# String Theory, Duality, and the Primordial Universe (Part 1)

## String Theory, Duality, and the Primordial Universe

For anybody reading the introduction, there will no doubt have been important questions which arise spontaneously and are left unanswered. For instance, why does what we have called the “pre- Big Bang scenario” emerge within string theory and not within the classical cosmological setup based upon Einstein’s equations? And what is string theory?

For the answer to the second question we refer the reader to Chap. 4. With regard to the ﬁrst question, there are many reasons (that we will discuss below), but it is probably appropriate to say that the fundamental argument pertains to the particular symmetries that are present in string theory and not in Einstein’s theory of gravity.

So let us consider the theory of general relativity. As is typical of classical physics, this theory enjoys the following key property of symmetry: any (fundamental, elementary) process described by such a theory is invariant when the sign of the time coordinate is changed (provided this is not in contradiction with the laws of relativistic causality, of course). This is the so-called time-reversal, or time-reﬂection, symmetry. It implies for instance that, if the equations of that theory admit a solution describing a particle moving at constant speed from left to right, then there must exist a solution describing the same process seen backward in time, i.e., describing an identical particle moving with the same speed but going from right to left.

Furthermore, for a given solution describing a decelerating particle which moves from left to right, there must exist a solution that describes the same particle accelerating in a motion from right to left. In other words, the theory should work like a video tape which allows us to play the recorded images both forward and backward.

It is worth stressing, in particular, that some vectorial quantities, such as velocity, are reversed in sign when time is reversed, while others, like the acceleration, remain unchanged. In fact, if we consider the previous example, we may note that the acceleration is always pointing from right to left. However, in the ﬁrst case (i.e., when the motion is from left to right) the acceleration is opposite to the direction of the motion, and thus decreases the speed of the particle, while in the second case (i.e., from right to left) the acceleration is pointing along the direction of the motion, and thus increases the speed of the particle.

Within classical cosmology, the time-reversal symmetry implies that, to any given cosmological solution describing an expanding Universe (with a growing spatial radius R, like ours), there must be an associated solution describing the same Universe but evolving backward in time, i.e., a contracting Universe (with de- creasing R). Also, if the expanding Universe is decelerated then the contracting (time-reversed) universe will be accelerated, just as pointed out in our previous example regarding the motion of a particle (see Fig. 3.1). FIGURE 3.1 A pictorial representation of time-reversal, or time-reﬂection, symmetry. The curve plotted in the positive range of the time coordinate (i.e., to the right of the origin t = 0) represents the spatial radius R(t) of a Universe which is expanding (as the radius is growing), but in which the expansion is decelerating (as shown by the concavity of the curve), and which is in a state of decreasing curvature, as the rate of change of R(t) (not plotted in the picture) is decreasing. By applying a time- reﬂection transformation, t → −t, one obtains the curve R(−t) plotted in the negative-time range (i.e., to the left of the origin t = 0). Thus, time reﬂection transforms expansion into contraction (and vice versa), deceleration into acceleration (and vice versa), and decreasing curvature into increasing curvature (and vice versa)

Since the behavior of the space-time curvature follows the absolute value of the speed measuring the rate of change of R(t), we may also say that a time-reversal transformation maps expansion into contraction and decreasing curvature into increasing curvature.

It should be stressed that the existence of a possible solution of the cosmological equations does not automatically guarantee that the corresponding scenario does actually occur in nature. In other words, we cannot conclude that, given the existence of our expanding Universe, a corresponding contracting Universe also exists; similarly, we cannot establish that for a given particle moving from left to right we would observe a particle moving from right to left. According to the theory this a possibility, but its occurrence is not mandatory.

General relativity is a classical theory of gravitation, based upon macroscopic observations (Newton’s law, the motion of planets around the Sun, and so on), and implicitly rooted in the fundamental concepts of classical relativistic mechanics, generalized to the case of a curved space-time framework. Years of study and joint effort by many research groups around the world have shown that this theory is unlikely to be compatible with quantum mechanics, i.e., with the theory which lies at the heart of a physical description of the microscopic world. This is the reason why general relativity, as a theory of the gravitational forces, has always strongly resisted any attempt to unify gravity with theories describing the other forces active in a microscopic context – namely, the theories describing nuclear (weak and strong) interactions, and the electro- magnetic interactions.

A key step forward in this direction does seem possible, how- ever, within the framework of string theory, which should provide a uniﬁed description of all the forces of nature, valid at all energies, and including gravity even in its quantum regime. The description of nature proposed by string theory, rather than being based upon point-like elementary objects (the well-known particles of classical physics), admits as building blocks objects that have a spatial ex- tension, albeit one-dimensional. Such objects can be either closed (in the case in which both ends coincide), or open, although some- times with their ends ﬁxed on some preferred spatial hyperplane: we may visualize them as ordinary strings of ﬁnite length and negligible thickness. Different vibrational modes of these strings may simulate the various types of particles and the fundamental interactions that we are currently able to observe.

A more explicit illustration of string theory will be given in the next chapter and in Chap. 10. As far as this chapter is concerned, we just need to point out that according to string theory Einstein’s gravitational equations ought to be generalized, and that such a modiﬁcation brings about two important consequences.

The ﬁrst is that the gravitational equations provided by string theory, besides the time-reversal symmetry, show another important kind of symmetry, dubbed duality. This symmetry has no counterpart in any type of classical or quantum theory of ﬁelds, since (as will become clearer in the next chapter) it is rooted in the fact that the fundamental objects of the theory are extended rather than point-like.

There are, in general, various types of duality symmetry (T, S, U duality), corresponding to different types of transformation that leave the form of the string theory equations unchanged.

For instance, T-duality states that if the equations of the theory admits solutions describing universes of radius R, then universes with the reciprocal radius 1/R are also possible solutions of the same theory. Similarly, according to S-duality, if the theory admit solutions describing a particle characterized by a charge of strength Q (not necessarily electric charge), then there must be solutions de- scribing a particle with charge 1/Q. Finally, U-duality is (roughly speaking) a combination of T and S duality.1 It is worth noting that a large value of the charge Q corresponds to a small value of 1/Q and vice versa; a large value of the radius R corresponds to a small value of 1/R and vice versa. In other words, duality relates large universes to small universes, and strong couplings to weak ones. Furthermore, it transforms expansion into contraction, and vice versa (see Fig. 3.2). Indeed, if the function R(t) grows in time, then a Universe of radius R expands, while its dual partner of radius 1/R contracts, as 1/R(t) decreases. FIGURE 3.2 A duality transformation that inverts the spatial radius, R → 1/R, transforms decelerated expansion into decelerated contraction and vice versa. The time behavior of the curvature, as well as the acceler- ation/deceleration properties of the original solution, are left unchanged by a duality transformation

However, since there is no reﬂection of the time coordinate, there is no change in the acceleration properties of the transformed solution. Thus, decelerated expansion transforms into decelerated contraction. Similarly, the time behavior of the absolute value of the rate of change of R(t) is invariant under the inversion of the radius. Thus, the decreasing-curvature (or growing-curvature) status of the original solution is preserved by a duality transformation.

The second important property of string theory that we would like to recall here is that the implementation of the duality symmetry necessarily requires the introduction of a new type of force into the gravitational equations, mediated by a neutral scalar particle (i.e., by a particle without electric charge and without intrinsic angular momentum). This particle is called a dilaton, and under a duality transformation the force ﬁeld associated with the dilaton is not invariant in general, whence different, duality-related solutions of the theory are characterized by different values of the dilatonic forces.

Within string theory, the gravitational importance of the dilaton is encoded into the fact that this ﬁeld determines the value of the effective Newtonian constant G which, in its turn, ﬁxes the strength of the gravitational interaction (as will be illustrated in Chap. 4). Applying a duality transformation that changes the value of the dilaton it is then possible to modify the value of the effective gravitational coupling. In such a context, the Newtonian constant loses the role of fundamental gravitational parameter, and the theory may describe physical situations where the gravitational force can be either weaker or stronger than we usually experience.

Furthermore, the strength of the interaction may not be constant, but vary in space and time, following the dilaton behavior.

This is certainly a big physical revolution introduced by string theory and, as we shall see, it may have important consequences in a cosmological context. It is true that the possibility of a variable gravitational constant, associated with the presence of a scalar ﬁeld in the gravitational equations, was previously suggested by Carl Brans and Robert Dicke in 1961 (well before string theory was introduced). However, it is only through string theory that such a variable-G scenario ﬁnds robust motivations, and actually becomes essential for the consistent formulation of a quantum theory of gravity.

In a cosmological context, the simultaneous implementation of the duality symmetry and the time-reversal symmetry allows us to obtain new cosmological solutions (i.e., models for the Universe) that were not contemplated by general relativity, as suggested by the theoretical physicist Gabriele Veneziano in 1991 and later elaborated by him, in collaboration with the present author.

Let us consider, for instance, the current Universe, assuming that it can be properly described by the solutions of the standard cosmological theory, and let us focus our attention on an expanding spatial section of spherical shape. The corresponding spatial radius R(t) increases with time, while the space-time curvature, being proportional to the square of the expansion velocity, decreases with time, as the expansion rate is slowing down according to the standard cosmological scenario. Let us then apply a time-reﬂection transformation to this solution, that is, let us reverse the time arrow. As previously pointed out, we will obtain a new solution in which the curvature is increasing, and the transformed Universe will contract, as the radius R(−t) is a decreasing function of time.

Finally, let us apply a duality transformation, and invert the radius: the curvature will keep increasing, while the Universe will become an expanding universe, since if R(−t) is decreasing in time, its reciprocal 1/R(−t) is increasing (see Fig. 3.3). FIGURE 3.3 Pictorial representation of the combined action of a time re- versal and duality transformation applied to a given initial solution R(t) (top right curve), describing the decelerated expanding radius of a standard (decreasing-curvature) model of the Universe. The ﬁnal result is the curve 1/R (−t), plotted in the bottom left of the ﬁgure, describing accelerated expansion and growing curvature

We are thus led to the following result. Thanks to the combined action of duality symmetry and time-reversal symmetry, with any cosmological solution describing an expanding Universe with decreasing curvature we can associate another possible solution describing an expanding Universe with increasing curvature.

In particular, if the two solutions are smoothly matched at the time t = 0, one obtains a cosmological model where the spatial radius of the Universe increases continuously from zero to inﬁnity, as illustrated in Fig. 3.4 (top panel). Let us compute the space-time curvature corresponding to this model, namely the absolute value of the speed measuring the rate of change of R (more precisely, the absolute value of the time derivative of the radius divided by the radius itself). The resulting plot will grow in the range of negative values of the time coordinate (left of the origin), and decrease in the positive range (right of the origin). Assuming that the standard solution and its dual partner join continuously across the origin, t= 0, we then ﬁnd the characteristic bell-shaped behavior for the curvature, as shown in Fig. 3.4 (bottom panel) and anticipated in Chap. 1 as typical of the pre-Big-Bang scenario. FIGURE 3.4 Time evolution of the spatial radius (top panel) and the space- time curvature (bottom panel), in a cosmological model based upon the duality symmetry of string theory. The curvature is parametrized by the absolute value of the Hubble parameter H, deﬁned as the time derivative of the radius divided by the radius itself. The standard cosmological phase is described by the curves plotted in the positive range of the time coordi- nate, i.e., to the right of the origin t = 0. The complementary pre-Big-Bang phase, plotted to the left of t = 0, is obtained by applying a duality and a time-reversal transformation to the standard solution

We should recall, also, that the time derivative of the radius divided by the radius itself deﬁnes the Hubble parameter H introduced in the last chapter. We can say, therefore, that duality and time-reversal transformations map an expansion with decreasing H, typical of the standard cosmological phase, into an expansion with increasing H, typical of the pre-Big-Bang phase. Also, and most importantly, the decelerated expansion of the standard solution is mapped into an accelerated expansion of the transformed solution.

This is a consequence of time-reversal symmetry, as already out- lined in the example of a moving particle. (The relevance of the fact that the dynamics of the pre-Big-Bang phase becomes accelerated will be explained in Chap. 5.) As we have already stressed, implementation of the duality symmetry requires inversion of the radius to be accompanied by a simultaneous transformation of the dilaton ﬁeld, according to the rules dictated by string theory. Since it is the dilaton which ﬁxes the value of Newton’s constant, it follows that the duality-transformed cosmological solutions will be characterized by different values of G, namely by different intensities of the corresponding gravitational forces. This certainly makes the transformed solutions unacceptable for a description of the current Universe. Indeed, high-precision observations establish that the current value of G is nearly ﬁxed in time, with possibly allowed annual percentage variations smaller than one part in a thousand billion.2 However, this constraint does not prevent that in the remote past; before the formation of galaxies and stars, and even before atomic nuclei were formed, the gravitational force could have had a different value.

This may be a ﬁrst hint that the dual of the standard cosmological solutions, rather than the current Universe, may describe the Universe in its early stages. However, there is more. There are also thermodynamical arguments according to which, by exploiting the duality symmetry, our Universe may have undergone an epoch with “specular” features in the past, as compared with the current one. This possibility was put forward at the end of 1980s by the pioneering work of some cosmologists and string-theory experts like Robert Brandenberger and Cumrum Vafa, followed by later work by the theoretical physicist Arcady Tseytlin (see also the string gas cosmological scenario discussed in the next chapter).