{ F_{n}(\rho _{vac},\rho _{n})}{V},\qquad n=1,2,\ldots N_i\; ; $$

each phase is conserved if and only if $F_{n}=0$. With this, the energy conservation equation turns to

$$ \frac{d\rho _{vac}}{dV}= \frac{ F(\rho _{vac},\rho _1,\rho _2,\ldots ,\rho _{N_i})}{V}\; , $$

where

$$\begin{aligned} F(\rho _{vac},\rho _1,\rho _2,\ldots ,\rho _{N_i})=\sum \limits _{n=1}^{N_i}\,F_{n}(\rho _{vac},\rho _{n})\; . \end{aligned}$$

(4.12)

The last two differential equations govern the evolution of the universe in this case; as before, it is convenient to use them in an autonomous form,

$$\begin{aligned} \frac{d\rho _{n}}{d\lambda }= -\left[ (1+w_{n})\rho _{n}+ F_{n}(\rho _{vac},\rho _{n})\right] ,\qquad n=1,2,\ldots N_i\; ; \nonumber \\ \frac{d\rho _{vac}}{d\lambda }= F(\rho _{vac},\rho _1,\rho _2,\ldots ,\rho _{N_i}):\qquad \lambda =\ln \left( V/V_{*}\right) \; ,\qquad \quad \end{aligned}$$

(4.13)

with F defined by the equality (4.12).

Generally, this is a nonlinear autonomous system of ODEs of the order $N_i+1\ge 3$, which allows for solutions with various behavior: even a strange attractor is possible, in principle, in the large time limit. This alone shows that the approach in which matter is represented by a single ‘dominant’ component (like radiation, $w=1/3$, in our early universe, or dark matter, $w=0$, later) might be insufficient no matter how small the abundances of other matter species are.

A usual regular limiting behavior at large times occurs when a cosmological solution goes to a rest point $P=\{\rho _1^{*},\rho _2^{*},\ldots ,\rho _{N}^{*},\rho _{vac}^{*}\}\equiv P\{\rho _{n}^{*},\rho _{vac}^{*}\}$ of the system (4.13). Such a rest point is described by the equations:

$$ \rho _{n}^{*}=(1+w_1)^{-1}F_{n}(\rho _{vac}^{*},\rho _{n}^{*}),\quad n=1,2,\ldots N_i; \quad \sum \limits _{n=1}^{N_i}\,F_{n}(\rho _{vac}^{*},\rho _{n}^{*})=0. $$

It is a physical equilibrium when $\rho _{n}^{*}\ge 0$, therefore in this case $F_{n}(\rho _{vac}^{*},\rho _{n}^{*})\ge 0$ for all relevant values of n. But then the second of the above equations implies then $F_{n}(\rho _{vac}^{*},\rho _{n}^{*})=0$ for all n, so from the first equation it follows that the only possible physical rest point is

$$\begin{aligned} \rho _{n}^{*}=0, \qquad F_{n}(\rho _{vac}^{*},0)=0, \qquad n=1,2,\ldots N_i; \qquad \rho _{vac}^{*}>0\; ; \end{aligned}$$

(4.14)

it corresponds again to a de Sitter universe. If an equilibrium point exists and is stable, then, in view of the expressions (4.11), a set of cosmological solutions of a non–zero measure tends to it at large times. However, this requires all interaction functions $F_{n}(\rho _{vac},0),\;n=1,2,\ldots N_i,$ have a common positive root $\rho _{vac}^{*}$.

This is a strong restriction, unless some serious physics underlies it; if it is not valid, then the densities of the interacting matter species do not all tend to zero at large times, and the DE density does not tend to a constant. So generically the large time behavior of cosmological solutions with several matter phases involved in the DE–matter interaction is more complicated than the usual one; this is a characteristic feature of the multiple phase interaction.

Of course, the governing system (4.13) cannot be explicitly integrated for a general set of interaction laws $F_{n}$. For this reason, below we explore two more particular models of interaction allowing for a detailed analysis and some new features.

4.2.1 Linear Interaction Laws

In a complete similarity with the case of single matter phase (see formula (3.35)) we consider linear interaction laws

$$\begin{aligned} F_{n}(\rho _{vac},\rho _{n})=-s_{n}\rho _{n}+\theta _{n}(\rho _{vac}-\rho _{\infty }),\quad s_{n},\theta _{n},\rho _{\infty }=\text{ const },\quad \rho _{\infty }\ge 0\; . \end{aligned}$$

(4.15)

The governing equations (4.13) become thus

$$\begin{aligned} \frac{d\rho _{n}}{d\lambda }= -\left[ (1+w_{n}-s_{n})\rho _{n}+\theta _{n}(\rho _{vac}-\rho _{\infty })\right] ,\qquad n=1,2,\ldots N_i\; ; \\ \frac{d\rho _{vac}}{d\lambda }= -\sum \limits _{n=1}^{N_i}s_{n}\rho _{n}+\Theta (\rho _{vac}-\rho _{\infty }),\quad \Theta =\sum \limits _{n=1}^{N_i}\theta _{n}\; . \qquad \qquad \end{aligned}$$

Introducing an $N_i+1$–dimensional vector function $\mathbf{z}(t)$,

$$\begin{aligned} \mathbf{z}(\lambda )=\{\rho _1(\lambda ),\;\rho _2(\lambda ),\;\ldots ,\;\rho _{N_i}(\lambda ),\;[\rho _{vac}(\lambda )-\rho _{\infty }]\}^T\; , \end{aligned}$$

(4.16)

we rewrite this system of the first order equations in a matrix form ($\delta _{jk}$ is the Kronecker symbol):

$$\begin{aligned} \frac{d\mathbf{z}}{d\lambda }=\mathcal{M}\mathbf{z}\; ;\quad \qquad \qquad \qquad \qquad \qquad \\ \mathcal{M}_{nj}=-(1+w_{n}-s_{n})\delta _{nj}-\theta _{n}\delta _{jN_{1}+1},\;n=1,2,\ldots N_i\;;\nonumber \\ \mathcal{M}_{N_{1}+1j}=-s_j,\;n=1,2,\ldots N_i;\quad \mathcal{M}_{N_{1}+1N_{1}+1}=\Theta \; .\quad \nonumber \end{aligned}$$

(4.17)

It is a linear system with constant coefficients, so its general solution is obtained as a linear combination of exponents of $\lambda$ (powers of V):

$$\begin{aligned} \mathbf{z}(\lambda )=\sum \limits _{k=1}^{N_i+1}A_k\mathbf{e}_k\exp (\mu _k\lambda )= \sum \limits _{k=1}^{N_i+1}B_k\mathbf{e}_k V;\qquad B_k=A_k/V_{*}^{\mu _k}\; . \end{aligned}$$

(4.18)

Here $\mu _k$ are the eigenvalues of the matrix $\mathcal{M}$, that is, the roots of the algebraic equation

$$ \text{ det }\left( \mathcal{M}-\mu I\right) =0\quad (I\;\mathrm{is}\;\mathrm{the}\;\mathrm{unit}\;\mathrm{matrix})\; , $$

and ${\mathbf{e}_k}$ are the corresponding normalized eigenvectors,

$$ \left( \mathcal{M}-\mu I\right) \mathbf{e}_k=0,\;\;\mathbf{e}_k=\{e_{k1},\;\ldots ,\;e_{kN_i+1}\}^T,\;\;\sum \limits _{n=1}^{N_i+1}e_{kn}^2=1,\;\; k=1,\ldots N_i+1\; $$

(for brevity, we consider only the generic case when all $\mu _k$ are different).

The matrix $\mathcal{M}$ is not symmetric, so its eigenvalues might be complex, coming in complex conjugate pairs. Since the physical solution must be real, real parts should be taken at the proper places of expression (4.18). Namely, suppose there are $N_c\ge 1$ pairs of complex eigenvalues $\mu _k$ and $\bar{\mu }_k$, with the eigenvectors $\mathbf{e}_k$ and $\bar{\mathbf{e}}_k$, respectively, $k=1,2,\ldots ,N_c$. The physical solution then becomes

$$\begin{aligned} \mathbf{z}(\lambda )=\sum \limits _{k=1}^{N_c}C_kV^{\eta _k}\left[ \mathbf{r}_k \cos (\nu _k\ln V)+ \mathbf{i}_k\sin (\nu _k\ln V)\right] +\sum \limits _{k=2N_c+1}^{N_i+1}C_kV^{\mu _k}\mathbf{e}_k;\nonumber \\ C_k=A_k/V_{*}^{\eta _k};\qquad \eta _k= \mathbf{Re}(\mu _k),\quad \nu _k= \mathbf{Im}(\mu _k)\;;\qquad \qquad \quad \\ \mathbf{r}_k=2{{\mathbf{Re}}}(\mathbf{e}_k)/V_{*}^{\eta _k},\,\mathbf{i}_k=-2\mathrm{Im}(\mathbf{e}_k)/V_{*}^{\eta _k}\;, \qquad \qquad \qquad \;\;\nonumber \end{aligned}$$

(4.19)

but it still requires two additional conditions to be met. First, all the densities must vanish at large times $(V\rightarrow \infty )$, so all the powers of V must be negative,

$$\begin{aligned} \eta _k=\mathbf{Re}(\mu _k)<0,\;\;1\le k\le N_c; \qquad \mu _k<0,\;\;2N_c+1\le k\le N_i+1\; ; \end{aligned}$$

(4.20)

this shows also that the expansion starts $(V\rightarrow +0)$ from singularity.

The second and more constraining condition comes from the fact that all the densities must be positive throughout the expansion. For the case of a single matter specie interacting with DE, corresponding to $N_i=2$, this condition never holds, as demonstrated in Sect. 3.1. For $N_i>2$ this condition might be possible to meet with some proper combination of parameters $w_{n},\;s_{n},\;\theta _{n}$, and the right choice of the arbitrary constants $C_k$. Additional restrictions are needed when some of the eigenvalues $\mu _k$ are indeed complex ($N_c\ge 1$). In this case the densities contain some terms oscillating around zero (the first sum in the expression (4.19)); those oscillations must be dominated by other strictly positive monotonic contributions. This can happen if one of the real negative eigenvalues $\mu _k$ is smaller than all $\eta _k,\;k=1,2,\ldots ,N_c$, and the other one is larger than them. If this is true, the oscillations are compensated at least near the initial singularity $(V\rightarrow +0)$ and towards the end of the expansion $(V\rightarrow \infty )$, with a possibility for the density to stay positive in between as well.

If all the mentioned conditions are fulfilled, then the physical solution is

$$\begin{aligned} \rho _n=\sum \limits _{k=1}^{N_c}C_kV^{-|\eta _k|}\left[ r_{kn} \cos (\nu _k\ln V)+ i_{kn}\sin (\nu _k\ln V)\right] +\sum \limits _{k=2N_c+1}^{N_i+1}C_k e_{kn}V^{-|\mu _k|};\nonumber \\ \rho _{vac}=\rho _{\infty }+\sum \limits _{k=1}^{N_c}C_kV^{-|\eta _k|}\left[ r_{kN_i+1} \cos (\nu _k\ln V)+ i_{kN_i+1}\sin (\nu _k\ln V)\right] +\qquad \\ +\sum \limits _{k=2N_c+1}^{N_i+1}C_k e_{kN_i+1}V^{-|\mu _k|}\; .\qquad \qquad \qquad \qquad \qquad \nonumber \end{aligned}$$

(4.21)

This solution drives to a de Sitter universe $\rho _{n}=0,\;\rho _{vac}=\rho _{\infty }$. If oscillations are present, then their frequency becomes infinitely large both at the initial singularity and the expansion end.

It is worthy to consider one particular case studied in detail in Sect. 3.1 for the single matter phase cosmology. In this case the linear interaction law does not depend on the DE density, i.e., $\theta _{n}=0$, $F_{n}(\rho _{vac},\rho _{n})=-s_{n}\rho _{n}$. Thus every density $\rho _{n}$ satisfies its own linear equation, making the answer rather simple:

$$\begin{aligned} \rho _n=\frac{C_n}{V^{1+w_{n}-s_{n}}},\;\;n=1,2,\ldots N_i;\qquad \rho _{vac}=\rho _{\infty }+\sum \limits _{k=1}^{N_i}\,\frac{s_{n}C_n}{V^{1+w_{n}-s_{n}}}\; ; \end{aligned}$$

(4.22)

it is an exact analog of the single matter phase solution (3.11), with all the properties described in Sect. 3.1. The solution (4.22) is physically meaningful under the condition

$$ 0<s_{n}<1+w_{n},\qquad n=1,2,\ldots N_i \; , $$

which is a generalization of the condition (4.19). It guarantees that all the densities, including $\rho _{n}$, are positive and monotonically decreasing with matter vanishing at infinity. However, the left inequality above is, in fact, necessary for only one value of n, say, $n=k$, corresponding to the maximum difference $(w_{n}-s_{n})$,

$$ w_k-s_k=\max _{1\le n\le N_i}(w_{n}-s_{n})\; , $$

providing that the DE density is positive at small times, near the singularity. Depending on the values of the positive constants $C_n$, some of other parameters $s_{n},\;n\not =k$, can be negative, with $\rho _{vac}$ remaining positive throughout the expansion. In this case it might be non-monotonic, having positive maxima and minima at some moments of time.

4.2.2 Quadratic Interaction Laws

Here we set

$$\begin{aligned} F_{n}(\rho _{vac},\rho _{n})=-(s_{n}/R)\rho _{n}(\rho _{vac}-\rho _{0}),\quad s_{n}, R,\rho _{0}=\text{ const },\quad R,\rho _{0}\ge 0\; . \end{aligned}$$

(4.23)

The governing equations (4.13) become:

$$\begin{aligned} \frac{d\rho _{n}}{d\lambda }+(1+w_{n})\rho _{n}=\frac{s_{n}}{R}\rho _{n}(\rho _{vac}-\rho _{0}),\qquad n=1,2,\ldots N_i\; ; \nonumber \\ \frac{d\rho _{vac}}{d\lambda }= -\frac{1}{R}\,\sum \limits _{n=1}^{N_i}s_{n}\rho _{n}(\rho _{vac}-\rho _{0})\; .\qquad \qquad \qquad \end{aligned}$$

(4.24)

As usual, we are interested only in its physical solutions, with all the densities positive and matter phase densities vanishing at the end of the expansion.

The autonomous system (4.24) of $(N_i+1)$ equations has a physical rest point $\rho _{n}=0,\rho _{vac}=\rho _{\infty }$ with any $\rho _{\infty }\ge 0$, i.e., the whole semi-axis $\rho _{vac}\ge 0$ consists of its equilibriia. They can attract solutions at large times; the corresponding asymptotic expressions for the case $\rho _{\infty }\not =\rho _{0}$ are ($t,V\rightarrow +\infty$):

$$\begin{aligned} \rho _{n}=\frac{D_n}{V^{1+w_{n}+\gamma s_{n}}}\left[ 1+o(1)\right] ,\;\;\gamma =\frac{\rho _{0}-\rho _{\infty }}{R},\;\;\rho _{\infty }\not =\rho _{0};\;\; n=1,2,\ldots N_i \; ; \qquad \\ \rho _{vac}=\rho _{\infty }-\frac{\gamma s_k D_k}{(1+w_k+\gamma s_k) V^{1+w_k+\gamma s_k}}\left[ 1+o(1)\right] ,\;\; w_k+\gamma s_k=\min _{1\le n\le N_i}(w_{n}+\gamma s_{n})\; ;\nonumber \end{aligned}$$

(4.25)

here $D_n>0$ is some constant. For the matter densities to vanish asymptotically, the following condition is required, for all n:

$$ 1+w_{n}+\gamma s_{n}>0\; , $$

which splits into two sets of inequalities:

$$\begin{aligned} (a)\,\rho _{\infty }>\rho _{0},\;s_{n}<\frac{R}{\rho _{\infty }-\rho _{0}}(1+w_{n});\quad (b)\,\rho _{\infty }<\rho _{0},\;s_{n}>-\frac{R}{\rho _{0}-\rho _{\infty }}(1+w_{n})\; . \end{aligned}$$

(4.26)

In both cases the signs of the parameters $s_{n}$ are not fixed: some of them can be positive, the other can be negative.

The large time asymptotics for the exceptional case of the attracting rest point with $\rho _{vac}=\rho _{0}$ is more complicated, except the obvious exact solution with the constant DE density:

$$\begin{aligned} \rho _{n}={D_n}/{V^{1+w_{n}}},\;n=1,2,\ldots N_i ;\qquad \rho _{vac}=\rho _{0}=\text{ const }\; . \end{aligned}$$

(4.27)

Here there is no interaction between dark energy and matter, thus all the matter species are conserved (by the formulas (4.27) and (4.11)). However, this solution, and thus the parameter $\rho _{0}$, can play a role in the initial behavior of solutions that may ‘branch’ from the above one out of the singularity. The asymptotic formulas describing such behavior are ($t,V\rightarrow +0$):

$$\begin{aligned} \rho _{n}=\frac{D_n}{V^{1+w_{n}}}\left[ 1+o(1)\right] ,\quad D-n>0,\quad n=1,2,\ldots N_i \; ; \qquad \qquad \\ \rho _{vac}=\rho _{0}+D_0\exp \left[ \frac{-s_kD_k}{(1+w_k)V^{1+w_k}}\right] \left[ 1+o(1)\right] ,\;\; w_k=\max _{1\le n\le N_i}w_{n},\;\; s_k>0\; .\nonumber \end{aligned}$$

(4.28)

The last inequality is needed because the correction to $\rho _{0}$ must vanish in the limit. For the first time this correction proves to be exponentially small; all other cosmological solutions obtained and discussed so far do not have this feature.

A solution with the small time asymptotics (4.28) is similar to the solution (4.27) in a sense that all matter in the universe described by it is born form a singularity, and the DE density is finite at the initial moment of time. If this solution also has the large time behavior described by the formulas (4.25), then the DE density evolves from one value, $\rho _{0}$, in the beginning, to some other, $\rho _{\infty }$, at the end of the expansion.

The governing system (4.24) of $(N_i+1)$ equations can be reduced to just two equations for any $N_i>1$, since its $N_i-1$ integrals are explicitly found. Indeed, the first Eq. (4.24) implies

$$ \frac{d\ln \rho _{n}}{d\lambda }+(1+w_{n})=\frac{s_{n}}{R}(\rho _{vac}-\rho _{0}),\qquad n=1,2,\ldots N_i\; , $$

allowing for the following $N_i-1$ combinations:

$$ s_n\frac{d\ln \rho _1}{d\lambda }-s_1\frac{d\ln \rho _n}{d\lambda }+\left[ s_n(1+w_1)-s_1(1+w_n)\right] =0, \quad n=2,3,\ldots N_i\;. $$

These equations can be immediately integrated to give the expressions for all interacting phase densities through the first one,

$$\begin{aligned} \rho _{n}=A_n\rho _1^{\theta _{n}}\exp (\beta _n\lambda )=B_n\rho _1^{\theta _{n}}V^{\beta _n},\quad B_n=A_n/V_{*}^{\beta _n},\quad n=2,3,\ldots N_i\; ;\nonumber \\ A_n,B_n>0;\qquad \theta _{n}=s_n/s_1,\qquad \beta _n=-(1+w_n)+(s_n/s_1)(1+w_1)\;.\; \end{aligned}$$

(4.29)

What remains is the system (4.24) of two equations for $\rho _1$ and $\rho _{vac}$, the second of them with the coefficient depending generally on the evolution variable ($\lambda$ or V):

$$\begin{aligned} \frac{d\rho _1}{d\lambda }+(1+w_1)\rho _1=\frac{s_1}{R}\rho _1(\rho _{vac}-\rho _{0}),\quad \frac{d\rho _{vac}}{d\lambda }= -\frac{ \mathcal{P}(\rho _1,\lambda )}{R}\,(\rho _{vac}-\rho _{0})\; ; \qquad \\ \mathcal{P}(\rho _1,\lambda )=\sum \limits _{n=1}^{N_i}s_{n}A_n\rho _1^{\theta _{n}}\exp (\beta _n\lambda ) =\sum \limits _{n=1}^{N_i}s_{n}B_n\rho _1^{\theta _{n}}V^{\beta _n}\; .\qquad \qquad \qquad \nonumber \end{aligned}$$

(4.30)

Here by the definition (4.29) $\beta _1=0$, $\theta _1=s_1/s_1=1$, and we set $A_1=1$; all the densities $\rho _2,\rho _3,\ldots ,\rho _{N_i}$ are replaced with their expressions (4.29).

A physically meaningful solution of the system (4.30) together with the expressions (4.29) provides the complete answer, i.e., a cosmological solution describing the universe with N matter species, of which $N_i>1$ interact with dark energy by the law (4.23).

We consider now one example where the equations integrate completely, that is, the second order system (4.30) proves to be explicitly integrable. This is the case when the Eq. (4.30) become autonomous, i.e., the independent variable is not involved in the second of them. We take (see formulas (4.30))

$$ \beta _n=0,\qquad s_{n}=s_1\,\frac{1+w_{n}}{1+w_1},\qquad n=2,3,\ldots N_i\; ; $$

note that all parameters $s_{n}$ are of the same sign. Using this in the definition (4.30) of the function $\mathcal{P}$ we find

$$\begin{aligned} \mathcal{P}(\rho _1,\lambda )=s_1\,\sum \limits _{n=1}^{N_i}\frac{1+w_{n}}{1+w_1}A_n\rho _1^{\frac{1+w_{n}}{1+w_1}}\equiv s_1Q(\rho _1)\; , \end{aligned}$$

(4.31)

so the Eq. (4.30) become:

$$\begin{aligned} \frac{d\rho _1}{d\lambda }+(1+w_1)\rho _1=\frac{s_1}{R}\rho _1(\rho _{vac}-\rho _{0}),\quad \frac{d\rho _{vac}}{d\lambda }= -\frac{ s_1}{R}\,Q(\rho _1)(\rho _{vac}-\rho _{0})\; . \end{aligned}$$

(4.32)

Dividing the first of them by the second one we arrive to the equation with the separable variables,

$$ \frac{d\rho _1}{d\rho _{vac}}=\frac{\rho _1}{Q(\rho _1)}\left[ \frac{(1+w_1)R}{s_1(\rho _{vac}-\rho _{0})}-1\right] \; , $$

whose integral, by virtue of the expression (4.31), is:

$$ s_1\,\sum \limits _{n=1}^{N_i}A_n\rho _1^{\frac{1+w_{n}}{1+w_1}}=K-\rho _{vac}+\frac{(1+w_1)R}{s_1}\ln |\rho _{vac}-\rho _{0}|\; ; $$

here K is a constant of integration. It is determined from the large time behavior of a physical solution: in this limit $\rho _1$ must vanish, and $\rho _{vac}$ must tend to some value $\rho _{\infty }\ge 0,\,\rho _{\infty }\not =\rho _{0}$. The l.h.s of the above equality goes to zero in this limit, therefore the same must happen with the r.h.s, which gives

$$ K=\rho _{\infty }-\frac{(1+w_1)R}{s_1}\ln |\rho _{\infty }-\rho _{0}|\; , $$

and the integral becomes

$$\begin{aligned} \sum \limits _{n=1}^{N_i}A_n\rho _1^{\frac{1+w_{n}}{1+w_1}}=\rho _{\infty }-\rho _{vac}+ \frac{(1+w_1)R}{s_1}\ln \left| \frac{\rho _{vac}-\rho _{0}}{\rho _{\infty }-\rho _{0}}\right| \; . \end{aligned}$$

(4.33)

It is convenient to treat this as an equation for the DE density as function of the matter density depending also on the limit value $\rho _{\infty }$. If it has a solution $\rho _{vac}=\rho _{vac}(\rho _1,\rho _{\infty })$ such that $\rho _{vac}(0,\rho _{\infty })=\rho _{\infty }$, then the first Eq. (4.32) reduces to integrating a known function and determining thus $\rho _1(\lambda )$, or $\rho _1(V)$, from the equation:

$$\begin{aligned} \int \limits ^{\rho _1}\frac{dx}{x\left\{ 1+w_1+\left( s_1/R\right) \left[ \rho _{0}-\rho _{vac}(x)\right] \right\} }=-\lambda =\ln \left( \frac{V_{*}}{V}\right) \; . \end{aligned}$$

(4.34)

If, in its turn, this equation has a solution $\rho _1=\rho _1(V)$ going to zero when $V\rightarrow +\infty$, then we have a consistent solution to the system (4.32). This is a cosmological solution if both densities are positive on the whole semi–axis $V>0$.

A simple enough graphic analysis of the transcendental equation (4.33) shows that its positive solution $\rho _{vac}=\rho _{vac}(\rho _1,\rho _{\infty })$ does exist under certain restriction on the parameter values. First of all, the large time limit of DE density should be smaller than $\rho _{0}$,

$$\begin{aligned} \rho _{\infty }<\rho _{0}\; . \end{aligned}$$

(4.35)

Under this condition there are two cases yielding solutions of a different type.

Case A Parameter $s_1$, and hence all $s_{n}$, are positive,

$$\begin{aligned} s_{n}>0,\qquad n=1,2,\ldots ,N_i\; . \end{aligned}$$

(4.36)

A single solution $\rho _{vac}$ to the Eq. (4.33) then exists that increases from the initial zero value to $\rho _{\infty }$ in the course of the expansion, while the density $\rho _1$ decreases from a finite initial value $\rho _{*}$ to zero (the value $\rho _{*}$ is found from the Eq. (4.33) with $\rho _{vac}=0$). This is not surprising, because the governing equations (4.24) show that in this case the interaction reduces the matter phases and produces heavy vacuum for $0<\rho _{vac}<\rho _{\infty }<\rho _{0}$. According to the expressions (4.29), all other interacting matter densities $\rho _{n},\, n=2,3,\ldots ,N_i$ are also finite at the beginning of the expansion.

However, Eq. (4.34) shows that the finite initial value of $\rho _1$ corresponds to a finite non-zero initial value of V, or of the scale factor, which does not make sense, unlike the situation described at the end of Sect. 3.3.2. So we need to extend the solution towards larger density $\rho _1$ (smaller values of V), but the DE density becomes negative there, for $\rho _1>\rho _{*}$. The solution thus has no physical meaning.

Case B Parameter $s_1$, and hence all $s_{n}$, are negative,

$$\begin{aligned} s_{n}<0,\qquad n=1,2,\ldots ,N_i\; . \end{aligned}$$

(4.37)

Here the positive solution $\rho _{vac}$ to the Eq. (4.33) decreases from the initial value $\rho _{0}$ to $\rho _{\infty }$, while the matter density $\rho _1$, singular at the beginning, decreases monotonically to zero. Dark energy permanently produces the interacting matter phases while being reduced accordingly, which production slows down the decay of matter densities, as compared to the absence of the interaction.

By the expressions (4.29), all interacting densities $\rho _{n},\, n=1,2,\ldots ,N_i,$ also emerge from the initial singularity; since the DE density is finite, this is a cosmology of a mixed, singular—non-singular, type. From the Eq. (4.34) we find that the initial behavior of the solution is given by the formulas (4.28), and its final behavior is described by the expressions (4.25). So each matter density is inversely proportional to some power of V, or the scale factor, at the beginning of the expansion, and to some other power at its end. The initial dependencies are the same as in the case without the interaction, because it becomes negligibly small when $\rho _{vac}\rightarrow \rho _{0}+0$.

Finally, we note briefly the general quadratic interaction law

$$\begin{aligned} F_{n}(\rho _{vac},\rho _{n})=a_n\rho _{n}^2+b_n\rho _{vac}^2+c_n\rho _{n}\rho _{vac}+d_n\rho _{n}+e_n\rho _{vac}\; , \end{aligned}$$

(4.38)

with some constants $a_n,\;b_n,\;c_n,\;d_n$, and $e_n$. The condition for a physical equilibrium point is

$$ b_n\rho _{*}^2+e_n\rho _{*}=0,\quad \rho _{*}\ge 0,\quad , n=1,2,\ldots ,N_i\; . $$

So an empty space is always a rest point, but the existence of a non-trivial de Sitter equilibrium requires

$$ \rho _{*}=-b_n/e_n>0,\qquad n=1,2,\ldots ,N_i\; , $$

giving $N_i$ relations on the $5N_i$ parameters involved. For small values of matter densities, i.e., in the large time limit, the solution is effectively governed by the general linear law. Otherwise the signature of the quadratic form in the r.h.s. of Eq. (4.29) is most important for the existence of physical solutions and their properties.

4.2.3 Non-singular Cosmologies

Non-singular cosmologies found in Sect. 3.3 for one matter specie exist in the multiple matter component case as well. They evolve according to the general picture of non-linear interaction described there, namely, as heteroclinic phase trajectories connecting one physical rest point, $\rho _{n}=0,\;\rho _{vac}=\rho _{0}>0$, of the system (4.13) with the other, $\rho _{n}=0,\;\rho _{vac}=\rho _{\infty }>0$, now in the $N_i+1$–dimensional phase space.

Note that if not all matter species interact with dark energy ($N_i<N$), then a ‘mixed’ type cosmology is obtained in this way: the interacting components and DE are non-singular, but the non-interacting ones start at a singularity. In this case there is no limitations on the spacetime curvature pointed out in Sect. 3.3.2, because the denisities of the conserved components dominate everything else, including the curvature contribution, at the expansion beginning. The corresponding ‘mixed type’ universe can be either open, or flat, or closed. When all the matter species are interacting, $N_i=N$, then an entirely non-singular universe is necessarily open. In this case all matter is born from heavy vacuum and pushed apart by its anti-gravity, as E.B. Gliner suggested back in 1965.

The semi–inverse method for constructing such solutions developed in the Sect. 3.4.1 also works in the general case. Indeed, in a complete similarity with the one-specie Anzatz (3.51) we assume that a heteroclinic trajectory $\mathcal{H}$ is described by the equations

$$\begin{aligned} \rho _{n}=h_{n}(\rho _{vac}),\;\;\rho _{\infty }<\rho _{vac}<\rho _{0},\;\; h_{n}(\rho _{0})=h_{n}(\rho _{\infty })=0;\quad n=1,2,\ldots N_i\; , \end{aligned}$$

(4.39)

where the functions $h_{n}$, positive inside their domain, are otherwise arbitrary. The appropriate calculations go the same way as in the Sect. 3.4.1.

Namely, the first $N_i$ Eq. (4.13) require certain values of the interaction functions $F_{n}(\rho _{vac},\rho _{n})$ on the heteroclinic curve which are found from the linear algebraic system (as usual, the prime denotes the derivative in $\rho _{vac}$):

$$\begin{aligned} -(1+w_k)h_k(\rho _{vac})&= F_k(\rho _{vac},h_k(\rho _{vac}))+h_k^{'}(\rho _{vac})\sum \limits _{n=1}^{N_i}\,F_{n}(\rho _{vac},h_{n}(\rho _{vac}))\;,\\&\ \ \,\rho _{\infty }<\rho _{vac}<\rho _{0},\qquad k=1,2,\ldots N_i\; . \end{aligned}$$

It allows for a simple explicit solution: by summing up all the equations, we first find the sum

$$\begin{aligned} S(\rho _{vac})\equiv F\biggl |_\mathcal{H}=\sum \limits _{n=1}^{N_i}\,F_{n}\biggl |_\mathcal{H}= -\frac{\sum \limits _{n=1}^{N_i}\,(1+w_{n})h_{n}(\rho _{vac})}{1+\sum \limits _{n=1}^{N_i}\,h_{n}^{'}(\rho _{vac})}\;, \end{aligned}$$

(4.40)

and then, from each of the above equations, functions $F_{n}$ on the curve $\mathcal{H}$:

$$ F_k\biggl |_\mathcal{H}=--(1+w_k)h_k(\rho _{vac})-S(\rho _{vac}),\qquad k=1,2,\ldots N_i\; $$

(we do not actually use them in what follows). They have no singularity on the interval $[\rho _{\infty },\;\rho _{vac}]$ under the condition

$$\begin{aligned} \min _{\rho _{\infty }\le \rho _{vac}\le \rho _{0}}\sum \limits _{n=1}^{N_i}\,h_{n}^{'}(\rho _{vac})>-1\; , \end{aligned}$$

(4.41)

and can be extended from the curve $\mathcal{H}$ to the whole phase space in a continuum of ways, as noted in Sect. 3.4.1; the inequality (4.41) is a direct generalization of the single-phase condition (3.53).

Now, by the formula (4.40), the last of the governing equations (4.13) on the heteroclinic curve $\mathcal{H}$ becomes

$$ \frac{d\rho _{vac}}{d\lambda }\biggl |_\mathcal{H}=F\biggl |_\mathcal{H}=S(\rho _{vac})=-\frac{\sum \limits _{n=1}^{N_i}\,(1+w_{n})h_{n}(\rho _{vac})}{1+\sum \limits _{n=1}^{N_i}\,h_{n}^{'}(\rho _{vac})}\; , $$

so determining $\rho _{vac}$ reduces to integrating the known function. The result, in terms of the variable V, is:

$$\begin{aligned} \exp \left[ \hat{H}(\rho _{vac})\right] =\frac{C}{V},\qquad \hat{H}(\rho _{vac})= {\Large \int \limits ^{\rho _{vac}}}\,\frac{1+\sum \limits _{n=1}^{N_i}\,h_{n}^{'}(x)}{\sum \limits _{n=1}^{N_i}\,(1+w_{n})h_{n}(x)}\, dx\; , \end{aligned}$$

(4.42)

with $C>0$ being a constant of integration. This is the analog of the Eq. (3.54) for determining the DE density. If this transcendental equation has a positive solution $\rho _{vac}=\rho _{vac}(V)$ decreasing monotonically from $\rho _{vac}=\rho _{0}$ to $\rho _{vac}=\rho _{\infty }$, then $\rho _{n}=h_{n}(\rho _{vac}(V))$, and these $N_i+1$ functions provide a solution of the system (4.13) corresponding to the heteroclinic trajectory $\mathcal{H}$ in its phase space. If exist, the densities of non-interacting species are given by the usual expressions (4.11), completing the solution describing a ‘mixed’ cosmology.

By specifying algebraic behavior of functions $h_{n}$ at the ends of the interval of their definition, like in the equalities (3.55), one can find the asymptotic behavior of the interacting densities at the beginning and end of the expansion, first as functions of V, as it is done in the Appendix C, and then as functions of time, as in the formulas (3.58), (3.59).

We here extend our calculations for just one special case, which leads to even more similarity with the results of Sect. 3.4.1, and hence to the set of particular exact solutions. Namely, we assume that the projections of the heteroclinic trajectory $\mathcal{H}$ on each of the planes $\{\rho _{n},\rho _{vac}\},\; n=1,2,\ldots ,N_i$ all have the same shape. That is, we assume that $h_{n}(\rho _{vac})$ differ from each other only by scaling:

$$\begin{aligned} h_{n}(rv)=\chi _n h(\rho _{vac}),&\qquad \chi _n>0\; ;\\ h(\rho _{vac})>0\;\;\text{ for }\;\;\rho _{\infty }<\rho _{vac}<\rho _{0},&\qquad h(\rho _{\infty })=h(\rho _{0})=0\; . \nonumber \end{aligned}$$

(4.43)

It is then straightforward to calculate, by the formula (4.42):

$$ \exp \left[ \hat{H}(\rho _{vac})\right] =\chi ^{-\frac{1}{1+\bar{w}}}\left\{ \check{h}(\rho _{vac})\exp \left[ H(\rho _{vac})\right] \right\} ,\quad H(\rho _{vac})={\Large \int \limits ^{\rho _{vac}}}\,\frac{dx}{\check{h}(x)}\; , $$

where

$$ \chi =\sum \limits _{n=1}^{N_i}\,\chi _n,\quad 1+\bar{w} =\frac{1}{\chi }\,\sum \limits _{n=1}^{N_i}\,(1+w_{n})\chi _n,\quad \check{h}(\rho _{vac})=\chi h(\rho _{vac})\; . $$

After some constant reassignment we can thus rewrite the resolving Eq. (4.42) in exactly the form of the resolving Eq. (3.54) of the single–phase case:

$$\begin{aligned} \check{h}(\rho _{vac})\exp H(\rho _{vac})=\rho _{*}\left( \frac{V_{*}}{V}\right) ^{1+\bar{w}}\; , \end{aligned}$$

(4.44)

with just h replaced with $\check{h}$, and w replaced with $\bar{w}$. So one can use the exact solutions of the examples from Sect. 3.4.2 obtained for the functions

$$\begin{aligned} \check{h}(\rho _{vac})&= (\rho _{0}-\rho _{vac})(\rho _{vac}-\rho _{\infty })/R, \quad \check{h}(\rho _{vac})=\theta (\rho _{0}-\rho _{vac})(\rho _{vac}-\rho _{\infty })/\rho _{vac}\;,\\&\qquad \quad \check{h}(\rho _{vac})=(\rho _{0}-\rho _{vac})(\rho _{vac}-\rho _{\infty })^2/R^2\; , \end{aligned}$$

as well as construct many other.

4.3 Three Matter Phases: A Model for Our Universe

To get closer to the only reality known by us, we finally consider a cosmological model with dark energy and three matter phases: dark matter (DM), $w=0$, normal matter, $w=0$, and radiation, $w=1/3$. There are many speculations about a possible relation between the dark energy and dark matter, which seem plausible intuitively. Following these ideas we here assume that only dark matter interacts with dark energy, and the other two matter phases are conserved, as in the usual cosmological models. This puts us in the case of the Sect. 4.1 with $N=3$; we denote $\rho _1=\rho _{dm}$ the DM density ($w_1=0$), $\rho _2=\rho _{m}$ the density of normal matter ($w_2=0$), and $\rho _3=\rho _{r}$ the density of radiation ($w_3=1/3$). The last two species are conserved, so their densities are given by the standard formulas:

$$\begin{aligned} \rho _{m}=C_m/V,\qquad \rho _{r}=C_r/V^{4/3},\qquad C_{m,r}>0 \; . \end{aligned}$$

(4.45)

Thus there is always the Big Bang in this model, but DE and DM are not necessarily involved in it. The behavior of $\rho _1$ and $\rho _{vac}$ is determined by the system (4.9), written as

$$ \frac{d\rho _1}{dV}= -\frac{\rho _1+ F(\rho _{vac},\rho _1)}{V},\qquad \qquad \frac{d\rho _{vac}}{dV}= \frac{ F(\rho _{vac},\rho _1)}{V}\; , $$

or, in terms of $\lambda =\ln (V/V_{*})$ and $\rho _{dm}=\rho _1$, as

$$\begin{aligned} \frac{d\rho _{dm}}{d\lambda }= -\left[ \rho _{dm}+ F(\rho _{vac},\rho _{dm})\right] ,\qquad \qquad \frac{d\rho _{vac}}{d\lambda }=F(\rho _{vac},\rho _{dm})\; . \end{aligned}$$

(4.46)

It is nothing else as the Eq. (3.6) with $w=0$, so we can use all results of Chap. 3 in the discussion of our model of the Universe.

We start with a special linear interaction law (3.9), $F(\rho _{vac},\rho _{dm})=-s\rho _{dm}$, when the rate of DE reduction is proportional to the dark matter density. The corresponding exact solution (3.11) reads:

$$\begin{aligned} \rho _{dm}=\frac{C_{dm}}{V^{1-s}},\qquad \rho _{vac}=\rho _{\infty }+\frac{s}{1-s}\,\frac{C_{dm}}{V^{1-s}}\; ; \end{aligned}$$

(4.47)

here $C_{dm}>0,\;\rho _{\infty }\ge 0$ are arbitrary constants, and the interaction parameter s is in the range (3.12), $0<s<1$.

The expressions (4.47) and (4.45) combine to give a cosmological solution that differs from the usual one, with the constant DE density, by the power in the dependence of $\rho _{dm}$, $V^{-(1-s)}$ instead of $V^{-1}$. However, this difference is essential from the point that, although both the dark and normal matter densities tend to zero at large times, their ratio

$$ \rho _{dm}/\rho _m\propto \left( C_{dm}/C_{m}\right) V^{s}\rightarrow \infty ,\qquad V\rightarrow \infty \; , $$

tends to infinity at the large time limit independent of the parameters involved. So, without any fine–tuning, dark matter dominates normal matter at later stages, as observed in our universe. Otherwise, radiation dominates the early universe, as usual, so the scale factor $a(t)\propto t^{1/2},\; t\rightarrow +0$; non-vanishing DE dominates all other components at later time providing the typical exponential time dependence of the the scale factor (see formula (3.13)).

Interestingly, this linear model of interaction between DE and DM in our universe was checked against the observational data in a recent paper [1]. The authors used the Planck 2013 data, the baryon acoustic oscillations measurements, the type-Ia supernovae data, the Hubble constant measurement, the redshift space distortions data and the galaxy weak lensing data to estimate the parameter s (denoted $\beta$ in the paper). One-sigma errors of the found estimates are larger than 100%. Generally, constraints on any interaction models are very important, but they are definitely a subject for further investigation(s).

Next, as demonstrated in Sect. 4.1, the general interaction law (3.35),

$$ F(\rho _{vac},\rho _{dm})=-s\rho _{dm}+\theta (\rho _{vac}-\rho _{\infty }),\quad s,\theta ,\rho _{\infty }=\text{ const },\quad \rho _{\infty }\ge 0\; , $$

does not allow for any continuous physical solution. However, a solution of the form (4.10) with a jump in the DE density is possible. In this case it is given by the following expressions:

$$\begin{aligned}&\rho _{vac}=\rho _{\infty },\;\; \rho _{dm}=0,\;\;\rho _m=C_m/V,\; \rho _{r}=C_r/V^{4/3}\quad \text{ for }\; 0<V<V_{*}\;;\nonumber \\&\rho _{vac}=\rho _{\infty }+{|s|}\rho _{*}\left[ Q_2\left( \frac{V_{*}}{V}\right) ^{|\mu _{2}|}+Q_1\left( \frac{V_{*}}{V}\right) ^{|\mu _{1}|}\right] ,\;\rho _{dm}=\rho _{*}\left[ \left( \frac{V_{*}}{V}\right) ^{|\mu _{2}|}-\left( \frac{V_{*}}{V}\right) ^{|\mu _{1}|}\right] ,\nonumber \\&\rho _m=C_m^{'}/V,\;\; \rho _{r}=C_r^{'}/V^{4/3}\quad \text{ for }\quad V_{*}<V<+\infty \; ;\\&\Delta \rho _{tot}\Bigl |_{V=V_{*}}=\left( \Delta \rho _{vac}+\Delta \rho \right) \Bigl |_{V=V_{*}}= \rho _{0}-\rho _{\infty }+\frac{C_m^{'}- C_m}{V_{*}}+\frac{C_r^{'}- C_r}{V_{*}^{4/3}}=0\nonumber \; ; \end{aligned}$$

(4.48)

all the parameters are restricted as in the formulas (4.10), and $\rho _{0}=\rho _{vac}({V_{*}+0})$.

In this cosmology matter and radiation are born from a singularity on the background of a finite DE density remaining constant, $\rho _{vac}=\rho _{\infty }$, until some moment of time $t_*,\;V_{*}=V(t_*)$. At this moment the two existing non-interacting species undergo an instant phase transition raising the DE density to the value $\rho _{0}>\rho _{\infty }$. After this the DE density relaxes all the time back to its initial value $\rho _{\infty }$, and dark matter appears whose density first grows, then reaches some maximum, and then declines to zero at infinity; the evolution of the two interacting species is depicted in Fig. 3.1.

As before, this example leads us to non-singular cosmologies appearing under non-linear interaction laws, i.e., to the results of Sects. 3.3.2 and 3.4, which all apply to our current model. Non-singular cosmological solutions discussed and explicitly found there correspond to heteroclinic curves in the phase plane $\{\rho _{vac},\rho _{dm}\}$ connecting two de Sitter equlibrium states with $\rho _{vac}=\rho _{0}$ and $\rho _{vac}=\rho _{\infty }$. So the dark energy density evolves from the initial value $\rho _{0}$ to the final value $\rho _{\infty }$. Dark matter appears at the start of the evolution, its density reaches a maximum (whose value depends on the model parameters, c.f. formulas (3.65), (3.70), (3.75)) at some moment of time, and then tends back to zero.

The conserved radiation and normal matter are born in a singularity, their densities evolve according to the usual expressions (4.45). At large times DM often dominates normal matter, since the former goes to zero slower than the latter. This is clearly seen from the asymptotic formulas (C4) in the cases (a) and (b), when at large times the ratio $\rho _{dm}/\rho _{m}$ tends to infinity independent of the model parameters. In the case (c) both densities have the same later times dependence $\propto V^{-1}$, so the DM dominance requires parameter tuning.

Of course, our universe can be also modeled with two or all the matter phases interacting with DE; the results of Sect. 4.2 apply to such models. When all matter phases interact with DE, there can exists entirely non–singular solutions describing matter born by heavy vacuum at the onset of cosmological expansion caused by DE anti-gravity.