Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in various dimensions: Low-dimensional $\mathrm{\Lambda }$-term case

In this paper we perform a systematic study of spatially flat $[(3+\mathrm{D})+1]$-dimensional Einstein-Gauss-Bonnet cosmological models with the $\mathrm{\Lambda }$-term. We consider models that topologically are the product of two flat isotropic subspaces with different scale factors. One of these subspaces is three-dimensional and represents our space and the other is $D$-dimensional and represents extra dimensions. We consider no Ansatz on the scale factors, which makes our results quite general. With both Einstein-Hilbert and Gauss-Bonnet contributions in play, the cases with $D=1$ and $D=2$ have different dynamics due to the different structure of the equations of motion. We analytically study equations of motion in both cases and describe all possible regimes. It is demonstrated that the $D=1$ case does not have physically viable regimes while $D=2$ has a smooth transition from high-energy Kasner to an anisotropic exponential regime. This transition occurs for two ranges of $\alpha$ and $\mathrm{\Lambda }$: $\alpha >0$ with $\alpha \mathrm{\Lambda }\le 1/2$ (including $\mathrm{\Lambda }<0$) and $\alpha <0$, $\mathrm{\Lambda }>0$ with $\alpha \mathrm{\Lambda }<-3/2$. For the latter case, if $\alpha \mathrm{\Lambda }=-3/2$, the extra-dimensional part has $h\to 0$ and so the size of extra dimensions (in the sense of the scale factor) is reaching a constant value. We report substantial differences between $D=1$ and $D=2$ cases and between these cases and their vacuum counterparts, describe features of the cases under study, and discuss the origin of the differences.

Figure 1

Behavior of $h(H)$ from Eq. (12) in the $D=1$ case. Panel (a) corresponds to $\alpha >0$ and both (b) and (c) to $\alpha <0$. In all panels the black curves correspond to the typical $\mathrm{\Lambda }>0$ behavior and grey to $\mathrm{\Lambda }<0$. The difference between the (b) and (c) panels is that for black curves in (b) we have $\alpha \mathrm{\Lambda }>-3/2$ while in (c), $\alpha \mathrm{\Lambda }<-3/2$. (See the text for more details.)

Figure 2

Behavior of $\dot{H}(H)$ and $\dot{h}(H)$ from Eqs. (13) in the $D=1$ case. In all panels the black curves correspond to $\dot{H}(H)$ and the grey curves to $\dot{h}(H)$. Panel (a) corresponds to $\alpha >0$ and $\mathrm{\Lambda }>0$ and panel (b) to $\alpha >0$ and $\mathrm{\Lambda }<0$. The remaining panels correspond to $\alpha <0$: $\alpha \mathrm{\Lambda }<-3/2$ in (c), $\alpha \mathrm{\Lambda }=-3/2$ in (d), and $\alpha \mathrm{\Lambda }>-3/2$ in (e). Finally, in panel (f) we present the typical $\alpha <0$, $\mathrm{\Lambda }<0$ behavior. (See the text for more details.)

Figure 3

Behavior of Kasner exponents from Eq. (15) in the $D=1$ case. On all panels the black curves correspond to $p_H$, the solid grey curves to $p_h$, and the dashed grey to the expansion rate $\sum p=3p_H+p_h$. Panel (a) corresponds to $\alpha >0$ and $\mathrm{\Lambda }>0$ and panel (b) to $\alpha >0$ and $\mathrm{\Lambda }<0$. The remaining panels correspond to $\alpha <0$: $\alpha \mathrm{\Lambda }<-3/2$ in (c), $\alpha \mathrm{\Lambda }=-3/2$ in (d), and $\alpha \mathrm{\Lambda }>-3/2$ in (e). Finally, in panel (f) we present the typical $\alpha <0$, $\mathrm{\Lambda }<0$ behavior. (See the text for more details.)

Figure 4

Behavior of the radicand from (19) in panels (a) and (b), reduced denominator (26) in panels (c) and (d), and $h(H)$ curves from (19) for different cases in panels (e)–(i) for the $D=2$ case. (See the text for more details.)

Figure 5

Behavior of $\dot{H}(H)$ (black curves) and $\dot{h}(H)$ (grey curves) in the $D=2$ case for $\alpha >0$ and $\zeta <{\zeta }_0$ in panels (a) and (b); $0>\zeta >{\zeta }_0$, $\zeta \ne -1/6$ in panel (c); $\zeta =-1/6$ in panels (d) and (e); $0<\zeta <15/32$ in panels (f) and (g); and $\zeta =15/32$ in panels (h) and (i). (See the text for more details.)

Figure 6

Continuation of Fig. 5 for $\alpha >0$ and $1/2>\zeta >15/32$ in panels (a) and (b); $\zeta =1/2$ in panels (c) and (d); $1>\zeta >1/2$ in panel (e); $\zeta =1$ in panels (f) and (g); $\zeta >1$ in panels (h) and (i). (See the text for more details.)

Figure 7

Continuation of Figs. 5 and 6 for $\alpha <0$ and $\zeta >0$ in panels (a) and (b); $0>\zeta >-5/6$ in panels (c) and (d); and $\zeta <-5/6$ in panels (e)–(h). (See the text for more details.)