Examples for bouncing solutions with positive curvature (left) or a negative potential (right, negative cosmological constant). The solid lines show solutions of effective equations with a bounce, while the dashed lines show classical solutions running into the singularity at where diverges.
Example for a solution of and , showing early loop inflation and later slow-roll inflation driven by a scalar that is pushed up its potential by loop effects. The left hand side is stretched in time so as to show all details. An idea of the duration of different phases can be obtained from Figure 3.
Movie showing the initial push of a scalar up its potential and the ensuing slow-roll phase together with the corresponding inflationary phase of .
Movie illustrating the Bianchi IX potential (37) and the movement of its walls, rising toward zero and and along the diagonal direction, toward the classical singularity with decreasing volume . The contours are plotted for the function .
Movie illustrating the Bianchi IX potential in the anisotropy plane and its exponentially rising walls. Positive values of the potential are drawn logarithmically with solid contour lines and negative values with dashed contour lines.
Approximate effective wall of finite height  as a function of , compared to the classical exponential wall (upper dashed curve). Also shown is the exact wall (lower dashed curve), which for smaller than the peak value coincides well with the approximation up to a small, nearly constant shift.
Movie illustrating the effective Bianchi IX potential and the movement and breakdown of its walls. The contours are plotted as in Figure 4.
Movie illustrating the effective Bianchi IX potential in the anisotropy plane and its walls of finite height, which disappear at finite volume. Positive values of the potential are drawn logarithmically with solid contour lines and negative values with dashed contour lines.
Discrete subset of eigenvalues of (left) for two choices of (and ), together with the approximation from Equation (25) and small- power laws. The classical divergence at small , where the behavior is strongly modified, is cut off. The right panel shows the dependence of the initial increase on .
Movie showing the coordinate time evolution  of a wave packet starting at the bottom and moving toward the classical singularity (vertical dotted line) for different values of an ambiguity parameter. Some part of the wave packet bounces back (and deforms) according to the effective classical solution (dashed), but other parts penetrate to negative . The farther away from the effective bounce happens, depending on the ambiguity parameter, the smaller the part penetrating to negative is. The coordinate time evolution represents a physical state obtained after integrating over .
© Max Planck Society and the author(s)