**The Universe Before the Big Bang: Cosmology and String Theory**

This basic lack of knowledge, however, is continuously being ﬁlled by the recent developments of theoretical physics, which have provided us with a very powerful tool: string theory. In principle, this theory (and its possible, though as yet not fully deﬁned completion, M-theory) allows a coherent merger of quantum mechanics and gravitation, and thefore provides a potentially consistent framework to describe the geometry of space-time in the regime of extremely high energy densities and curvatures. It has thus become possible to study the evolution of the Universe near the Big Bang, and even beyond it, by means of a robust and consistent theory, valid at all energies. It is as though, in the above analogy with ancient times, someone had built a more solid and reliable ship that would allow some brave explorers to sail the seas beyond the Columns of Hercules. In this way, it has been found that the extension of space-time is not necessarily con- strained by an initial singularity, and questions about the possible state of the Universe before the Big Bang are fully legitimate and well posed.

Anticipating the demand of the curious reader, and as an intro- duction to the content of the following chapters, let us immediately give some idea of what the Universe would look like according to the indications provided by string theory, if we could look back in time to the epoch of the Big Bang, and even beyond the Big Bang itself. Such remote epochs cannot be traced using objects like stars and galaxies, which formed only very recently on the time-scale of cosmic evolution. These structures were not yet formed at the onset of the Big Bang, and neither did they exist before it. Instead, we need to exploit some geometrical properties of the Universe that are always valid, like space-time curvature. Let us therefore ask about the past evolution of space-time curvature, and represent its behavior graphically as a function of time.

According to the so-called standard cosmological model (which will be introduced in next post, and which is the model providing the grounds for the hypothesis of the Big Bang as the singular beginning of “everything”) the Universe expands and the curvature decreases in time in a continuous and decelerating fashion. Hence, going backward in time, we reach epochs characterized by progressively increasing curvature. This monotonic growth proceeds continually until the inﬁnite curvature state is reached (corresponding to a singularity, and conventionally ﬁxed at the initial time t = 0). Beyond that point, no classical description is possible (see Fig. 1.1).

However, as already pointed out, a singularity can often be interpreted in a scientiﬁc context as a signal that we are applying some physical laws outside their realm of validity. Concerning this point, it is interesting to quote the opinion of Alan Guth, one of the fathers of modern inﬂationary cosmology. In his recent book he makes the following remarks about the initial singularity:

It is often said – in both popular-level books and in textbooks – that this singularity marks the beginning of time itself. Perhaps it’s so, but any honest cosmologist would admit that our knowledge here is very shaky. The extrapolation to arbitrarily high temperatures takes us far beyond the physics that we understand, so there is no good reason to trust it. The true history of the universe, going back to “t = 0”, remains a mystery that we are probably still far from unraveling.

In other words, according to Guth, there is little hope of describing the initial phase of the Universe within the standard cosmological model. Indeed, as we have already pointed out, in the presence of arbitrarily high curvature, energy and density, the Einstein theory of gravitation ceases to be valid, and the associated description of the space-time geometry becomes meaningless.

Beside the singularity problem, however, there are also other issues concerning the standard cosmological model that hint at the need for a modiﬁcation near the initial time, even before reaching the quantum gravity regime. Such a modiﬁcation requires in particular that, at some point during its primordial evolution, the Universe should undergo a phase of highly rapid expansion, dubbed inﬂation. We are giving here just a glimpse of what will be illustrated in more detail in Chap. 5. For the purposes of our fast-track, time-reversed journey, it will be enough to point out that during an inﬂationary phase of conventional type the evolution of the Universe is expected to be determined by the energy density of a “strange” particle – dubbed the inﬂation – that generates a scalar- type ﬁeld strength.

Going further backward in time, the potential energy of this ﬁeld progressively increases, and eventually becomes so strong as to be able to “freeze out” the space-time curvature. Then, as shown in Fig. 1.2, the curvature of the Universe stops increasing and levels off to an almost constant value. During this initial inﬂationary phase, the geometry of the Universe thus approaches that of the de Sitter space-time (named after the cosmologist who found the solution describing a spacetime with constant curvature). The primordial Universe, in that case, closely resembles a tiny, four- dimensional hypersphere with constant radius.

However, there is also a problem in this case: a phase in which the Universe expands while the curvature stays ﬁxed at a constant value cannot be extended backward in time without limit. In fact, for a physical (stable) particle, moving according to the laws of general relativity within this type of geometry, it would take a long, but certainly ﬁnite amount of time to reach us starting from the moment when the radius of the Universe was zero. In order to obtain a “complete” model of space-time, the initial Universe should exist in a contracting phase, at least according to the de Sitter solution to the equations of general relativity. However, as has been shown by some cosmologists (in particular, Arvind Borde, and Alexander Vilenkin), within the framework of an inﬂationary model based upon the potential energy of some scalar ﬁeld, a transition between a contracting and an expanding phase is not allowed, at least according to general relativity and the physical laws that we currently believe to be valid.

Hence, the inﬂationary scenario at constant curvature is also unable to provide a complete model for the evolution of our cosmos. As Guth himself points out in his book:

Nonetheless, since inﬂation appears to be eternal only into the future, but not the past, an important question remains open: How did it all start? Although eternal inﬂation pushes this question into the past, and well beyond the range of observational tests, the question does not disappear.

Apart from the possibility of experimental tests (which will be discussed later in the book) it seems undeniable, as outlined by Guth, that a constant curvature, expanding phase cannot be arbitrarily extended backward in time, and does not lead to a complete description of the origin of our Universe. A possible solution to this problem relies upon the possibility that a Universe with the appropriate, expanding de Sitter geometry might spontaneously emerge from the vacuum at some very early epoch (but not inﬁnitely distant in time), through a typical quantum mechanical effect.

Leaving aside this possibility for the moment and limiting our options to a classical context, it is clear that if the curvature cannot remain indeﬁnitely constant, then we are only left with two alternatives in order to extend our cosmological description back in time. The ﬁrst possibility is that at some point the curvature starts to increase again. In this case, however, the singularity would persist, with the only difference that the position of the Big Bang would be moved backwards in time with respect to the standard cosmological model. The second possibility is that, going backwards in time, the curvature starts to decrease, becoming smaller and smaller as we go back in time. This second case is exactly the model of the Universe suggested by string theory, represented in Fig. 1.3, which will be discussed in detail in the following chapters.

In fact, as will be shown in Chap. 3, string theory suggests that the plot of the cosmological curvature scale versus time could have a specular reﬂection symmetry with respect to the time co- ordinate t = 0. It also suggests models in which there is no singularity in the space-time curvature, and time can be arbitrarily extended to inﬁnity, in both the backward and the forward directions. Within these models – dubbed string cosmology models, as opposed to standard cosmology or conventional inﬂationary models – the curvature starts from arbitrarily small values, increases up to a maximum value (dictated by string theory), and eventually decreases until it joins the behavior typical of standard cosmology and the current epoch.

The typical trend for the cosmological space-time curvature, in this context, is thus described by a bell-shaped curve, eventually joining the curve that represents the standard-model curvature.

Indeed, at recent times standard cosmology works well, so string cosmology should not predict signiﬁcant differences. However, as shown in Fig. 1.3, the string cosmology curve matches the standard one at higher curvatures (and hence earlier) than the curve corresponding to the inﬂationary de Sitter model. This occurs because the highest value of curvature reached in a string cosmology con- text (viz., the top of the bell curve drawn in Fig. 1.3) is in general higher than the value reached during de Sitter inﬂation. This feature may have some key phenomenological consequences, as will be discussed in the following chapters.

The moment at which the curvature reaches its (high, though ﬁnite) maximum value at the top of the bell replaces the singularity and corresponds to the position of the Big Bang in the standard cosmological scenario. It is then natural to refer to the cosmological phase characterized by increasing curvature (the left-hand side of the bell in the graph) as the pre-Big-Bang phase, describing the initial evolution of the Universe starting from an initial “vacuum” state. In the same way, the right-hand side of the bell corresponds to the post-Big-Bang phase, characterized by decreasing curvature and representing the typical evolution of the current Universe, in agreement with standard cosmological predictions.

According to this class of string cosmology models, the Uni- verse at the epoch of the Big Bang was not a newly born baby, but a rather aged creature, midway through an evolution of probably in- ﬁnite duration. Furthermore, the Big Bang itself is not viewed as a singular point, but as a transition – certainly violent and explosive, but of ﬁnite duration and intensity – between two phases characterized by different physical and geometrical properties. What happens, therefore, is that a traditional representation of our cosmos is somehow overturned, whence a comparison with the well-known Copernican revolution may seem natural. With Copernicus, the Earth lost its role as the center of the Universe, or focal point of the physical space. Similarly, within string cosmology, the Big Bang may lose its role as the beginning of the Universe, or focal point of physical time. A sort of Copernican revolution – in time, rather than in space – even though the Big Bang, in contrast to the Earth for Copernicus, does not completely lose its privileged role in the cosmic scene.

At this point, we are beginning to outline a potentially interesting cosmological scenario. However, before we proceed further, we cannot avoid asking the following question, which is of fundamental importance if we hope to keep working in a scientiﬁc con- text: What is the observational evidence that could either prove or disprove this scenario for the primordial evolution of the Universe?

The answer to this question is quite similar to the one an archaeologist would give to anyone asking him about the evidence supporting the existence of ancient civilizations. He would argue that, by studying old remains and available relics, one can attempt to trace back to original sources and reconstruct the past. In the same way, a cosmologist may be viewed as an archaeologist who studies the relics of the various cosmic epochs, in order to piece together the evolution of the Universe.

For a further clariﬁcation of this point let us go back to an example concerning unstable particles, and consider again the process of neutron decay introduced previously. In a sense, the particles left by the decay represent the relics of the decay process: analyzing such decay products, a physicist is able to trace back to and reconstruct the properties of the initial state. In particular, by studying the proton, the electron, and the neutrino that have emerged from the decay, and using his knowledge of the theory of weak interactions, he may deduce the prior existence of a neutron and compute, for instance, the neutron mass, even without having directly observed the initial particle.

In a similar fashion, the evolution of the Universe and the transitions between the various cosmological epochs are generally characterized by intense emission of a very large amount of radiation, of all kinds and in all allowed frequency bands. Part of such radiation has subsequently been transformed, by exploiting the relativistic equivalence between mass and other forms of energy. However, there is also a fraction of this radiation which reaches us today, retaining its original features. So by studying its properties, it is then possible to gain direct information about the past evolution of our Universe.

It is worth mentioning that, also in the context of the standard cosmological scenario, the hypothesis of a primordial explosion has been experimentally conﬁrmed by the observation of relic electro- magnetic radiation, the so-called cosmic black-body background, discovered by Penzias and Wilson. In the same way, as will be discussed in Chap. 6 and thereafter, other types of relic cosmic radiation – gravitational, dilatonic, or axionic backgrounds – could experimentally corroborate or rule out various scenarios describing the Universe before the Big Bang.

Finally, to complete this short introduction, let us just mention a few other topics that will be covered in subsequent chapters. A ﬁrst and quite essential issue pertains to the motivations for introducing the cosmological phase already dubbed the pre-Big-Bang phase. Our aim, in particular, will be to explain how such a phase appears to emerge naturally within string theory, and not in the context of Einstein’s gravitational theory.

Another point concerns the kinematical aspects of the primordial cosmological evolution. Despite their differences with respect to conventional inﬂationary models, string cosmology models are also initially characterized by a very fast, accelerated expansion. It follows that they are inﬂationary models too, in every respect, and may thus provide solutions for the shortcomings of the standard cosmological scenario.

But the point which is likely to be the key issue concerns the phenomenological consequences of these models, and the problems concerning their observation and their possible use as tests of string theory itself. We will therefore attempt to present a detailed description of the physical effects marking the differences between string models and more conventional cosmological models, paying particular attention to the possibility of either direct or indirect experimental observations of such differences.

To this end, it is interesting to observe that the phenomeno-logical consequences of a cosmic phase preceding the Big Bang can be divided into three classes: type I, II or III. Type I includes observations that will be carried out in the near future (20–30 years from now); type II consists of observations relative to the immediate future (some years from now); and type III refers to observations (whether performed or not) that are already accessible to current technology. It was the very existence of experimentally testable consequences, emerging with growing evidence from the development of string cosmology models, that has encouraged and motivated many researchers to pursue the study of string cosmology and its various possibilities.

In order to provide the reader with a better introduction to the problems of cosmic evolution, and to the primordial epoch close to the birth of the Universe, it seems adequate to start with a concise overview of the standard cosmological model and its underlying theoretical background (the theory of general relativity). The next chapter will be devoted to this purpose.

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