Archimedean-type force in a cosmic dark fluid. II. Qualitative and numerical study of a multistage universe expansion

In this (second) part of the work we present the results of numerical and qualitative analysis, based on a new model of the Archimedean-type interaction between dark matter and dark energy. The Archimedean-type force is linear in the four-gradient of the dark energy pressure and plays a role of self-regulator of the energy redistribution in a cosmic dark fluid. Because of the Archimedean-type interaction the cosmological evolution is shown to have a multistage character. Depending on the choice of the values of the model-guiding parameters, the Universe expansion is shown to be perpetually accelerated, periodic or quasiperiodic with a finite number of deceleration/acceleration epochs. We distinguished the models, which can be definitely characterized by the inflation in the early Universe, by the late-time accelerated expansion and nonsingular behavior in intermediate epochs, and classified them with respect to a number of transition points. Transition points appear, when the acceleration parameter changes the sign, providing the natural partition of the Universe’s history into epochs of accelerated and decelerated expansion. The strategy and results of numerical calculations are advocated by the qualitative analysis of the instantaneous phase portraits of the dynamic system associated with the key equation for the dark energy pressure evolution.

Figure 1

Perpetually accelerated universe: the acceleration parameter $-q(\tau )$ is non-negative for arbitrary $\tau$. Since there are no transition points, in which $-q(\tau )$ changes the sign, the history of this universe includes only one epoch; since at least five extrema can be recognized in Fig. 1f, the history consists of six or more eras. Here the dark solid line relates to the following set of the parameters: $\xi =0.35$, $\sigma =-0.99$ (i.e., $3\xi +\sigma \simeq 0.06>0$), ${\mathcal{V}}_{(0)}=1$, $E_{(0)}=0.0205$, ${\lambda }_{(0)}=1$ (i.e., the DM is initially medium-relativistic), ${\rho }_{*}=0.333\times {10}^{-4}$, and ${\Pi }^{\prime }(1)=-1$.

Figure 2

Periodic universe with infinite number of acceleration/deceleration epochs. This model is a limiting case, when the parameter $3\xi +\sigma$ tends to its critical value $0_{+0}$. For this model not only the sum $\rho +E$ but the DE energy density $\rho$ itself remain non-negative. Oscillations in the plots of the Hubble function $H(\tau )$ and of the scale factor $a(t)$ are the most explicit (see dark solid line), when the parameters of the model are the following: $\xi =0.1$, $\sigma =-0.299999$ (i.e., $3\xi +\sigma \simeq {10}^{-6}>0$), ${\mathcal{V}}_{(0)}=1$, $E_{(0)}=0.0205$, ${\lambda }_{(0)}=1$, ${\rho }_{*}=0.333\times {10}^{-4}$, and ${\Pi }^{\prime }(1)=0.01$.

Figure 3

The model with one transition point and $-q(\tau )$ curve of the Heaviside step-function type. The first (deceleration) epoch and the second (acceleration) epoch are not divided into eras. The epoch with the accelerated expansion is of the de Sitter–type, characterized by $\rho +\Pi =0$. The sum $\rho (\tau )+E(\tau )$ is non-negative for arbitrary $\tau >0$. The parameters of the model are the following: $\xi =0.1$, $\sigma =50$, ${\mathcal{V}}_{(0)}=1$, $E_{(0)}=0.0205$, ${\lambda }_{(0)}=1$, ${\rho }_{*}=0.333\times {10}^{-4}$, and ${\Pi }^{\prime }(1)=-5$.

Figure 4

The model with one transition point and two eras in both decelerated and accelerated epochs. The model describes the transition from one de Sitter–type universe to another one with different Hubble constants $H_0\ne H_{\infty }$. In contrast to DM energy density, the DM pressure grows in the early universe, reaches the maximum, and then decreases quickly in the deceleration epoch. The parameters of the model are the following: $\xi =0.1$, $\sigma =0$, ${\mathcal{V}}_{(0)}=1$, $E_{(0)}=0.0205$, ${\lambda }_{(0)}=1$, ${\rho }_{*}=0.333\times {10}^{-4}$, and ${\Pi }^{\prime }(1)=-17$.

Figure 5

The model with two transition points, two acceleration epochs, and one deceleration epoch. The plots for the DE energy density, DE pressure, and Hubble functions are analogous to the curves given in Fig. 1. The DM pressure has a maximum at the end of the first era of the first acceleration epoch in analogy with the curve given by Fig. 4d. The parameters of the model are the following: $\xi =0.1$, $\sigma =-0.08$, ${\mathcal{V}}_{(0)}=1$, $E_{(0)}=0.0205$, ${\lambda }_{(0)}=1$, ${\rho }_{*}=0.333\times {10}^{-4}$, and ${\Pi }^{\prime }(1)=-15$.

Figure 6

The model with two epochs of decelerated expansion and two epochs of accelerated evolution. The plot of the function $-q(\tau )$ is of the so-called $\mathcal{N}$ type. The second accelerated epoch is of the de Sitter type. The parameters of the model are the following: $\xi =0.1$, $\sigma =1$, ${\mathcal{V}}_{(0)}=1$, $E_{(0)}=0.0205$, ${\lambda }_{(0)}=1$, ${\rho }_{*}=0.333\times {10}^{-4}$, and ${\Pi }^{\prime }(1)=-5$.

Figure 7

The intermediate model with a big but finite number of transition points. The oscillatory regime is produced by the DE component of the dark fluid, and it is switched on at the moment when the DM energy density and DM pressure become vanishing. Late-time expansion is, evidently, accelerated. The dark solid lines relate to the following parameters: $\xi =0.1$, $\sigma =-0.29$ (i.e., $3\xi +\sigma \simeq 0.01>0$), ${\mathcal{V}}_{(0)}=1$, $E_{(0)}=0.0205$, ${\lambda }_{(0)}=1$, ${\rho }_{*}=0.333\times {10}^{-4}$, and ${\Pi }^{\prime }(1)=-1$.

Figure 8

Phase portrait of the autonomous dynamic system with stable node at $X_{(1)}=0$, $Y_{(1)}=-0.15$, and saddle point at $X_{(2)}=0.2125$, $Y_{(2)}=-1$. Integral curves in the northern and western domains, which are cut out by the separatrices of the saddle point, tend asymptotically to the node. Integral curves in the southern and eastern domains of this saddle point can be asymptotically characterized by infinite values of $X$ and/or $Y$.

Figure 9

Phase portrait of the autonomous dynamic system with stable focus and saddle point. The domain of quasiperiodic motion is situated in the southwestern domain, which is cut out by the separatrices of the saddle point at $X_{(1)}=0$, $Y_{(1)}=0.1875$. The focus is situated at $X_{(2)}=-0.59375$, $Y_{(2)}=-1$.

Figure 10

The plot illustrates the appearance of a pair of (quasi)critical points, when the cosmological time increases. The function $f(X)\equiv X-\beta \log X$ has the minimum $f_{(\min )}=\beta \log \frac{e}{\beta }$ at $X=\beta$; the point of minimum divides the plot of this function into two branches: the first of them is situated at $0<X<\beta$, and the second one relates to the interval $X>\beta$. Clearly, Eq. (32) has no solutions, when $\mathcal{K}({\tau }_0)<\beta \log \frac{e}{\beta }$; it has only one (double) root, when $\mathcal{K}({\tau }_0)=\beta \log \frac{e}{\beta }$; there are two different roots, when $\mathcal{K}({\tau }_0)>\beta \log \frac{e}{\beta }$, the corresponding points indicated as $M_{(*)}$ and $M_{(**)}$. The curve on the plot corresponds to the value $\beta =1$.

Figure 11

In the left panel a fragment of the plot is presented, which illustrates the inflationary-type growth of the scale factor $a(t)$ for the early Universe, given in terms of cosmological time $t$. In the right panel one can see a typical late-time behavior of $E(\tau )$ and $P(\tau )$; this rescaled fragment illustrates the features of the DM evolution at $\tau >1$, which cannot be recognized on the plots given by Figs. 1b, 1d, 2b, 2d, 3b, 3d, 4b, 4d, 5b, 5d, 6b, 6d, 7b, 7d.

Figure 12

The left panel contains the plot of the ratio $w(\tau )=\frac{P(\tau )}{E(\tau )}$, which presents the effective equation of state for the DM component of the dark fluid. Maximal value of this ratio is about $1/3$ in the first epoch of the Universe’s evolution, when the DM can be considered as an effectively ultrarelativistic substrate. In the early Universe the DM state evolves quasiperiodically; starting from $\tau \simeq 4$ the function $w(\tau )$ tends to zero linearly in $\tau$, i.e., the DM behaves effectively as a cold gas (dust). The right panel displays the cross-points of the $\rho$ and $E$ plots. This example illustrates the so-called coincidence problem: why the energy densities of the DE and DM components of the dark fluid are of the same order today (72% and 23%, respectively).