A note on splitting solutions in $4+1$ dimensional quadratic gravity

Abstract

In the present paper we consider anisotropic cosmological vacuum solutions in (4+1) dimensional general quadratic gravity. In particular, we present a solution with 3 equal and 1 different Hubble parameters, and study its stability. We show that for a certain range of coupling constants this solution is stable. This means that initially totally anisotropic 4-dim Universe can evolve naturally to a product of 3-dim isotropic subspace and 1-dim space. By numerical integration of equations of motion we construct bassin of attraction of this solution which covers part of the initial conditions space with non-zero measure.

Figures (4)

• Figure 1: Stable region in parameter space. Panel a) Plane $\alpha\times \gamma$. For each white point all the real parts of the eigenvalues are negative, and each black point the real parts can be zero or positive. Zone of negative real part of all the eigenvalues is the "good" region which is marked by white. For this plot we chose $H=0.5201$ and $h=-0.2407$, which for each pair of values $\alpha,\,\gamma$ sets the value for $\beta$ and $\Lambda$\ref{['sol.de.sitter']}. Panel b) Now we fix $\alpha=0.02$ and $\gamma=-16$, which is a white point from panel a), and now a grid of points in $h,\,H$ which according to \ref{['sol.de.sitter']} fixes the values for $\beta$ and $\Lambda$.
• Figure 2: Stable (3+1)-splitting solution \ref{['sol.de.sitter']}. This orbit is a white point from Figure \ref{['fig1']}a). In Panel a) it is shown the 3 expanding dimensions together with the stable solution $H=0.5201$ in dotted. Panel b) shows the shrinking dimension with the stable $h=-0.2407$ in dotted. Once $\alpha=0.017$ and $\gamma=-16$ are chosen from Figure \ref{['fig5']}a), while $H=0.5201$ and $h=-0.2407$ from Figure\ref{['fig5']}b), $\beta=48.0061$ and $\Lambda=-0.505859$ are uniquely specified from \ref{['sol.de.sitter']}
• Figure 3: A typical singular orbit. Curvature scalars increase and derivatives of Hubble parameters also increase, which characterizes a curvature singularity. Now this orbit is a black point from Figure \ref{['fig1']}.
• Figure 4: Basin plot in initial condition space for Hubble parameters. In both plots the coupling constants are fixed to $\alpha=0.017$, $\beta=48.0061$, $\gamma=-16$, $\Lambda=-0.505859$ and each black point corresponds to a "good" initial condition, which asymptotes the constant 3+1 splitting solution denoted as the red point, while each white point goes to the singularity. Panel a)The plane $H_1 \times H_2$. Panel b) the plane $H_1\times h_1$. From Panel b) it is possible to see that it is not necessary to have initially one contracting dimension and 3 expanding. Although of rather small measure region, initially sufficient anisotropy can be enough to reach the (3+1) splitting solution.