String Theory, Duality, and the Primordial Universe (Part 2)

The Universe Before the Big Bang

String Theory, Duality, and the Primordial Universe

After this discussion, it is probably more evident to the reader why the existence of cosmological solutions that expand with increasing curvature is a peculiar property of string theory: the appearance of these solutions, as well as their close link to the solutions of the standard scenario, is a direct consequence of duality, namely of a new “stringy” symmetry which is absent in the pure Einstein theory. Obviously, as already pointed out, the existence of allowed solutions of a theory does not necessarily imply that the scenario they describe actually occurs in nature. However, the presence of these dual solutions suggests a possible answer to one of the key questions implicit in standard cosmology (as well as in inflationary cosmology, discussed in Chap. 5), i.e., to the question of the initial state of the Universe (assuming that the singularity is somehow avoided).

String theory suggests that, initially, the state of our Universe might correspond to the state determined by applying a duality and a time-reversal transformation to the current cosmological state.

The cosmological scenario that completes the standard evolution by adding the phase dubbed “pre-Big-Bang”, introduced in Chap. 1, does indeed emerge from the assumption that the evolution of the Universe should be self-dual and time-symmetric, i.e., simultaneously invariant under the combined action of duality and time- reversal transformations (as in the case of the cosmological model shown in Fig. 3.4).

Under such a hypothesis the current Universe, which is characterized by an almost flat space-time geometry and by an average energy density and temperature much lower than their standard macroscopic values, should have had, in its very early past, a dual counterpart similar to its present state. Hence, the Universe should have undergone a very early regime associated with an al- most flat, empty and cold state which, going backward in time, gets progressively more and more flat and empty until it corresponds, asymptotically, to the state called the perturbative vacuum of string theory. In the primordial cosmological phase, that we now identify with the epoch preceding the Big Bang, the growth of the curvature has led the Universe towards progressively more curved and denser states, until the radiation produced at a microscopic level became dominant, causing the primordial explosion that finally led to the current (standard) decreasing-curvature regime.

For a true self-dual scenario, however, it is essential that the evolution of the geometry is accompanied by the evolution of the dilaton field, which is present in all string theory models. In the current cosmological state the gravitational force has a nearly constant strength, controlled by the Newtonian constant G; hence the dilaton, which fixes this strength, must be constant. In the primordial state representing the dual counterpart of the present one, it turns out – according to the rules of the duality transformations – that the dilaton has to increase with time, thus describing an in- creasing gravitational coupling. It follows that the current value of the Newtonian constant is reached after starting from an almost zero initial asymptotic value, as illustrated in Fig. 3.5 (right panel).

string-theory-duality-and-the-primordial-universe-pic-5
FIGURE 3.5 Time evolution of the curvature (represented by the Hubble parameter H) and the Newtonian constant G (determined by the dilaton), for a typical self-dual solution of the string cosmology equations.
The initial cosmological configuration – approaching the so-called string perturbative vacuum at very large negative times – is characterized by a nearly flat space-time geometry and the vanishingly small intensity of all interactions

This feature of the dual solutions has an important physical consequence. Indeed, in all unified models based upon string theory – i.e., in all models which, using strings, try to incorporate a somehow unified description of all forces of nature – the dilaton must determine not only the value of G, but also the coupling strengths of the other fundamental forces (see Chap. 4). Hence, in the context of a self-dual cosmological scenario, one finds that, in the initial state of the Universe, all the coupling parameters tend to approach zero. In other words, all the forces and all the inter- actions are asymptotically suppressed, following the behavior of G shown in Fig. 3.5. This implies that the initial evolution of the Universe, starting from the asymptotic state called the perturbative vacuum, can be correctly described using the string equations in the semi-classical limit and weak-coupling regime; the quantum (higher-order) corrections are expected to become relevant only later when, as the time of the Big Bang gets closer (i.e., around t = 0), the various forces become sufficiently strong.

It should be stressed at this point that, if the self-dual symmetry of the cosmological evolution were to be exact, then any instant in the life of the current Universe would have its dual counterpart in the past. However, we have learned many times in the history of physics that the symmetries of our theories, when implemented in nature, are not always exact. Frequently, they are only valid in some approximate limit, since there are physical effects responsible for their breaking (through a spontaneous, or more involved, mechanism). As far as cosmology is concerned, we know in advance that the duality symmetry cannot be exactly implemented at all times. Indeed, the current state of the Universe contains matter and thermal radiation characterized by a high entropy level, contrary to the asymptotically empty and cold initial state. This means that, in a realistic cosmological scenario, the behavior of the space-time curvature will not be represented by an exactly symmetric curve (like the ones shown in Figs. 3.4 and 3.5); rather, it will have some distortions caused by an approximate implementation of duality and time-reversal symmetry. This by no means contradicts the basic idea suggested by duality, i.e., the fact that our Universe could have undergone in the past an accelerated phase with increasing curvature, complementary to the current one.

Within modern theoretical physics, duality is a quite important symmetry. There is a common belief that the duality symmetry is somehow present in nature at a fundamental physical level.

The pre-Big-Bang scenario, based upon this symmetry, is thus sup- ported by rather solid theoretical motivations. It should be recalled, in fact, that the practice of exploiting the symmetries of the theory in order to formulate predictions about as yet unobserved phenomena, is a common working method employed by physicists, a method which in the past has led to many important discoveries.

It is probably appropriate to draw an analogy here with another important symmetry of modern theoretical physics, known as supersymmetry.

According to supersymmetry, any physical state with statistical properties of bosonic type (e.g., a particle whose intrinsic an- gular momentum is measured, in quantum units, by an integer number) must correspond to a supersymmetric “partner” whose statistical properties are of fermionic type (e.g., a particle with half-integer intrinsic angular momentum). Given the presence in nature of known bosonic particles (like the photon, the graviton, and so on), and using supersymmetry, we may then infer the existence of new fermionic particles associated with the previous ones (and suggestively called the photino, the gravitino, and so on), even if such particles have not yet been observed. The validity of super- symmetry as a true symmetry of nature has still to be confirmed experimentally, but its predictions have been taken so seriously that in many laboratories around the world (including CERN, the large European laboratory for particle physics and nuclear research in Geneva), studies are currently being undertaken to detect some supersymmetric partners, using the most powerful particle accelerators presently available.

In a similar fashion, duality symmetry (with the help of time- reversal symmetry) establishes that any expanding geometrical configuration characterized by decreasing curvature corresponds to a dual partner characterized by expansion and increasing curvature. On the other hand, our Universe is currently associated with a post-Big-Bang, decreasing-curvature state. Assuming that the duality symmetry is in fact realized in nature, and in particular (even if approximately) during the course of cosmological evolution, we may then expect the Universe to have undergone in the past a phase characterized by an increasing-curvature expansion.

At this stage, an obvious observation springs to mind. A phase of increasing curvature, if unbounded, could bring the Universe to- wards a state of infinite curvature, thus introducing the pathology associated with the presence of a singularity, as in the standard model – with the difference here that the singularity, rather than being in the past, would be located in the future. The answer is that, in contrast to models based upon general relativity, such a pathology is not necessarily present in string cosmology thanks to another important feature of string theory, namely, the presence in this theory of a typical fundamental length Ls.

The value of this length, at least according to the most conventional string-theoretical schemes unifying all interactions, is expected to be about 10−32 cm, i.e., almost an order of magnitude bigger than the so-called Planck length LP, which characterizes the length scale at which quantum gravitational effects become important. In any case, this length Ls determines the minimum characteristic size for the spatial extension of any physical system, and therefore also for the space-time curvature radius of the Uni- verse, or, equivalently, for the Hubble radius c/H, as these two quantities are proportional.

During the initial pre-Big-Bang evolution the Universe, starting from an empty and nearly flat state, becomes progressively more and more curved, so that the absolute value of the Hub- ble parameter grows in time, while its reciprocal, representing the curvature (or the Hubble) radius, shrinks monotonically. The geometry, on a cosmological scale, may thus evolve towards the state corresponding to a curvature radius of the same order as Ls.

We may expect, however, that this threshold cannot be crossed, i.e., that the growth of the curvature has to stop when it reaches the string scale, since a greater value of the curvature would correspond to a Hubble radius smaller than Ls, and this seems to be meaningless for a model based upon strings. Hence, after reaching this threshold configuration, the curvature should either remain constant or start decreasing.

The detailed mechanism by which the increasing curvature is tamed – smoothing out the singularity, and eventually decreasing according to standard model behavior – is not yet fully understood, mainly due to technical issues related to the presence of quantum effects (and higher-order string theory corrections) near the maximum curvature regime (see Chap. 8). We may recall in particular that, when the curvature radius of the Universe reaches Ls, the gravitational equations are drastically modified: an infinite sum of terms has to be added up, and even the classical notion of space-time becomes inadequate to describe processes that take place in such a regime. However, even in that case, the duality symmetry may play an important role, and this reinforces the key importance of such a symmetry within models of pre-Big-Bang evolution.

However, duality alone cannot account for a consistent formulation of the cosmological scenario described in the previous paragraphs. Another important element concerns the properties of the string perturbative vacuum, the asymptotic state characterized by flat geometry and zero coupling constants, which duality suggests as a possible representation of the initial configuration of our Universe.

If such a state were stable, it could not represent the initial stage of the evolution since the Universe, once in such a configuration, would be eternally trapped there, remaining forever flat, cold, and empty, i.e., radically different from the Universe that we currently observe. Instead, the perturbative vacuum is unstable, i.e., it tends to decay spontaneously exactly like an atom or a molecule which, starting from an excited state, tends to reach another configuration made more favorable by the forces involved.

In particular, the perturbative vacuum tends to evolve towards a non-static configuration where the curvature and the dilaton both increase with time, just as expected in the context of a self-dual pre-Big-Bang scenario.

This instability of the initial state is linked to the fact that the expansion of the Universe provides a negative (gravitational) contribution to the total energy of the system, while the increase of a field like the dilaton yields a positive (non-gravitational) contribution. The two energies compensate each other and, as a con- sequence, the simultaneous increase of the spatial volume (due to the expanding geometry) and the growth of the coupling constants (due to the dilaton evolution) mutually sustain each other, and can be “spontaneously” ignited without any variation of the total energy of the cosmological system, i.e., without feeding this process with external energy sources.

More details about the decay process of the string perturbative vacuum will be provided in Chap. 5. Here we limit ourselves to a somewhat more intuitive description of its instability, considering the motion of some strings within the primordial Universe, and using a simple model where no other matter sources or fields are present than these strings and the dilaton. Hence, the string distribution will be responsible for determining the gravitational field (i.e., the geometry) on a cosmological scale while, at the same time, the behavior of the geometry and the dilaton will determine the evolution of strings and their dynamics.

Let us suppose we want to solve the equations of motion for the full system consisting of gravitational field, strings, anddilaton, taking into account their mutual correlations. Let us impose – as particular initial conditions – that the system starts evolving from a state which, going sufficiently far back in time, tends to coincide with the perturbative vacuum. Finally, let us ask whether our system tends to go back towards the initial, asymptotically free configuration (i.e., towards the perturbative vacuum), or whether it tends to go away from it, beginning a one-way journey towards the high-curvature Big Bang regime. In the former case the perturbative vacuum would be stable, while in the latter it would be unstable.

To answer this question it may be useful to recall that, in the dual solutions describing the state of the Universe before the Big Bang, the cosmological geometry is characterized by an increasing curvature and thus also, as repeatedly stressed (and illustrated in Figs. 3.4 and 3.5), by a monotonic growth of the Hubble parameter H. The reciprocal of the Hubble parameter defines the quantity c/H introduced in Chap. 2, and called the Hubble radius, or Hubble horizon; actually, it represents the spatial size of what is technically called an event horizon, since it measures the maximum distance within which exchange of signals, and consequently causal interactions, are allowed. What is relevant in this context is that, if H increases during the regime of pre-Big-Bang evolution, then its reciprocal decreases, so that the Hubble horizon tends to shrink with time.

Within a standard gravitational theory like general relativity, which is based upon classical field theory, and in which the sources of the force fields as well as the fundamental test bodies are point- like objects, the occurrence of a shrinking horizon does not cause any trouble, either practical or conceptual: the number of point particles included within the causal horizon may decrease, but there is no traumatic consequence for the system. However, for a theory in which the fundamental objects are extended, like the strings under consideration, the shrinking of the horizon may lead to a potentially pathological configuration.

Indeed, since the horizon tends to shrink with time, sooner or later a very long string will become greater than the horizon itself, i.e., there will be a situation where parts of the string are within the horizon, while other parts are outside it, without any chance of causal contact, and hence without any possibility of exchanging information between these two parts. It is like a man having his whole body intact but with an invisible barrier, impassable to any signal, cutting the body at the level of the stomach: the head could not know what the feet were doing, and would not even know whether the feet and the legs still existed, and vice versa.

Such a situation, certainly bizarre and somehow unthinkable in the context of classical theories based upon the notion of a point, is possible in a theory containing extended fundamental objects like strings. In particular, when the proper length of a string be- comes larger than the horizon, and the propagation of causal signals from one end to the other is blocked, the string is said to be frozen.

In that case the string somehow loses its own life, stops oscillating, and starts passively following the evolution of the surrounding geometry, as would happen to a tiny object that was frozen into an ice-block which is part of an iceberg: when the iceberg goes adrift, the object inside the block passively follows its movements.

Even frozen outside the horizon, however, the strings may have an important cosmological effect, since an ensemble of such strings behaves like a gas with negative pressure (as shown by a series of studies initiated by Norma Sanchez, Gabriele Veneziano, and the present author, and later pursued with the collaboration of Massimo Giovannini and Kris Meissner). A negative pressure does not in fact oppose, but tends to favor the increase of both the curvature and the dilaton. As a consequence, it accelerates the shrinking of the horizon, triggering a back-reaction mechanism which renders the initial configuration (the perturbative vacuum) highly unstable.

We may indeed compare the initial state to a small ball on the top of a steep hill: as soon as a breeze moves the ball from its privileged position, it starts rolling towards the bottom of the hill.

In a similar fashion, in our simple model of a universe filled with strings, as soon as the curvature starts to increase (for instance as a consequence of unavoidable quantum fluctuations), the horizon starts to shrink, so that more and more strings become larger than the horizon and get frozen, whence their negative pressure brings the Universe even further into the pre-Big-Bang phase, accelerating its race towards states of progressively increasing curvature and shrinking horizon.

Concluding this chapter, we may say that string theory suggests, in various ways, the possibility that our Universe emerges from a primordial state which is unstable, empty, and flat, and has no interactions. The Big Bang, within this scenario, is interpreted as a moment of violent and explosive transition from an increasing- curvature phase to a decreasing-curvature phase, thus corresponding to an intermediate stage in the history of our Universe rather than to the beginning.

If we accept such ideas, at least as a working hypothesis, and take seriously the possibility that the past Universe may have undergone a phase which is (at least approximately) related by duality to the present one, a particular question comes to mind: Is such a phase (so radically different from the standard one) compatible with the subsequent cosmological evolution, i.e., is it possible to join this phase consistently with the standard-model cosmology? We are asking in particular whether the time evolution of the pre-Big-Bang geometry, despite the great differences with respect to the conventional one, has the right properties to solve the kinematical problems which (as we shall see) are present in the standard model of cosmological evolution.

The answer to those questions will be discussed in the following chapters, after a short interlude devoted to those readers wishing to learn a little more about string theory. In the next chapter, in fact, we will try to give the reader a more precise idea of why a theory of strings may help to avoid the initial singularity, and why such a theory necessarily leads to a modification of the equations of general relativity, providing a different and more complex gravitational theory.

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