Nonsingular bounces catalyzed by dark energy

We investigate classically nonsingular bounces caused by dark energy. In the presence of positive spatial curvature, vacuum energy, either in the form of a cosmological constant or a scalar field potential, allows for an open set of initial conditions leading to nonsingular bounces, without any violation of the null energy condition. We study anisotropic Bianchi IX cosmologies and demonstrate that they can even have multiple bounces, accompanied by intricate evolutions of the anisotropies that provide a nonsingular analogue of mixmaster crunches. The relation of these solutions to more complete cosmological models, as well as to the recently proposed swampland criteria, are briefly discussed.

Figure 1

The anisotropy potential $U({\beta }_{+},{\beta }_{-})$. The minimum is at $U(0,0)=-3$. Around the minimum the potential has an approximate circular symmetry, while at larger values of the anisotropy parameters, it has the symmetries of an equilateral triangle. The potential asymptotes to zero from below in the “corner” directions. In this plot, only the region where $U<0$ is displayed, for reasons explained in the main text.

Figure 2

This plot shows the evolution of the anisotropies ${\beta }_{\pm }$ as a function of time. Time is height in the graph, the plotted ranges are $-7/10<{\beta }_{+}<11/10$, $-9/10<{\beta }_{-}<9/10$, and $-300<t<300$, for $\mathrm{\Lambda }=3\times {10}^{-4}$ so that the Hubble radius is $1/H=100$ Planck lengths. The bounce occurs in the middle, with zero derivatives $\dot{a}={\dot{\beta }}_{+}={\dot{\beta }}_{-}=0$ at $t=0$. There is a general focusing towards smaller values of the anisotropies away from the bounce. The colored curves show the evolution of the anisotropies near the bounce for solutions that evolve to a large universe asymptotically. The color changes as a function of the distance from the isotropic (pure de Sitter) solution located at the center of the plot; see also the next figures and the text for more details.

Figure 3

Time slices through the previous figure: clockwise from top left at $t=0$, 50, 100, 500. The anisotropy parameters are rescaled by a factor of 100. Gray dots in the $t=0$ slice indicate values where the potential $U$ is positive (cf. Fig. 1), and where no bounce can occur. Black dots mark the anisotropy values for which the bounce is followed by a rapid recollapse. As one can see, the rapid recollapse region surrounds the conditions for a bounce in all anisotropy directions. Overall, the triangular shape of the anisotropy potential is easily recognizable, and the later time slices show how various solutions reflect off the potential walls, while overall there is a general focusing effect towards smaller anisotropy values away from the bounce.

Figure 4

These graphs indicate the number of extrema of the scale factor $a(t)$, as a function of the anisotropy values at $t=0$. A value of 1 corresponds to a single bounce, while 5 for instance implies three bounces separated by two local maxima of $a$. An $R$ marks a bouncing solution that rapidly recollapses to a singularity, while 0 means that no bounce is possible at all. The plot on the right is a zoom in near the edge of the region of recollapse.

Figure 5

These two graphs show the evolution of the scale factor and the anisotropy parameters for the time symmetric solution with ${\beta }_{+}=-11/30$, ${\beta }_{-}=1/6$ at the bounce. For this solution, there are a total of seven bounces occurring in succession, while the anisotropy parameters undergo elaborate reflections off the walls of the anisotropy potential. Multibounce solutions, such as this one, occur near the edges of the allowed parameter space, as evidenced in Fig. 4.

Figure 6

This plot shows the evolution of the anisotropies ${\beta }_{\pm }$ as a function of time. Time is height in the graph, the plotted ranges are $-7/10<{\beta }_{+}<11/10$, $-9/10<{\beta }_{-}<9/10$ at the bounce, and $-300<t<300$, for $\mathrm{\Lambda }=3\times {10}^{-4}$ so that the Hubble radius is $1/H=100$ Planck lengths. Initial conditions are imposed at $t=0$, where $\dot{a}=0$ and ${\dot{\beta }}_{+}={\dot{\beta }}_{-}=1/200$ (left graph) and ${\dot{\beta }}_{+}={\dot{\beta }}_{-}=1/80$ (right graph).

Figure 7

The number of extrema of the scale factor $a(t)$ for the solutions plotted in Fig. 6.

Figure 8

These graphs show the evolutions of the scale factor and of the anisotropy parameters as a function of time, for the solution with ${\beta }_{+}(0)={\beta }_{-}(0)=0$, but with large velocities ${\dot{\beta }}_{+}(0)={\dot{\beta }}_{-}(0)=1/80$. Although the kinetic energy at $t=0$ is larger than the value that could lead to a bounce, away from $t=0$ two bounces nevertheless occur since the energy in the anisotropies is slightly reduced there.

Figure 9

The left plot depicts the anisotropy potential $U({\beta }_{+},{\beta }_{-}=0)$ along one of the axes of symmetry, here chosen to be the axis ${\beta }_{-}=0$. The range over which nonsingular bounces can occur is marked in red. Bounces occur for arbitrarily large positive anisotropies in this direction, and the solutions are strongly focused towards zero away from the bounce. There is a minimum value ${\beta }_{+}=\frac{1}{3}\ln (\frac{1}{3})\approx -0.366$ below which the solutions rapidly recollapse. Below ${\beta }_{+}=\frac{1}{3}\ln (\frac{1}{4})\approx -0.462$, the anisotropy potential is positive an no bounce can occur at all.

Figure 10

These two plots show the evolution of the scalar field as a function of time, for solutions that bounce at $t=0$ at a specified value of $a=10$, $\varphi =1/2$, for a range of values of $\dot{\varphi }$ at the bounce. The left panel is for a potential $V=e^{\varphi /10}$, showing solutions with $\dot{\varphi }(t=0)$ up to values of 0.90 while the right panel is for $V=e^{\varphi /2}$, showing solutions with $\dot{\varphi }(t=0)$ up to values of 0.48. Lighter curves correspond to larger velocities at the bounce.

Figure 11

The evolution of the scale factor and the scalar field for a range of initial velocities leading to nonsingular bounce solutions. The potential is taken to be $\varphi =e^{\mathrm{phi}/10}$ here. The initial values for the scale factor and scalar field are $a_i=9$, ${\varphi }_i=23/50$, and the range of initial velocities leading to nonsingular bounces is found to be $0.058⪅\dot{\varphi }⪅0.064$, represented by the curves ranging from black to yellow, respectively.

Figure 12

Same as Fig. 11, but for the potential $V=e^{\varphi /100}$. The initial values for the scale factor and scalar field are $a_i=9$, ${\varphi }_i=23/50$, and the range of initial velocities leading to nonsingular bounces is found to be $0.0024⪅\dot{\varphi }⪅0.095$, represented by the curves ranging from black to yellow, respectively.

Figure 13

Evolution of the anisotropy parameter $\beta$ as a function of time for Kantowski-Sachs bounces. For all $\beta \mathrm{s}$, there exists a bounce (here with $\dot{\beta }=0$ at the bounce).