**The Universe Before the Big Bang: Other Relics of the Primordial Universe**

In addition to gravitational waves, there are other types of signal that could reach us from a primordial epoch preceding the Big Bang, and that could be (directly or indirectly) available to our present observation. Actually, as pointed out in the last chapter, the transition from an accelerated to a decelerated phase not only ampliﬁes the ﬂuctuations of the geometry (thus producing a gravitational-wave background), but also enhances the ﬂuctuations of other ﬁelds. In this chapter we discuss in particular three important effects, all of them peculiar to string cosmology: the production from the vacuum of primordial magnetic ﬁelds, dilatons and axions. Let us start with the magnetic ﬁelds.

It is a well-known fact that all celestial bodies – the Earth, the Sun, the planets, up to galaxies and clusters of galaxies – have magnetic ﬁelds of various intensities. The origin of such cosmic magnetic ﬁelds is one of the issues that modern astrophysics has not yet completely resolved. We know, in particular, that magnetic ﬁelds can be produced by the rotation of electrical charges – as happens for instance in electric motors exploiting the dynamo effect. Since all celestial bodies rotate, their magnetic ﬁeld could somehow be the result of a dynamo effect.

Even if this were true, however, the explanation would still be incomplete. In order to ignite the dynamo and trigger the mechanism producing the observed cosmic magnetic ﬁelds, the presence of a small initial ﬁeld – also dubbed a magnetic seed – is essential.

For the dynamos that we use in everyday life, (e.g., the small box with a rotating head producing the current for our bicycle light), the starting magnetic ﬁeld is provided by a magnet. With regard to the dynamo that produces the magnetic ﬁeld in a galaxy, the origin and the main features of the corresponding seed ﬁeld are still largely speculative.

The simplest and most natural explanation is that the origin of the cosmic seed ﬁelds has to be traced to the vacuum ﬂuctuations of the electromagnetic ﬁeld, which are ampliﬁed during the accelerated evolution of our Universe (as ﬁrst suggested by Michael Turner and Lawrence Widrow in 1988). Indeed, according to quantum mechanics (in particular, according to Heisenberg’s un- certainty principle), the electromagnetic ﬁeld in vacuum is never exactly zero but – as with all other ﬁelds – has some spontaneous, small-amplitude oscillations. An inﬂationary phase, during which the Universe expands in an accelerated fashion, could in principle amplify those oscillations and produce the magnetic seeds able to ignite the dynamo. After all, it is in exactly this way that we obtain the background of cosmic gravitons discussed in the previous chapter: the relic gravitational radiation is the result of the ampliﬁcation of the vacuum oscillations of the gravitational ﬁeld ﬁlling the Universe.

However, there is an additional issue for the magnetic ﬁelds.

The Maxwell equations governing their evolution are characterized by a symmetry known as conformal symmetry. The presence of this symmetry implies that, for the evolution of the electromagnetic ﬂuctuations, an isotropic, homogeneous and spatially ﬂat space-time like the cosmological space-time is perfectly indistinguishable from a completely ﬂat space-time. In other words, the electromagnetic vacuum oscillations are unable to “feel” the expansion of the Universe. Hence, they are not ampliﬁed, even if the expansion is accelerated, i.e., inﬂationary. The seed magnetic ﬁelds cannot therefore be produced in this way.

The above result refers to a classical cosmological framework based upon the Maxwell and Einstein equations. According to string theory, however, the electromagnetic ﬁeld should also be coupled – in addition to gravity – to the dilaton, a crucial and un- avoidable component of all string cosmology models. The vacuum oscillations of the electromagnetic ﬁeld do not feel the expansion, but they are inﬂuenced by the dilaton evolution and can be efﬁciently ampliﬁed under the effect of a growing dilaton, as shown by Massimo Giovannini, Gabriele Veneziano, and the present author, and by the two French astrophysicists David and Martin Lemoine (brothers).

In fact, during the pre-Big-Bang phase the amplitude of the magnetic oscillations can increase, “dragged” somehow by the dilaton acceleration, until it reaches a “freezing” regime similar to the one described in the last chapter. During the subsequent phase of standard evolution, the ﬂuctuations in the magnetic ﬁeld turn out to be ampliﬁed with an increasing spectrum (as happens to gravitons), and begin to oscillate, re-entering the horizon. At re-entry, such ﬂuctuations provide a homogeneous magnetic ﬁeld within a causally-connected region of space. This ﬁeld could act precisely as the seed for the dynamo that generates the galactic ﬁelds.

Explicit computations have shown that a pre-Big-Bang phase, as predicted by the minimal self-dual scenario, is able to produce primordial magnetic ﬁelds on cosmological scales with an amplitude more than appropriate for seeding the currently observed magnetic ﬁelds. There are also other, more or less contrived, explanations for the origin of the seed ﬁelds, but they require the ad hoc introduction of new ﬁelds and/or new types of couplings and/or new phases of cosmological evolution (the list of all models and all authors is too long to be reported here). The dilaton coupling, and its primordial accelerated growth, are instead natural predictions of a fundamental theory like string theory and its basic symmetries, i.e., duality.

This result may lead us to conclude that the seed production effect described above represents an undeniable success for string cosmology (and for the pre-Big-Bang scenario in particular). Pushing the argument forward as far as possible, we could even say that the existence of cosmic magnetic ﬁelds can be seen as an indirect proof for the occurrence of a cosmological phase dominated by the kinetic energy of the dilaton, and preceding the Big Bang. In this spirit, the analysis of the cosmological magnetic ﬁelds can provide indirect hints pertaining to the pre-Big-Bang phase, and can be used to test string models at the experimental level.

We should recall, in fact, that the coupling of the dilaton to the electromagnetic ﬁeld may have different intensities, depending on the string model we are considering. As we shall see in Chap. 10, there are indeed ﬁve possible models of superstrings, with different physical properties. The pre-Big-Bang ampliﬁcation of the electromagnetic and gravitational ﬂuctuations for different string models has been compared by Stefano Nicotri and the present author. It has been found that for some models (for instance, type I superstrings) a large production of magnetic seeds is easily compatible with the production of a graviton background strong enough to be detected by near-future experiments. For other models (like heterotic superstrings), however, the two effects – large enough seeds and detectable gravitons – seem to be hardly compatible, at least in the context of the minimal pre-Big-Bang scenario. The cross-correlated study of magnetic and gravitational backgrounds – to be performed when gravitational antennas reach the required sensitivity – could therefore give us experimental information able in principle to discriminate between the various string models.

Let us now focus on another and quite exclusive feature of string cosmology, that has no counterpart in either the standard or the inﬂationary cosmology: dilaton production (studied by the present author soon after the formulation of the pre-Big-Bang scenario). In addition to gravitational waves, the transition from pre- Big-Bang to post-Big-Bang also ampliﬁes dilatonic waves, i.e., the quantum oscillations of the dilaton ﬁeld in vacuum. The outcome of such an ampliﬁcation is the production of a cosmic background of neutral scalar particles (without charges, and with zero intrinsic angular momentum). Those particles, dubbed dilatons, should be characterized by a primordial spectral distribution very similar to the graviton distribution.

Actually, when computing the spectrum, one should take into account the fact that the dilaton ﬂuctuations – unlike the gravitational ﬂuctuations – are not freely oscillating, as they are coupled to both the scalar ﬂuctuations of the geometry and the matter ﬂuctuations. The corresponding equations are quite complicated and, up to now, have been solved only in some special cases. The out- come is a primordial spectrum, valid in the high-frequency regime, which increases exactly like the graviton spectrum, with a slope that is a model-dependent parameter. Such a parameter does not affect in any crucial way the total energy density of the dilatonic background, obtained by summing over all frequencies.

There is, however, an important difference with respect to the graviton case: the dilatons present in our Universe today could have a non-zero rest mass. If they have a mass, then the dilaton spectrum turns out to be modiﬁed at frequencies (i.e., energies) that are low with respect to the oscillation frequency associated with the rest mass of the dilaton. It is found, in particular, that this low-frequency part of the dilaton spectrum can be much more intense and ﬂatter than the graviton (massless) spectrum. But why should dilatons be massive? In order to answer this question let us recall that one of the milestones of Einstein’s gravitational theory is the equality be- tween inertial and gravitational mass. As a consequence of this equality all bodies should fall with the same acceleration (we may recall the mythical experiment carried out by Galileo Galilei at the leaning tower of Pisa). The fact that the motion of a test body within a gravitational ﬁeld does not depend upon its mass – according to the so-called weak equivalence principle – has been checked with extreme precision, for both the terrestrial and the solar gravitational ﬁeld. The various experiments have been carried out over distances ranging from the astronomical unit (corresponding to the radius of the Earth orbit, about one hundred million kilometers) down to the millimeter scale.

However, according to string theory, the gravitational interaction should always be accompanied by a second interaction carried by the dilaton ﬁeld. The dilaton interaction may produce a force whose intensity should be of the same order as the gravitational force, at least to a ﬁrst approximation. Hence, test bodies should feel both forces, and should be accelerated accordingly. However, while the response to the gravitational force is universal – since it depends on the ratio between the inertial and gravitational masses, which are the same for all bodies – the response to the dilatonic force depends upon the “dilatonic charge” of the body itself, which could be different for bodies characterized by different internal structures. For instance, aluminum and gold (or platinum) objects – like those used in high-precision tests of the equivalence principle carried out by Roll, Krotkov, and Dicke at Princeton in 1967, and by Braginskii and Panov at Moscow in 1971 – could fall with different accelerations due to the dilaton interaction.

On the other hand, an effect of this type has never been observed. Hence, string theory is consistent with present observations only if the effects of the dilatonic force somehow disappear at the macroscopic scales where the above-mentioned experiments have been carried out. In this respect, there are in principle two possibilities.

A ﬁrst possibility is that the dilatonic force is short-range. If its effective range were to be smaller than the millimeter scale, in particular, such a force would not be observable in any of the experiments carried out up to now. On the other hand, the range of a force is inversely proportional to the mass of the carrier particle (in this case, the dilaton). If the particle is massless, the range is inﬁnite: in order to have a ﬁnite range it is therefore necessary for the dilaton rest mass to differ from zero. In particular, to have a range smaller than the millimeter scale, the rest energy must be bigger than about 10−4 eV, i.e., the corresponding mass must be bigger than 10−37 grams (about one ten billionth of the electron mass).

A second possibility (suggested by the theoretical physicists Thibault Damour and Alexander Polyakov) is that the interaction between the dilaton and the macroscopic bodies is much weaker than the gravitational interaction. (This is not impossible, in principle, though more difﬁcult to explain within string theory than the presence of the dilaton mass.) In this case the mass could be arbitrarily small and the dilatonic force, although long range, would not have been observed on a macroscopic scale simply because it would be too weak with respect to the current experimental sensitivities. Nevertheless, if the dilaton is sufﬁciently light, there could be interesting dilatonic effects on a cosmological scale (as we shall discuss in Chap. 9).

In this chapter we will mainly concentrate on the ﬁrst scenario, where the dilaton coupling has a strength of gravitational intensity and, for the experimental consistency of string theory, the current value of the dilaton mass is sufﬁciently high to avoid detectable violations of the equivalence principle. Note that this last condition could be violated during the primordial evolution of the Universe (for instance, during the production of the cosmic dilaton background), since the dilaton could have acquired its mass only later at lower energy scales, through a symmetry-breaking process.

Anyway, a non-zero value of the present dilaton mass modiﬁes the present dilaton spectrum (rendering it different from the graviton spectrum), and yields two main consequences.

First of all, the dilaton speed tends to slow down after their production so that, if they are massive, all dilatons eventually be- come non-relativistic. In fact, even if the initial mass is quite small, as the Universe expands and cools down during the phase following the Big Bang, the kinetic energy of the resulting dilatons progressively decreases until it becomes smaller than their rest-mass energy. From that moment on the relic dilatons behave as a gas of almost static particles, with zero or negligible pressure, and their energy density (which becomes proportional to their mass) starts to increase with respect to the radiation energy density. It is then mandatory to impose constraints on the intensity of their spectrum (i.e., on the number and energy of the resulting dilatons), in order to prevent their energy from becoming so high that it would block the onset of the standard cosmological evolution, and the subsequent formation of the cosmological state that we now observe.

The second important consequence concerns the fact that, if dilatons are massive, they must decay, producing radiation (in particular photons, i.e., electromagnetic radiation). Such a radiation production would increase the entropy of the Universe and could affect nucleosynthesis (i.e., the production of nuclear matter), and even baryogenesis (i.e., the very early processes breaking the equilibrium between matter and antimatter, thus allowing the birth of the Universe in its present form, with a negligible amount of anti- matter). The entropy produced through dilaton decay is inversely proportional to the square of the dilaton mass, and since such an entropy must be tamed – to allow baryogenesis and nucleosynthesis to occur as predicted by the standard model – it follows that the dilaton mass must be sufﬁciently high.

We thus arrive at a quite complicated, though interesting, scenario. On the one hand the dilaton mass ought to be sufﬁciently small to avoid the energy density of the non-relativistic dilatons exceeding a critical value, which would dominate the Universe, while on the other hand the dilaton mass must be sufﬁciently high to avoid violations of the equivalence principle, and to produce a sufﬁciently small entropy upon decay. This scenario is completed by the fact that the intensity of the dilaton background is not arbitrary, but depends (like the graviton background) upon the value of the string mass Ms.

The outcome is the existence of two bands of allowed values or, to use the jargon, two possible windows for the dilaton mass.

These windows also depend upon the value of the dilaton charges, i.e., on the intensity of the coupling between dilatons and matter. In the case we are considering here – the case of couplings of gravita- tional intensity – the dilaton mass must be either greater than about 10 TeV (i.e., about 10^−20 g, ten thousand times the hydrogen mass) or smaller than about 10 keV (i.e., about 10^−29 g, one hundredth of the electron mass). In any case, there is the additional lower limit of 10^−37 g, always valid, ﬁxed by tests of the equivalence principle (see Fig. 7.1). The critical mass value of about one hundred MeV (i.e., 10^−25 g), ﬁxed by the process of dilaton decay, lies just between these two mass windows. Such a decay mass scale corresponds to a dilaton lifetime which is just of the same order of magnitude as the present Hubble time (which is the typical time-scale of our Universe in the present cosmological conﬁguration).

The case of very massive dilatons (i.e., the mass window on the right of Fig. 7.1) is the least interesting one, from an observational point of view. In that case, in fact, all the dilatons of the resulting cosmic background would already have decayed (their lifetime is inversely proportional to the cubic power of their mass), and there would not be any (directly) observable dilatonic trace of the pre-Big-Bang phase. This seems to be the preferred case in the context of supersymmetric extensions of the fundamental gauge interactions, where the dilaton mass is directly linked to the mass of its supersymmetric partners (believed to be quite heavy).

If, on the contary, the dilaton mass lies in the left-hand mass window of Fig. 7.1, the resulting dilatons would still be “alive” today, and would represent a peculiar relic of the epochs preceding the Big Bang, with no counterpart in other kinds of cosmological scenario. In that case the energy density of the dilaton background could be very near to, or could even saturate, the critical density limit (see the small shaded triangle Fig. 7.1), thus representing a consistent fraction of the so-called dark matter density. The existence of dark matter (i.e., of matter that cannot be seen by ordinary optical telescopes and which ﬁlls the Universe on a cosmological scale) seems to be necessary to explain some discrepancies between the theory and the present astrophysical observations (as we shall discuss in Chap. 9). The detection of this kind of matter has been pursued for many years in various ways, but up to now without any decisive result.

This discussion of the possible cosmological effects of the dilaton unavoidably leads us to the following important question: If dilatons exist, how can they be detected? The answer depends on the strength of their coupling and on the value of their mass.

If their coupling to macroscopic matter has gravitational strength, and their mass is not too big, one exploitable effect could be the violation of the equivalence principle. As previously stressed, string theory does indeed predict that the various kinds of elementary particles may have different dilatonic charges. In contrast to the electric charge – which is universal, apart from the sign – the dilatonic charge of the proton, for instance, seems to be 40–50 times bigger than the dilatonic charge of the electron, as shown by studies carried out by Tom Taylor and Gabriele Veneziano in the 1980s. The total dilatonic charge per unit mass of a macroscopic body would then depend upon its internal structure (i.e., on the number density of protons and electrons). The possible fractional variations of the dilatonic charge between different bodies, due to their different atomic structures, are certainly small, of the order of one per thousand. However, if the range of the dilatonic force is not too much smaller than the millimeter scale, some experiments may hopefully observe such effects in the not so distant future.

Another possibility, in the same range of masses, is the experimental study of the mutual conversion between photons and dilatons in the presence of an intense magnetic ﬁeld, in the laboratory. This possibility, however, does not seem to be within the reach of present technology, at least if the strength of the dilaton coupling is not much greater than the gravitational coupling.

We could also consider the use of the gravitational antennas illustrated in the last chapter. However, these can efﬁciently respond to dilatons only if the frequency associated with the dilaton mass is not higher than the frequencies of maximum sensitivity of the detector. Given the sensitivity range of current detectors (see Chap. 6), and observing that the frequency of one kilohertz corresponds to a mass of about 10^−12 eV, we can easily conclude that such a detection method can be efﬁciently applied only in the case of ultra-light dilatons, coupled weakly enough to matter to satisfy present phenomenological constraints.