List of Figures

View Image Figure 1:
Examples of bouncing solutions with positive curvature (left) or a negative potential (right, negative cosmological constant). The solid lines show solutions of equations with a bounce as a consequence of quantum corrections, while the dashed lines show classical solutions running into the singularity at a = 0 where φ diverges.
View Image Figure 2:
Example of a solution of a(t) and φ(t) showing early loop inflation and later slow-roll inflation driven by a scalar that has 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.
Watch/download Movie Figure 3: (mpg-Movie; 1154 KB)
Movie The initial push of a scalar φ up its potential and the ensuing slow-roll phase together with the corresponding inflationary phase of a.
Watch/download Movie Figure 4: (mpg-Movie; 1700 KB)
Movie An illustration of the Bianchi IX potential (40View Equation) and the movement of its walls, rising toward zero p1 and p2 and along the diagonal direction, toward the classical singularity with decreasing volume V = ∘ |p1p2p3|-. The contours are plotted for the function 1 2 2 1 2 W (p ,p ,V ∕(p p )).
Watch/download Movie Figure 5: (mpg-Movie; 1705 KB)
Movie An illustration of 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.
View Image Figure 6:
Approximate effective wall of finite height [80] as a function of x = − β+, compared to the classical exponential wall (upper dashed curve). Also shown is the exact wall 1 1 1 2 W (p ,p ,(V ∕p ) ) (lower dashed curve), which for x smaller than the peak value coincides well with the approximation up to a small, nearly constant shift.
Watch/download Movie Figure 7: (mpg-Movie; 1639 KB)
Movie An illustration of the effective Bianchi IX potential and the movement and breakdown of its walls. The contours are plotted as in Figure 4.
Watch/download Movie Figure 8: (mpg-Movie; 1699 KB)
Movie An illustration of 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.
View Image Figure 9:
Discrete subset of eigenvalues of ^d(p) (left) for two choices of j (and l = 3 4), together with the approximation d(p)j,l from Equation (26View Equation) and small-p power laws. The classical divergence at small p, where the behavior differs strongly from eigenvalues, is cut off. The right panel shows the dependence of the initial increase on l.
View Image Figure 10:
Internal time evolution for expectation values and spread of two bouncing states, one, which is unsqueezed (solid lines), and one, which is being squeezed (dashed).
View Image Figure 11:
Ratio of fluctuations before (Δ −) and after (Δ+) the bounce in the form |1 − Δ ∕Δ | − + as a function of a state parameter A in relation to the Hamiltonian H = ⟨ ˆH ⟩ for H = 1000 in Planck units. For larger H, the curve becomes even steeper, showing how precisely one has to tune the state parameter A in order to have a near symmetric state with |1 − Δ − ∕Δ+ | = 0. See [74] for further details.
Watch/download Movie Figure 12: (mpg-Movie; 866 KB)
Movie The coordinate time evolution [103] 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 μ. The farther away from a = 0 the effective bounce happens, depending on the ambiguity parameter, the smaller the part penetrating to negative μ is. The coordinate time evolution represents a physical state obtained after integrating over t [103].