The Universe Before the Big Bang: Quantum Cosmology (Part 2)

The Universe Before the Big Bang: Quantum Cosmology

The reflection of a particle from a barrier is part of our everyday experience and – unlike the tunneling effect – it is obviously also present in the context of classical mechanics. However, within quantum mechanics, there also exists the possibility of a new, truly “quantum”, reflection effect relative to particles which, despite having the energy required to climb over the barrier, are instead pushed backwards, in a completely unexpected way from a classical perspective. It is just as if a bullet, shot from a gun against an easily perforated target – like paper, for instance – were to bounce back instead of passing through the target, an impossible effect in the context of classical mechanics, but not within quantum mechanics.

FIGURE 8.2 Qualitative sketch of the quantum transition from pre-Big- Bang to post-Big-Bang, represented as a quantum mechanical reflection of the wave function in a mini-superspace whose coordinates correspond to the dilaton and to the spatial radius of the Universe. The incident wave (bottom left) describes the initial evolution of the Universe from the string perturbative vacuum towards the high curvature regime. Part of this incoming wave is not stopped by the barrier and is classically transmitted to the region of ever increasing dilaton, running towards the singularity (top right). Another part is reflected to the region of decreasing dilaton and standard post-Big-Bang evolution (top left)

In the pre-Big-Bang scenario, the barrier is represented by the region of very high energy and curvature that divides the post- Big-Bang phase from the singularity in mini-superspace. Without the presence of quantum effects, the Wheeler–DeWitt wave would easily overcome this barrier, ending up in the infinite-curvature pit that leads to the singularity (as predicted by the classical cosmological dynamics of the low-energy solutions). Instead, thanks to the possibility of quantum mechanical reflection, there is a finite probability that the wave reaching the barrier will be bounced back, thus describing a Universe that moves towards a quiet end after having happily reached the standard, post-Big-Bang evolutionary phase (see Fig. 8.2).

The transition from the pre-Big-Bang to the post-Big-Bang regime, which somehow represents the birth of the Universe in the form we are currently observing it, can then be described according to quantum string cosmology as a process of scattering and reflection of the Wheeler–DeWitt wave function in mini-superspace. It may be noted that, whereas in the standard cosmological scenario quantum effects are required for the Universe to enter the inflationary regime, in string cosmology such effects are required for the Universe to be able to leave the inflationary pre-Big-Bang regime, and enter the phase of standard evolution. Instead of a tunneling effect, there is a reflection process. The conceptual differences are evident, but the methods and the formalism are the same.

It must be pointed out, however, that if the transition occurs via either a tunneling or a quantum reflection process, then the final oscillation amplitude of the wave function turns out to be significantly reduced with respect to the initial amplitude (see Fig. 8.1). This means that the transition probability is quite small, i.e., the transition mechanism is not effective. Then, it would be hard for the Universe to exit from the pre-Big-Bang phase and to fulfill its standard evolution up to now.

However, the above mechanism is not the only way the transition can proceed (more technically stated, the tunneling/reflection process is not the only decay channel of the string perturbative vacuum). There are other, more effective processes (studied by the present author) in which the wave function, instead of being sup- pressed, is strongly enhanced through the high-curvature regime.

For such processes there is a quantum mechanism that acts inversely with respect to the one producing tunneling (it is in fact called the anti-tunneling effect). This latter effect is quite similar to the one described in Chap. 6, which amplifies the quantum fluctuations, with a subsequent production of particle pairs from the vacuum.

The crucial difference with respect to the process of Chap. 6 is that the oscillations now being amplified are those of the Wheeler– DeWitt wave function, representing the evolution of the Universe.

Hence, the process can once again be described (in the context of third quantization of the Wheeler–DeWitt wave function) as a process of pair creation. However, the resulting pairs are not particles, but rather pairs of universes, directly produced from the string perturbative vacuum, i.e., from the initial state of the pre-Big-Bang scenario (see Fig. 8.3). In each pair, one of the created universes is absorbed by the singularity (falling in the portion of mini-superspace where curvature and dilaton growth is unbounded, see Fig. 8.2), and disappears from our present experience. The other Universe evolves towards the opposite, low-curvature regime, thus entering into the post-Big-Bang phase, eventually to approach the current regime.

In the same way as the particles emerging from the vacuum are produced in pairs and characterized by opposite physical proper- ties (i.e., opposite charges, momentum, angular momentum, etc.), to avoid violations of conservation laws, the universes are also produced in pairs, and are characterized by opposite kinematic properties. One of the two universes expands while the other shrinks.

However, the shrinking Universe behaves as if it were traveling backwards in time with respect to the coordinate playing the role of time in mini-superspace.

FIGURE 8.3 Schematic view of the quantum transition from pre-Big-Bang to post-Big-Bang represented as an anti-tunneling effect of the wave function, i.e., as a creation of pairs of universes from the string perturbative vacuum. The Wheeler–DeWitt wave function is amplified during this process. The opposite charges of the resulting universes are to be interpreted as opposite kinematic states, corresponding to either expansion or contraction in mini-superspace

It is well known, on the other hand, within the context of second quantization, that a particle moving backwards in time ought to be interpreted physically as an antiparticle, with opposite charge, moving forward in time. Thus, in a third quantization context, the shrinking Universe must be reinterpreted as an anti-universe which is expanding, and the anti-tunnelling process must be seen as pair production of universes and anti-universes, both expanding, one towards the singularity and the other towards the current low- energy regime. Unlike the quantum reflection process, such a process can be quite efficient, as long as the dilaton interaction can provide the potential energy required for the occurrence of pair production in mini-superspace (in Fig. 8.3 this potential energy is represented by the barrier that amplifies the wave function).

To conclude, we can say that in the framework of quantum cosmology the transition from the pre-Big-Bang to the post-Big- Bang phase can be described in probabilistic terms, even without any detailed knowledge of the kinematics and dynamics of the high-curvature, strong-coupling regime. However, the use of the Wheeler–DeWitt equation obtained from the low-energy classical description is still an approximate approach. The existence of a minimum length within string theory does indeed imply that, close to the region of maximum curvature, the equations for the classical fields are modified by corrections including the square, the cube, and all higher powers of the curvature (see Chap. 4). Those corrections could also modify the Wheeler–DeWitt equation and the geometry of the mini-superspace.

According to string theory, a fully exact (to all orders) and consistent description of the Universe in the quantum regime should probably abandon concepts such as fields and geometry, and rely solely upon the motion of strings and the feature of conformal symmetries. Actually (as pointed out in Chap. 4), it is precisely from these symmetries that modifications to the equation of general relativity arise. Hence, it is precisely from them that the possibility of avoiding the initial singularity of standard cosmology may originate (as suggested by various studies and many authors). It is therefore possible that present quantum cosmology models will be improved by future developments of string theory, and eventually by its completion within the framework of membrane theory and M-theory, a recently born theoretical framework whose development looks promising (see Chap. 10).

The Universe Before the Big Bang: Quantum Cosmology (Part 1)