**Cosmology and String Theory: Quantum Cosmology**

The possible cosmological scenarios outlined in this book have been approached so far from a purely classical perspective, using concepts and ideas typical of the macroscopic world, like space, time, geometry, gravitational forces, and so on. The aim of this chapter is to introduce a possible alternative description of the cosmological evolution based on a quantum point of view, and using a framework where the Universe can be represented as a wave propagating in an abstract, multidimensional space dubbed superspace (no connection with the previously mentioned supersymmetry).

A detailed and rigorous explanation of this approach would clearly require some knowledge of quantum mechanics, which is not necessarily part of the scientiﬁc background of the typical reader, and whose introduction is beyond the scope of this book.

Hence, our discussion will be grossly qualitative and approximate. Nevertheless, we hope to provide the reader with an appropriate overview of the methods and goals pertaining to the ﬁeld of quantum cosmology.

Amongst the motivations suggesting the use of a quantum cosmology approach within string models of the Universe, we should mention ﬁrst of all the difﬁculties we currently face when we try to give a quantitative and fully consistent description of the transition between the pre-Big-Bang and post-Big-Bang phases. Whereas the obstacles appear to be of a formal nature, they nevertheless have a physical origin. Indeed, they are rooted in the fact that the instability of the initial state (the string perturbative vacuum) yields to a phase (the pre-Big-Bang phase) in which the curvature and the strength of the gravitational force (and of all other forces) increase in an accelerated manner.

Hence, in order to make a transition to the standard cosmological phase where the Universe decelerates and becomes radiation- dominated, and where all natural forces are stabilized, we need a mechanism that curbs the initial increase of both the curvature and the dilaton. Otherwise, the Universe would necessarily reach a singular state with inﬁnite curvature (like the one occurring in the standard scenario). Such a singularity would then completely detach the pre-Big-Bang phase from the current one. Without any physical connection, it would no longer make sense to relate the properties of the current Universe with those characterizing an epoch preceding the Big Bang. Furthermore, it would not be legitimate to hunt for possible experimental traces of such a primordial epoch in the cosmological backgrounds of relic radiation, as dis- cussed in previous chapters.

The dynamics of a possible mechanism able to stop the in- crease of the curvature and to provide a transition from the pre- to the post-Big-Bang phase is quite complicated, as shown by all studies and computations so far performed. Actually, in addition to the above-mentioned effects, such a mechanism should be able to turn the kinetic energy associated with the geometry and the dilaton into thermal radiation. Furthermore, if the pre-Big-Bang phase is higher-dimensional, such a mechanism should be able to “freeze” the extra spatial dimensions, and possibly break the symmetries between the various forces.

According to string theory, those effects can hardly take place when the curvature is small and the couplings are weak. The transition seems to require a phase where the gravitational forces are so strong that the resulting particles are themselves able to modify the geometry, yielding what are known as back-reaction effects, introducing quantum corrections into the classical equations (the quantum loop corrections introduced in Chap. 4). Moreover, when the curvature is quite high, other corrections (the so-called α� corrections, see again Chap. 4) are induced by the fact that it is no longer legitimate to approximate string behavior by point-like objects. In addition, the dilaton may start to develop a strong self-interaction, generating a large potential energy density.

Taking into account all these effects, the full equations of string cosmology become so complicated that – up to now – it has been impossible not only to ﬁnd their exact solutions, but even to write them down in a closed form (apart from some special cases). However, all results obtained so far (in some particularly simple cases that we can deal with) are encouraging, since they seem to suggest that the quantum corrections just provide damping corrections to the classical, accelerated evolution, and tend to favor the transition to the phase described by the standard cosmological model.

The relevance of the quantum corrections suggests that the obstacles we encounter when we attempt to describe the transition could be surmounted by abandoning the classical, geometric approach, where we follow the space-time evolution point-by-point, moment-by-moment. Since the equations describing this evolution are not fully known, in general, it could be convenient to adopt the probabilistic approach of quantum cosmology which does not re- quire full knowledge of all the intermediate evolutionary stages, but only of the initial and ﬁnal states.

It is worth noticing here that, even in the cosmological frame- work based upon Einstein’s equations, there are open issues which it seems appropriate to address with quantum mechanical methods.

We may recall for instance that, within the standard inﬂationary scenario, the primordial Universe approaches a state of exponential expansion and constant curvature. Such a state, described by the de Sitter geometry, cannot have lasted indeﬁnitely in the past (see Chap. 1). Hence, we cannot avoid facing the problem of how this state might have emerged.

A possible solution to this problem was suggested independently during the 1980s by some Soviet cosmologists (Alexander Vilenkin, Andrei Linde, Valery Rubakov, Yakov Zeldovich, and Alexei Starobinski). Their solution relies upon the idea that the initial de Sitter state may emerge “from nothing”, i.e., it may be spontaneously produced from the vacuum thanks to an effect called quantum tunneling. The tunneling effect is a well-known process in elementary particle physics, where a particle, represented by a quantum mechanical wave, is able to overcome a potential barrier even if its energy is inadequate at the classical level. (Very naively, it is as if a cyclist, who does not have enough energy to climb a small hill, unexpectedly ﬁnds a tunnel at the bottom of the hill that allows him to get through.) In a cosmological setup, the description of the birth of the Universe in terms of the tunneling effect requires the introduction of a peculiar inﬁnite-dimensional space, the so-called superspace, whose points represent all possible geometric conﬁgurations of the Universe. For practical reasons it is possible to use a reduced space, dubbed mini-superspace, and characterized by a ﬁnite number of dimensions (associated, for instance, with the radii of a spatial section of the Universe measured along the different spatial axes). The motion of a wave from one point to another of this mini-superspace represents the transition of the Universe from one geometrical state to another, and is governed by the so-called Wheeler–DeWitt equation, named after two theoretical physicists (John Archibald Wheeler and Bryce DeWitt) who ﬁrst proposed it in the 1960s.

The Wheeler–DeWitt equation is the exact analogue of the Schroedinger equation of ordinary quantum mechanics, the only difference being that its solutions, instead of describing the possible values of the position and momentum of a given physical system (for instance a particle), represent the possible geometrical states of the Universe. A Universe described by the Wheeler–DeWitt equation thus becomes a fully quantum mechanical Universe, subject to all possible quantum effects. We know, for instance, that the so-called second quantization of the Schroedinger wave function leads to the formalism of quantum ﬁeld theory, where it is possible to describe the creation and annihilation of particles. Similarly, quantization of the Wheeler–DeWitt wave function gives rise to the so-called third-quantization formalism, where it is possible to describe the creation and annihilation of universes.

By an appropriate choice of initial conditions, it is possible in particular to ﬁnd solutions to the Wheeler–DeWitt equations describing the birth of our Universe as a tunneling process, thus providing a solution to the classical problem of the origin of the inﬂationary de Sitter space. One ﬁnds that if the state of the Universe after the tunneling process is described by the de Sitter geometry, and is thus characterized by a constant Λ representing the vacuum energy density, then the bigger the value of Λ, the higher the tunnelling or transition probability. In this way, the Universe is created just in the appropriate inﬂationary state, which does in- deed require a high enough value for the parameter Λ (also called the cosmological constant).

Barring a number of formal and technical problems, the most unsatisfactory feature of this scenario is probably the fact that the initial conditions for the tunneling process are to be chosen ad hoc, since they are not unique. There are also arguments supporting the choice of different initial conditions (as discussed, again during the 1980s, by other theoretical physicists including James Hartle and Stephen Hawking), which lead to different scenarios. The reason for this arbitrariness (dubbed the boundary condition problem) is rooted in the fact that within standard cosmology the ﬁnal state – i.e., the cosmological conﬁguration we aim to obtain – is well known, whereas the initial state is completely unknown. Indeed, the very name of the tunneling process, “tunneling from nothing”, already automatically stresses the lack of knowledge about the initial state. The standard classical theory is not helpful at all, since it just predicts the Big Bang singularity as initial state, i.e., the state that the quantum mechanical approach would like to avoid.

Within the self-dual pre-Big-Bang scenario, the situation is radically different. The initial state, assumed to be the perturbative vacuum of string theory, is completely known, fully justiﬁed, and fully appropriate to be described – in the low energy regime – by the Wheeler–DeWitt wave function. Given the initial state, the computation of the transition probability towards the ﬁnal state, i.e., the current Universe, is no longer arbitrary.

It is therefore interesting to note that, by computing the probability that a transition occurs between the perturbative string vacuum and a post-Big-Bang Universe equipped with a cosmological constant, the outcome is quite similar to the result obtained in standard cosmology assuming the validity of the “tunnelling from nothing” scenario (as shown by Gabriele Veneziano, Jnan Maharana, and the present author). This could suggest that the ad hoc prescription for the boundary conditions, needed to obtain the tunnelling effect, somehow simulates the presence of the perturbative vacuum as initial state. It would then be more appropriate to talk about “tunneling from the string perturbative vacuum”, rather than “tunneling from nothing”. Figure 8.1 provides a qualitative representation of this result.

There is, however, a conceptual difference between string cosmology and standard cosmology. The quantum mechanical transition from pre-Big-Bang to post-Big-Bang described by the Wheeler–DeWitt equation, in a two-dimensional mini-superspace where the coordinates are represented by the spatial radius of the Universe and the dilaton, does not correspond to a tunnelling effect, but rather to a quantum reﬂection effect.

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