Six-dimensional cosmological models with conformal extensions

TL;DR

This work studies six-dimensional cosmologies with conformal extensions built from Weyl tensor cubed invariants, focusing on a $4+2$ spacetime split and the dynamical stabilization of the two extra dimensions. By isolating the highest-derivative terms, the authors formulate a constrained, numerically tractable dynamical system and analyze anisotropic background solutions in both flat and curved inner spaces. They find stable attractors where the three large spatial dimensions expand while the inner dimensions stabilize, with stability depending on the Weyl-cubed couplings and a tuned bare cosmological constant. The results show that conformal extensions can drive dimensional reduction and yield de Sitter–like behavior in anisotropic settings, providing a platform for further exploration of anomaly-induced actions and higher-dimensional gravity. Appendices supply the explicit Weyl contractions and variations that underlie the computed dynamics.

Abstract

We consider the background cosmological solutions in the $6D$ (six-dimensional) model with one time and five space coordinates. The theory of our interest has the action composed by the Einstein term, cosmological constant, and two conformal terms constructed from the third powers of the Weyl tensor. It is shown how the highest derivative terms in the equations of motion can be isolated that opens the way for their numerical integration. There are flat anisotropic solutions which make one of the flat isotropic subspaces to be static. Depending on the value of bare cosmological constant, either two-dimensional or three-dimensional subspace can be static. In particular, there is a physically favorable solution with three ``large'' space coordinates and two extra inner dimensions stabilized. This solution is stable for a wide range of coupling constants, but this requires a special value of the bare cosmological constant.

Figures (6)

• Figure 1: For this plot it is chosen $\Lambda=100$ which results in $\dot{\beta}_{\hbox{\tiny de Sitter}}\approx 4.08$ given by (\ref{['i.cond']}) while $\dot{\gamma}=0$. It can be seen that after a time interval, the solution converges as expected to the equally de Sitter inflating $S^2$ and $S^3$ attractor $\dot{\beta}_{\hbox{\tiny de Sitter}}=\dot{\gamma}_{\hbox{\tiny de Sitter}} \approx 3.16$ given by (\ref{['same.H']}).
• Figure 2: In this plot the choice of coupling constants is $\theta_{1}=-2.235$ and $\theta_{2}=-4$. The initial condition is not chosen exactly as the de Sitter solution, \ref{['de SitterC3']} for $\dot{\beta}_{\hbox{\tiny de Sitter}}\simeq 0.9411$, while $\dot{\gamma}=0$. The initial condition for the solution plotted as a solid line is chosen as $\dot{\beta}=\dot{\beta}_{\hbox{\tiny de Sitter}}$ and $\dot{\gamma}=0+0.03$. In panel a) it is shown that $\dot{\beta}\rightarrow \dot{\beta}_{\hbox{\tiny de Sitter}} \simeq0.9411$ the inflating $E^3$ asymptotically, while in panel b) it is shown that $\dot{\gamma}\rightarrow 0$ the static $E^2$ also asymptotically.
• Figure 3: This plot refers to the stability of the solution \ref{['de SitterC3']} with a de Sitter inflating $E^3$ with a static $E^2$ in the coupling constants $\theta_1 \times \theta_2$ plane. For every white point all the eigenvalues have negative real part. While for every gray point, at least one eigenvalue has positive real part. The black points represent the region for which the solution \ref{['de SitterC3']} is imaginary, is not physically relevant.
• Figure 4: For this plot it is chosen $\theta_{1}=1.88$, $\theta_{2}=2$ and the de Sitter inflating $E^2$ specified by \ref{['de SitterC3_S2']} results in $\dot{\gamma}_{\hbox{\tiny de Sitter}}\simeq 1.0541$ with static $E^3$, $\dot{\beta}=0$. The solid line in the plot shows an orbit with initial condition near this solution with $\dot{\gamma} =\dot{\gamma}_{\hbox{\tiny de Sitter}}$ and $\dot{\beta}=0+0.02$. Panel a) shows that the orbit approaches $\dot{\gamma}\rightarrow\dot{\gamma}_{\hbox{\tiny de Sitter}}$ asymptotically while panel b) shows that $\dot{\beta}\rightarrow 0$ asymptotically.
• Figure 5: For the dotted plots, the initial condition is chosen as $\theta_{2}=-0.5$ and $\Lambda=0.6$ and the de Sitter space $R\times E^3$, $\dot{\beta}_{\hbox{\tiny de Sitter}}=0.2$ with static $S^2$, $\dot{\gamma}=0$ which according to \ref{['sol:deSitterE3staticS2']} results in $\gamma_{\hbox{\tiny{$S ^2$}} } \simeq 1.231$ and $\theta_1\simeq 0.721$. The solid lines show an orbit with initial condition near this doted line solution, with the only difference that $\dot{\gamma} =0+0.001$ instead of $\dot{\gamma}=0$. Panel a) shows that the orbit approaches $\dot{\beta}\rightarrow\dot{\beta}_{\hbox{\tiny de Sitter}}$ asymptotically, while panel b) shows that $\dot{\gamma}\rightarrow 0$ asymptotically.