Cosmology and String Theory
Recent Developments: Brane Cosmology Scenarios
About ten years ago, at the end of the last century, we witnessed the appearance of two new players on the scene of physical and astronomical science. We are just beginning to appreciate their dramatic impact upon basic cosmology, and we are probably far from a full understanding of all their implications. One of these two players brings observational novelties concerning the present accelerated evolution of our four-dimensional space-time on a macroscopic scale. The other brings novelties (probably more speculative) concerning a series of improvements in our theoretical understanding of the dimensionality of our world, culminating in the discovery of a possible mechanism for the confinement of gravity within four space-time dimensions, and opening new perspectives for the geo- metric description of higher-dimensional universes.
Both novelties are rich in cosmological applications and con- sequences. The first, discussed in the last chapter, has drastically changed our expectations about the future evolution of the cosmos.
The second, which will be the subject of this chapter, has paved the way to new possible scenarios for the primordial Universe, in particular, for the description of the phase preceding the Big Bang and for models of the Big Bang itself.
As already pointed out in previous chapters, there are indeed many deep reasons – arising in the context of modern unified theories of fundamental interactions – for believing that the world in which we live is characterized by a number of spatial dimensions greater than three (at least nine, according to superstring theory).
If we accept this view, however, we are left with the problem of explaining why only three spatial dimensions seem to be accessible to our ordinary experience, and to physical exploration through the instruments provided by current technology.
The obvious answer to this question, until a few years ago, was that all physical objects of our world (ourselves included) are characterized by a higher-dimensional extension along all possible available spatial dimensions (more than three, in general).
However, only three of these dimensions have expanded over macroscopic (and larger) scales, giving rise to the currently observed Universe. The remaining dimensions (also called extra or internal dimensions) have instead evolved towards an extremely small and compact size, so as to become effectively invisible to all experiments so far performed.
We may think, for instance, of a long thin electric wire suspended between two high-tension steel pylons. The wire is actually a three-dimensional object but, if we look at it from far enough away, it will look like a one-dimensional object, simply because the size of its transverse sections are much smaller than its length.
The above approach to the dimensionality problem is quite reasonable, and still valid. Recently, however, a different approach has been put forward, which seems to offer an alternative explanation of why we are unable to perceive the extra dimensions. For a quick anticipation of what will be discussed in detail later, we only remark here that, according to this alternative explanation, all components of our physical world (ourselves included) have a pure three-dimensional spatial extension, in spite of being embedded in a higher-dimensional spatial manifold. The extra dimensions are not constrained to have a very small thickness as in the previous case.
In fact, they may have a large, possibly infinite, extension. They are invisible simply because all forces and interactions through which we can explore our physical world are strictly “confined” to three spatial dimensions, being unable to propagate in the additional “orthogonal” directions.
As a naive illustration of such a situation we may think of a small bug like an ant, climbing up a curtain hanging on the wall of a room. The space inside the room is certainly three-dimensional, and all dimensions are extended over distances of comparable size (very large with respect to the size of the ant observer). However the ant – being unable to fly out from the curtain – is confined to move only vertically and horizontally along the curtain: hence, it will effectively experience only two spatial dimensions, in spite of being embedded in a fully three-dimensional space.
This particular view of a higher-dimensional Universe might seem, after all, a rather obvious and trivial explanation of the observed dimensionality of our effective space-time manifold. But from a conceptual point of view it represents a truly innovative result, the fruit of the most recent progress in theoretical physics, and full of important consequences. Hence, it seems appropriate to present here a detailed explanation of how (and why) we are led to such a higher-dimensional scenario.
To this end, let us take a step backwards, returning to the starting point of this chapter, i.e., to the fact that we expect to live in a Universe with more than four space-time dimensions. What are the physical motivations leading us to such (in principle unnatural) expectations?
The motivations are at present only of a theoretical nature. In fact, it is fair to say that we do not yet know of any direct experimental result forcing us to consider the existence of extra (either space-like or time-like) dimensions. The theoretical motivations, on the other hand, are certainly not a recent issue. They have a long history, starting almost a hundred years ago with the work of two theoretical physicists, Theodore Kaluza and Oskar Klein.1 At that time, stimulated by the success of general relativity (which pro- vided an elegant geometric description of all gravitational forces), many theoretical physicists were trying to incorporate not only gravity but also the other known fundamental interaction, i.e., the electromagnetic interaction, into the space-time geometry.
The revolutionary proposal of Kaluza and Klein was to generalize the four-dimensional space-time of Einstein’s theory by adding a fifth dimension, of space-like character, and to interpret the additional degrees of freedom of that extended geometry as quantities directly related to the electromagnetic interactions. In this context, for instance, the extra components of the curvature were related to the electromagnetic field strengths, the moment along the fifth dimension was related to the electric charge, and so on. According to their model, now universally known as the Kaluza–Klein model, the new spatial dimension was wrapped onto itself (i.e., following the usual language, it was compactified) to form a microscopic circle, whose radius turned out to be fixed by Newton’s gravitational constant and the fundamental unit of electric charge. The presence of the electromagnetic interaction was then “explained”, in that context, as a consequence of the symmetry associated with the coordinate transformations in the fifth dimension.
The five-dimensional model of Kaluza and Klein represented a consistent and successful unification of electromagnetism and gravity, free from problems affecting many other attempts at geometric unification (some of them proposed by Einstein himself).
However, it was soon abandoned with the appearance in physics of new (strong and weak) interactions, active at the nuclear level, which seemed to be inconsistent with such a unification scheme.
The model was reconsidered, many years later, after the development of the so-called gauge theories, which associate with each interaction a well-defined symmetry group. By increasing the number of extra dimensions, and assuming an appropriate geometric structure for such dimensions (in general more complicated than a simple product of Kaluza–Klein circles), it is possible to obtain manifolds admitting a group of non-Abelian symmetries and reproducing the symmetry group typical of strong, weak, and electromagnetic interactions. In this way all fundamental forces of nature can be unified into a single (higher-dimensional) geometric description. The tiny volume of the extra, compactified dimensions, as in the original model of Kaluza and Klein, may prevent a direct experimental detection of those parts of space extending along the extra “internal” dimensions.
It should be noted, at this point, that such an effective unification scheme might be regarded as somewhat contrived, since the extra spatial dimensions have been introduced ad hoc (they are not a compelling prediction of the underlying theory). In addition, gravity is included in this unified scheme only as a classical interaction (differently from the other interactions, which can be consistently quantized). It is known, on the other hand, that a consistent quantization of gravity can be successfully implemented in a string theory context. It is thus quite remarkable that, within string theory, the higher-dimensional unification scheme of gauge interactions also finds its theoretical consecration, together with a potential phenomenological efficacy.
In fact, for a consistent description of the string motion at the quantum level, the string has to be embedded in an external space-time (the so-called target manifold), which has dimension D (known as the critical dimension) greater than four. The quantum states associated with the discrete oscillation levels of the quantized string are thus represented by higher-dimensional fields, describing the forces present in such a higher-dimensional space-time. To obtain our physical four-dimensional world, on the other hand, the extra D−4 dimensions present in the model must be compactified. If the geometry of the extra compactified dimensions admits the appropriate symmetry group – more precisely, if its metric is invariant under the appropriate group of coordinate transformations, called isometries – then our reduced four-dimensional world automatically acquires the various gauge symmetries reproducing the forces that we currently observe.
We are thus led to the following important question: How many spatial dimensions are required by quantum string theory? The answer depends on the type of string we are considering.
The first string models, proposed during the 1970s, were based on the analogy with a classical vibrating object, and were formulated in terms of “bosonic” variables representing the coordinates of the string in the higher-dimensional external space-time. This type of string, called a bosonic string, can be consistently quantized in an external Minkowski space-time with a critical dimension of D= 26 (one time-like and 25 space-like dimensions).
This number of dimensions guarantees, for both closed and open bosonic strings, that the quantized theory is free from the so-called ghost problem, that is, the appearance of states of negative norm (i.e., of imaginary length in the space of states), also associated with negative probabilities (which are a mathematical nonsense). However, the bosonic string theory, quantized in D = 26 dimensions, cannot avoid the presence in its spectrum of states with negative squared masses (i.e., imaginary masses), called tachyons. These states describe particles that should always move faster than light, thus apparently violating the basic causality principles at the foundation of the modern quantum theory of fields and particles. It is probably appropriate to recall that no tachyon particle has ever been observed.
Hence, tachyon states must be eliminated from the quantum string spectrum. The most popular (and presently also more effective) way of doing this is to generalize the bosonic string model by assuming that strings vibrate not in ordinary space but in the so- called superspace, a virtual manifold spanned by a set of coordinates containing an equal number of bosonic and fermionic degrees of freedom. In this way the string model automatically becomes supersymmetric (see Chap. 3 for a definition of supersymmetry).
More precisely, each bosonic coordinate determining the string position in the external D-dimensional space acquires a fermionic partner, transforming as a real spinor (i.e., as a massless, spin-half fermion) under coordinate transformations in the two-dimensional world-sheet surface spanned by the string (see Fig. 4.1), and trans- forming as a vector under Lorentz transformations in the external D-dimensional space.
In this way, one arrives at the so-called superstring theory, which eliminates not only ghost states but also tachyons from the physical spectrum. In fact, in a supersymmetric theory, the lowest allowed level in the squared mass spectrum has to be zero (negative eigenvalues are forbidden). In addition, a superstring model automatically introduces into the theory the fermionic variables required to represent the fundamental matter fields (quarks and leptons, whose various combinations can reproduce all forms of matter currently observed).
The consistency of superstrings with the basic principles of quantum mechanics and relativity (i.e., the absence of ghost and tachyon states in the spectrum) requires an external space-time manifold with D = 10 dimensions (one time-like and nine space- like dimensions). Three of these spatial dimensions correspond to our ordinary macroscopic space. The other six dimensions, wrapped onto themselves and confined to such a compact volume that they have so far escaped direct detection, are in principle enough to contain all the symmetries required to reproduce the interactions observed in our low-energy four-dimensional world.
We can say, therefore, that superstring models seem to provide not only a consistent framework for the quantization of gravity, but also a coherent and compelling scheme for a unified description of all fundamental forces of nature. Should this be confirmed, it would mean that two among the boldest and longest-sought goals of theoretical physics have been achieved in one shot.
There is a problem, however, due to the fact that the geometry of the compact extra dimensions is highly non-trivial. It may repro- duce not only the particles and the interactions typical of our low- energy Universe, but also many other types of interactions which apparently have nothing to do with our world. This is the so-called landscape problem, arising from the many possible effective theories existing as a low-energy limit of the exact, high-energy string model, and due to our present ignorance of the mechanism driving our world to adopt a particular low-energy realization of string theory. This tends to weaken the unifying power of the theory, as the theory gives us too many models of low-energy interactions, and we have to select ad hoc the most appropriate one for a realistic description of nature.
In addition, we should recall here that superstring theory is not unique. There are indeed five possible superstring models, physically different from each other but nevertheless satisfying all required properties of quantum consistency. So which is the “right” model to describe our Universe?
This difficulty, unlike the previous one concerning the land- scape, seems to have been solved. To understand how, let us first briefly introduce the five different types of superstring, starting with the so-called type II model, describing closed strings. In a closed superstring, the oscillations of its bosonic and fermionic components can propagate along the string either in the clockwise or in the counter-clockwise direction. Hence we have two possibilities, automatically defining two associated subtypes of the model, type IIA and type IIB. For their explicit definition we must intro- duce the notion of chirality, which we may think of as a vector pointing in a preferred direction in an appropriate virtual space.
In the type IIA superstring model, the oscillations of the fermionic fields propagating around the string in opposite directions are characterized by opposite values of chirality, i.e., they point in opposite preferred directions in the chirality space. In a type IIB superstring model, on the other hand, the fermionic oscillations propagating in opposite directions are characterized by the same chirality value, so that they cannot be distinguished by looking at the properties of the chirality space. These properties are not of purely formal character, as they have an important physical counterpart in the different particle and field contents of the two models.
Another superstring model is the so-called type I model, de- scribing closed and open superstrings. The two types of string in this model are non-oriented, i.e., they are invariant under exchange of the ends of the strings. In other words, there is no preferred direction along the spatial path joining two ends of the string.
Open strings, on the other hand, can carry charges (of all types) on their ends. The type I model (in contrast to the type II model) can thus contain in its spectrum the fields describing strong, weak, and electromagnetic interactions generated by the charges located on the ends of open strings. The fields associated with gravitational interactions (the graviton, the dilaton, etc.) are instead contained in the closed string spectrum.
Finally, there is the so-called heterotic superstring model, de- scribing closed oriented strings in which only half the physical degrees of freedom are supersymmetrized (for instance, those associated with modes moving clockwise), while the other half keeps its bosonic properties, and it is quantized without fermionic partners. The procedure is consistent because, for closed strings, modes moving clockwise and counter-clockwise are decoupled, and can be treated independently.
The bosonic part of the heterotic string, on the other hand, can be consistently quantized in an external space-time with 26 dimensions, while the quantization of the supersymmetric part of the string requires only 10 space-time dimensions, as remarked previously. The additional 26 − 10 = 16 spatial dimensions present in this model can then be compactified, to obtain an effective ten- dimensional theory. However, it turns out that the compactified dimensions are in principle compatible with two different symmetry groups, associated with different sets of higher-dimensional gauge fields and interactions. Hence, we have two possible models of heterotic superstrings, referred to as type HO and type HE.
These two types, together with the other three types (I, IIA, IIB) give a total of five different models.
The five superstring models, apparently so different, are closely linked to each other through the action of the duality symmetries introduced in Chap. 3. In fact, using the appropriate combination of duality transformations, it seems possible to switch from one superstring type to another. In particular, changing the sign of the dilaton φ, i.e., inverting the string coupling parameter exp φ, and then switching from the strong coupling to the weak coupling regime, we may pass from a superstring model in which the perturbative approximation is no longer valid to the dual model in which this approximation is in fact valid.
The existence of such a duality network has suggested the conjecture that the five superstring models may simply represent five different approximate versions (valid in different regimes) of a unique, more fundamental theory, called M-theory. The complete formulation of such a theory seems to require one additional space-like dimension than superstring theory, and hence a space- time with D = 11 dimensions. What is presently known about M-theory is that, at low enough energy (and small enough curvature), it can be approximated as a supergravity theory – i.e., as a supersymmetric theory of gravity – in its maximally extended version, discovered by Eugene Cremmer, Bernard Julia, and Joel Scherk in the 1980s. (This does indeed require eleven space-time dimensions for its formulation.) At high energies, on the other hand, we have at present no precise information about M-theory so that, following a popular joke, we can say that the letter M of the name stands for mystery (or mother of all theories, or monster theory, along with other suggestions). But the more appropriate interpretation of the name is probably membrane theory, since M-theory describes, besides strings, the dynamics of extended objects like membranes.
To understand why membranes may naturally appear in the M-theory context we should recall here the important theoretical results obtained a decade ago by Petr Horawa and Edward Witten at the University of Princeton. They showed that the growth of the coupling parameter of a superstring (i.e., the growth of the strength of all interactions) can be equivalently described by adding a new spatial dimension to the space-time, and then gradually increasing the size of this dimension, following the growth of the coupling. In this way, we pass from ten to eleven dimensions, so that a string acquires a transverse extension. It becomes a two-dimensional object, i.e., a membrane, with a transverse size directly controlled by the strength of the coupling (see Fig. 10.1).
This means that the fundamental (one-dimensional) building- blocks of string theory can be interpreted, in the M-theory context, as two-dimensional membranes (or two-branes) embedded in an eleven-dimensional external space-time. In the limit in which the coupling strength of the theory goes to zero, the size of the eleventh dimension becomes smaller and smaller, until the two-branes eventually degenerate to strings (see Fig. 10.1). It should be stressed that the eleventh dimension does not represent an additional direction along which a string can oscillate, so that there is no inconsistency between M-theory and the fact that a superstring requires precisely ten space-time dimensions.
The illustration of Fig. 10.1 refers to the case in which the coupling is growing, but the energy of the system remains low (and the space-time curvature remains small) with respect to the string scale. At higher energies, however, we may expect the theory to contain not only oscillating strings and membranes, but also higher-dimensional objects: three-dimensional extended bodies (called three-branes), four-dimensional extended bodies (called four-branes), and so on, up to nine-branes. In general, we will call an elementary object extended along p spatial dimensions a p-brane (a string is a one-brane, a particle is a zero-brane, and so on).
The important physical property of these higher-dimensional objects is that, in the weak coupling regime, they become very heavy, as their mass is proportional to the reciprocal of the coupling strength (and their mass grows with the number of spatial dimensions). However, it is obvious that, the heavier an object is, the more difficult it will be to produce it in a physical process, and the less important will be its contribution to a unified theory of fundamental interactions. Hence, strings are the most fundamental extended objects in the weak coupling regime.