The Future of our Universe (Part 2)

Cosmology and String Theory: The Future of our Universe

To explain why such discovery is so revolutionary and “ex- plosive”, and understand the subtle link that could exist between such acceleration and the pre-Big-Bang cosmology described in the previous chapters, it is wise to proceed step by step. Let us start by recalling that the standard cosmological model, using the equations of general relativity and the assumptions of homogeneity, isotropy, and perfect fluid sources (see Chap. 2), provides us with a very sharp description of the current Universe. It should be noted, incidentally, that the use of the classical theory of general relativity is certainly legitimate for describing geometrical configurations with a curvature much smaller than the Planck curvature (like the current Universe), since in that case the possible modifications of the theory due to quantum effects are completely negligible.

According to the equations of general relativity and the assumptions of the standard cosmological model, it is found, in particular, that the current evolution of the cosmological geometry – once the constant value of the spatial curvature has been fixed – is fully described by a unique time-dependent quantity: the spatial radius (or scale factor) R(t). The time behavior of this quantity will tell us whether the future Universe will always be expanding (in a decelerated or accelerated fashion), or become static, or start to shrink, eventually collapsing towards a future singularity.

One the most fascinating features of cosmological models – apart from their ability to describe the current state of the Universe and to trace its past history – is their ability to make predictions concerning future evolution. Obviously, the more accurate the knowledge of the current cosmological state, the more accurate and reliable such predictions become. As long as the number of observations increases, with progressively better experimental sensitivity, our knowledge of the current Universe is continually subject to revision and updates. Hence, predictions about the future may also need frequent revision and improvement.

In this chapter we focus on a discovery that can be counted amongst the most important made at the end of the last century, a discovery that has radically changed an already consolidated view, bringing a new perspective into our ordinary expectations about the future of the Universe: rather than slowing down as predicted by the standard model, the cosmological expansion is currently accelerating!

A plausible solution to those problems suggests that the dark energy is not represented, in the Einstein equations, by a cosmological constant, but rather by a time-dependent term associated with the energy of some cosmic fluid or field. The simplest field to be considered is then a neutral scalar field, without intrinsic angular momentum and charge, and self-interacting, i.e., equipped with an appropriate potential energy. Scalar field models that may be appropriate for representing dark energy effects were suggested in pioneering work by Bharat Ratra, James Peebles, and Christof Wetterich, well before the experimental discovery of the cosmic acceleration, and later studied by many astrophysicists (Michael Turner, Martin White, Robert Caldwell, Rahul Dave, Paul Steinhardt, Ivaylo Zlatev, Li-Min Wang, and others).

This scalar field, however, should have properties so peculiar as to make it difficult to identify it with one of the scalar particles already present in the standard model. As anticipated in Chap. 2, a new term has indeed been appositely coined for this field: quintessence, a name which stresses its exotic character (it recalls the elusive fifth element conjectured by the ancient philosophers and still sought by alchemists in the Middle Ages). So what are the strange properties this cosmic field should have?

First of all, in order to have negative pressure, its energy density must be dominated by potential energy (the kinetic energy corresponds in fact to a positive pressure). Moreover, even if it varies in time, its total energy must be negligible compared with that of the other cosmological components for most of the past history of the Universe, in order not to affect all the successful predictions of the standard cosmological model (the formation of nuclei, the gravitational aggregation of non-relativistic matter, the subsequent production of galactic structures, and so on). It is only recently that its contribution ought to become relevant.

In addition, in order to play a dominant cosmological role in the present Universe, the mass of the particle associated with this field must be extremely tiny. In fact, the range of the corresponding force (which is inversely proportional to the mass) must be at least of the order of the Hubble radius of the present horizon, namely about ten billion light years. This value corresponds to a very tiny mass, about 10^−66 g. (Recall that the electron, one of the lightest known particles, has a mass of about 10^−27 g.) Such a long range in turn generates other problems. If this particle carries a force among macroscopic bodies over such great distances, why has it not been found in any lab experiments so far performed? An explanation could be that the force is extremely weak, much weaker than all forces so far discovered, hence even much weaker than the gravitational force present in ordinary mat- ter. Or maybe most matter is “neutral” with regard to the type of force carried by the quintessence field, and thus unable to feel it.

On the other hand, in order to solve the cosmic coincidence problem, the quintessence field should interact with dark matter, at the cosmological level, with a force whose intensity should be al- most equal to the intensity of the gravitational field (as suggested by Luca Amendola for scalar-type models of dark energy, and by Luis Chimento, Alejandro Jakubi, and Diego Pavon for fluid-type models). To avoid contradictions with existing gravitational experiments, it therefore seems necessary to endow quintessence with a further peculiar property: the possibility of different couplings to different kinds of matter. In particular, the couplings should be stronger in the case of the (still unknown) dark matter particles, while they should be much weaker in the case of the known particles (protons, neutrons, etc.) composing ordinary matter.

It is worth noting that all the properties just mentioned as typical of the quintessence field can be satisfied by a scalar field which has not been introduced ad hoc to explain the current astronomical observations, but which must necessarily exist in a fundamental theory like string theory: the dilaton field. Even in the context of the pre-Big-Bang scenario, where the initial dilaton is massless and has negligible potential energy, there are in fact quantum effects able to generate a dilaton potential as soon as the Universe enters the strong coupling regime. Such a potential is known to go rapidly to zero at weak couplings, but its behavior in the opposite, strong coupling regime is not yet well understood. The cosmic coincidence problem can in this case be explained in two ways, depending on the behavior of the potential at strong couplings.

In fact, as repeatedly stressed in the previous chapters, the crucial property of the dilaton field is that it determines the strength of the various natural forces (including the gravitational field). During the phase preceding the Big Bang, the dilaton is subject to a rapid and intense variation that brings the strengths of all forces from initial values that are almost zero to final values approximately equal to what is currently observed. It follows that during most of the standard cosmological phase following the Big Bang, up to the present epoch, the strengths of the various forces must have been kept stable, fixed at nearly constant values. (Any possible variation, if it exists, is so small as to have escaped unambiguous detection.) Hence, the dilaton potential generated in the post-Big-Bang phase must have the appropriate form to guarantee such stabilization, to avoid contradictions with present observations. This may happen in essentially two ways.

A first possibility is that the potential develops a series of local minima when approaching the strong coupling regime and that the dilaton, during the phase of post-Big-Bang evolution, is “trapped” inside one of those minima (exactly like a ball reaching the bottom of a hole, see Fig. 9.1). After some oscillations backward and forward, the dilaton “falls asleep” and, remaining fixed at a value φ0, it also fixes the strengths of all forces at the values that we now observe.

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FIGURE 9.1 Possible plot of the dilaton potential for two different values of the dilaton mass, with m1 > m2. The potential goes to zero in the weak coupling regime, and has a local minimum at the beginning of the strong coupling regime, where the dilaton is trapped for the whole duration of the radiation era. Note that the amplitude of the potential depends on the mass and that, as the mass decreases (lowest curve), the depth of the potential well decreases too, so that it becomes progressively easier for the dilaton to escape from the equilibrium position φ0. Hence, the dilaton mass has to be large enough to avoid being shifted away from the minimum, and small enough to correspond to a small potential energy which becomes significant only in late cosmological epochs

In this scenario (studied by the present author) nothing else happens as long as the Universe is radiation dominated. However, as soon as the Universe enters the phase dominated by matter (see Chap. 2), the dilaton gets a kick that somehow wakes it up, tending to push it away from the equilibrium configuration. The kick is due to the fact that the dilaton couples to both the energy density and the pressure of the cosmic fluid sources, and that this coupling acts as a force in the dilaton equation of motion. In the radiation case, energy density and pressure compensate to give a null total effect, while in the matter case the pressure is zero, compensation is impossible, and a force appears which tends to accelerate the dilaton.

If the dilaton were too light and too strongly coupled to matter, it could escape the potential well, and run back towards the large negative values typical of the pre-Big-Bang phase, dragging the coupling constants of the various forces towards progressively smaller values. As this did not happen, it follows that the dilaton is sufficiently heavy to remain confined at the bottom of the potential well (see Fig. 9.1). The dilaton mass, on the other hand, also controls the intensity of the dilaton potential and, in particular, its minimum potential energy, which should become the dominant source of the present acceleration. Hence, the mass must be sufficiently light to correspond to a potential which does not affect the Universe’s evolution too early on. These two opposing effects leave us with only a limited range of values for the mass and the potential energy of the dilaton, thus alleviating – if not completely solving – the problem of explaining why the value of the dark energy density is such that it becomes dominant just at the current epoch.

An alternative solution to the coincidence problem (suggested by Federico Piazza, Gabriele Veneziano, and the present author, and subsequently studied by Luca Amendola and Carlo Ungarelli) can be obtained even if the dilaton does not remain trapped at a minimum of the potential, but keeps running toward higher and higher values, even during post-Big-Bang epochs. Of course, this is possible provided that the potential does not represent an in- surmountable obstacle to the motion of the dilaton: after reaching a maximum, the potential must decrease towards zero, not only at large negative values of the dilaton (typical of the pre-Big-Bang regime), but also at large positive values of the dilaton, as in the example shown in Fig. 9.2.

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FIGURE 9.2 Possible plot of a non-perturbative dilaton potential which is smoothly decreasing in the strong coupling regime. In this case the dilaton, starting from the extremely large negative value of the initial state of the pre-Big-Bang scenario, can increase unconstrained towards arbitrarily large positive values

Even in this case the strength of the various forces and couplings can be stabilized at constant finite values, with the difference (from the previous scenario) that the equilibrium position of the dilaton is now located at infinity (instead of being inside a potential well). In this case, as the dilaton grows towards higher and higher values (and, together with the dilaton, the bare value of the string coupling parameter also grows), the generated quantum corrections become stronger and stronger, but they tend to saturate, i.e., to compensate one another, in such a way as to fix all couplings at a final constant value. And it is just within such a compensation mechanism – which becomes more and more efficient as the dilaton grows, i.e., as time goes on – that we can find a key for understanding the cosmic coincidence. As the dilaton couples “non-universally” to the various kinds of matter (see Chap. 7), it may happen that its coupling to ordinary matter becomes progressively smaller (and eventually negligible), while the coupling to the dark matter particles tends to stabilize at a higher value. Due to this effect, the Universe tends to evolve towards a final regime where the dark matter and the dilaton (which represents the dark energy) interact strongly together, while ordinary baryonic matter is decoupled.

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FIGURE 9.3 The figure shows on a logarithmic scale the time dependence of the radiation energy density (dotted curve), baryon energy density (thin solid line), dark matter and dilaton energy density (thick solid lines). They are obtained in a model where the dilaton, at late enough times, becomes strongly coupled to dark matter (but not to ordinary baryonic matter) and where, after a “dragging” phase, the Universe reaches a final accelerated “freezing” phase, in which the dark matter and dilaton energy densities are of the same order of magnitude and evolve in time in the same way.
Note that, in this final regime, the baryon matter density is diluted faster than the dark energy and dilaton energy densities

Thanks to this coupling, the dilaton energy density, even if initially quite small, is progressively “dragged” towards the dark matter energy density, until a final regime is reached where the two densities are of the same order of magnitude, and evolve in time in the same way (see Fig. 9.3). This final “freezing” regime, reached at large enough values of the dilaton, is thus characterized not only by the stabilization of the various coupling strengths, but also by the stabilization of the dark energy over dark matter density ratio. In addition, if the dilaton potential is sufficiently strong, the cosmic expansion in this regime becomes accelerated, just as presently observed. The cosmic coincidence can thus be explained, in this context, by assuming that today we are already inside (or very near to) the freezing configuration, whence we may already observe that dark matter and dark energy have energy densities of the same order of magnitude.

There are various important differences between this second dilatonic dark energy scenario (which we shall call the coupled quintessence scenario) and the first one (more similar to conventional models of uncoupled quintessence).

First of all, in the coupled scenario the dark matter density will always remain of the same order as the dark energy density in the future, whereas in the decoupled scenario, the dark matter density will progressively become smaller and eventually negligible with respect to the dark energy density. What is diluted in time, in the coupled scenario, is the ratio between the baryonic and dark matter densities. This occurs because the dark matter density, thanks to the coupling to the dilaton, does not decrease proportionally to the reciprocal of the volume, as happens to the baryon density, but much more slowly (see Fig. 9.3). Incidentally, this effect could even explain why baryonic matter is today so much less abundant than dark matter, in spite of having the same equation of state.

As a further important difference, we should mention the fact that the accelerated regime of the coupled scenario may start at much earlier epochs than in the decoupled scenario. We do not yet know which scenario is the most realistic, and we do not even know whether one of them does effectively correspond to the effects we observe. Future observations will tell us. In fact, rather fortunately, the differences between the various scenarios mentioned above are observable (at least in principle).

Notwithstanding, we may safely argue that the dilaton represents (at least at the moment) a plausible candidate to play the role of the quintessence field, i.e., the dark energy that is accelerating the expansion of our Universe.

To conclude this chapter we note that the standard cosmological model, although it is effectively able to provide (probably for the first time in human history) an accurate and scientifically consistent quantitative description of the Universe and the physical processes determining its current state, cannot be extrapolated either too far backward or too far forward in time. Current observations point to the need for modifications at small times (as dis- cussed in previous chapters) as well as at large times (as outlined in this chapter).

Probably, from a purely conceptual perspective, it would be desirable for such modifications not to be detached from one another, but rather that they should originate from a unique theoretical framework or model. String cosmology, and in particular the pre-Big-Bang scenario, may provide a positive response to this requirement thanks to the presence of the dilaton, which is some- thing of a general factotum in string theory; not satisfied with the modifications to general relativity and the primordial history of the cosmos, it also seems able to determine the future of the Universe, progressively becoming the most relevant form of energy.

If this were true, then the future Universe would be characterized by a strict relationship with the pre-Big-Bang Universe, and the current experimentally observed acceleration would already represent an indirect confirmation of the primordial scenario de- scribed in previous chapters.

Finally, if the cosmological dynamics is so tightly controlled by the dilaton, it is also possible to envisage a new and interesting scenario for the future of our Universe. Let us suppose, in fact, that the post-Big-Bang growth of the dilaton does not continue forever up to infinity. At some point the dilaton is stopped because its potential – which we do not yet know, unfortunately – forces it to bounce back towards negative values, i.e., towards the weak coupling regime. This would reproduce initial conditions appropriate to a pre-Big-Bang phase, and the curvature could start increasing again, thus initiating a new cycle of self-dual evolution. With a suitable form of the dilaton potential this sequence of events could repeat itself an arbitrary number of times, eventually implementing a cyclic scenario (similar to that obtained in the context of the ekpyrotic model, to be illustrated in the next chapter).

The Future of our Universe (Part 1)

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