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

When the latter conditions are satisﬁed, there are in principle interesting prospects for the future detection of a cosmic back- ground of massive dilatons. In fact, provided their spectrum is ﬂat enough, they could induce an enhanced response to the non- relativistic part of the spectrum in both resonant mass and interferometric detectors, as shown by the studies of Carlo Ungarelli and the present author (with the collaboration of Eugenio Coccia for the spherical antennas). More precisely, the signal induced in two cross-correlated detectors could grow with the observation time much more rapidly than the signal produced by a massless background (like the graviton background). This effect could make a background of massive, non-relativistic dilatons detectable – in spite of the weakness of their coupling to matter – provided that the total energy density of the background is close enough to the critical value, and the dilaton mass lies within the sensitivity range of the antennas.

Given the various uncertainties, it seems wise to say that, at least for the time being, a direct experimental search for cosmic dilatons looks less promising than the corresponding search for gravitons, even though the history of physics has taught us that unexpected surprises are always possible.

The third effect we would like to discuss in this chapter is associated with the production of a cosmic background of neutral pseudo-scalar particles with interactions of gravitational strength, dubbed axions. In contrast to the case of graviton and dilaton production, such an effect could lead to already observed phenomena (as happens for the magnetic ﬁelds). The existence of axions is not peculiar to string theory. However, the axions produced during the pre-Big-Bang phase are characterized by a spectrum which – un- like that of gravitons, dilatons and photons – could be “ﬂat” (or very weakly dependent on the frequency). As a consequence, ax- ions could represent an indirect source of the observed anisotropy in the CMB radiation.

The cosmic microwave background (CMB) of electromagnetic radiation, frequently mentioned in this book, is a relic of high- temperature cosmological epochs, and its energy distribution fol- lows a thermal (or black body) spectrum corresponding to a present temperature of almost three degrees kelvin. This background consists of a sea of photons (or electromagnetic waves) originating from the epoch when matter and radiation started to decouple, at a temperature of about three thousand degrees kelvin. As the temperature dropped below that value, the mean-free-path of pho- tons became greater than the Hubble radius, so that the Universe became transparent to the electromagnetic radiation, and it has sur- vived up to now undisturbed, providing us with a faithful imprint of that early epoch.

Those photons ﬁll the whole currently observable space in an almost uniform fashion, apart from tiny variations of the local temperature whose fractional average value is of the order of one part in one hundred thousand. These temperature ﬂuctuations make the background anisotropic, and are characterized by an angular distribution which is approximately ﬂat (i.e., constant) at large an- gular scales (corresponding to distances of the order of the current Hubble radius), while it is oscillating at smaller angular scales.1 In the conventional inﬂationary scenario, the observed anisotropy of the CMB temperature is directly caused by the gravitational ﬂuctuations of the geometry – in particular by their scalar part – ampliﬁed during the phase of accelerated expansion. The oscillations of the gravitational ﬁeld can in fact induce a small breaking of the spatial homogeneity and isotropy at the time of decoupling between matter and radiation, thus affecting the ﬁnal temperature distribution. Such breaking would then propagate until today thanks to the so-called Sachs–Wolfe effect (from the names of the two astrophysicists who discovered it in the 1960s).

The above mechanism can successfully explain the observed anisotropies provided that the primordial gravitational ﬂuctuations are ampliﬁed with a spectrum which is sufﬁciently close to being ﬂat (also called the Harrison–Zeldovich spectrum), a requirement which is easily satisﬁed within the conventional inﬂationary scenario, as discussed in the last chapter. A phase of pre-Big-Bang inﬂation, on the other hand, ampliﬁes the gravitational ﬂuctuations (and also their scalar part) with a spectrum which tends to grow very rapidly with the frequency, and which cannot represent an efﬁcient source of the observed CMB anisotropy. This is an issue for pre-Big-Bang models, since the anisotropy exists and must be explained.

A possible solution to this problem relies upon the fact that the anisotropy of the CMB radiation (i.e., the local ﬂuctuations in its temperature) could be due not to the primordial geometric oscillations directly ampliﬁed by inﬂation, but rather to quantum oscillations of some other ﬁeld.

In fact, if the ﬂuctuations of such ﬁelds are ampliﬁed with a ﬂat enough spectrum, acquire a mass, and subsequently decay (early enough), they can in turn generate a “secondary” background of geometric oscillations, characterized by the same spectral slope as the primordial background, and satisfying all the required properties to act as a source for the observed anisotropies. The subsequent process of anisotropy production is just the same as that of conventional inﬂation, the only difference being the secondary – rather than primordial – character of the geometric ﬂuctuations.

This mechanism, dubbed the curvaton mechanism (as it generates perturbations in the geometry, and then in the curvature properties of the space-time), has been proposed by independent and almost contemporaneous studies carried out by Kari Enqvist and Martin Sloth in Finland, David Lyth and David Wands in England, and Takeo Moroi and Tomo Takahashi in Japan.

But what ﬁeld could play the role of the curvaton in a string cosmology context? Certainly neither the electromagnetic nor the dilaton ﬁeld, since they are produced with a spectrum that increases too rapidly with frequency (like the spectrum of gravitational ﬂuctuations). However, string theory contains other fundamental ﬁelds and, in particular, there is a ﬁeld represented by an antisymmetric tensor whose ﬂuctuations, the axions, could be ampliﬁed with the right (i.e., sufﬁciently ﬂat) spectrum to source the observed anisotropy. The possibility of a ﬂat axion spectrum was pointed out by the pioneering work of a group of theoretical physicists and astrophysicists including Edmund Copeland, Richard Easther, James Lidsey, and David Wands.

How can the axion spectrum be so different (ﬂatter, in particu- lar) from the other spectra so far analyzed? The answer relies upon the fact that the coupling of the dilaton to the axion ﬁeld is exactly the reciprocal of the dilaton coupling to the gravitational ﬂuctuations. Hence, whereas the accelerated expansion of the geometry tends to amplify the axion ﬂuctuations, the accelerated growth of the dilaton tends to tame them, and the net result is a spectrum which increases much more slowly than the others. In particular, the ﬁnal axion spectrum depends upon the number of expanding (or contracting) spatial dimensions, and if such expansion/contraction is isotropic then one ﬁnds that the spectrum is ﬂat just for a cer- tain “magic” number of dimensions equal to the so-called critical number of superstring theory (see Chap. 10).

At this stage, it is probably appropriate to point out that the number of spatial dimensions (three) that we normally consider (in both our everyday life and all physical experiments carried out so far) could not coincide with the number of spatial dimensions characterizing the Universe during the earliest remote epochs (in particular, during the phase preceding the Big Bang). This issue will be discussed in more detail in Chap. 10. For this chapter it will sufﬁce to anticipate that the models trying to include all natural forces in a uniﬁed scheme (including gravity) agree on the fact that this uniﬁcation, quite hard to achieve within a four-dimensional space-time, is actually easier to implement in spaces with a higher number of dimensions (superstring theories, for instance, require nine spatial dimensions, plus one temporal dimension).

So if the uniﬁed higher-dimensional theories are correct, how can we explain the fact that the Universe in which we live seems to have no more than three spatial dimensions? The standard answer is that an exact symmetry between all the natural forces, implemented in a multi-dimensional environment and at the high-energy scales typical of the primordial expanding Universe, cannot survive forever. At some time (i.e., below some energy scale) this symmetry breaks down, and three dimensions keep expanding while all the others (also called internal dimensions) “wrap around” them- selves, becoming compact and so tiny that they are practically invisible in all lab experiments carried out so far. (This process is called spontaneous – or dynamical – dimensional reduction.) As a consequence, we live today in a Universe which effectively seems to have only three spatial dimensions, and where there are many types of force apparently quite different one from another (but see Chap. 10 for other possible explanations of why our space looks three-dimensional).

In the initial Universe, when the forces were united into a single root, all spatial dimensions were on the same level, and equally accessible. This legitimates the study of the primordial vacuum ﬂuctuations in space-times with more than three spatial dimensions. One then ﬁnds that some spectra, like the graviton spectrum, are insensitive to the number of dimensions, while others, like the axion spectrum, depend strongly on the dimensionality of space, and may become ﬂat in an appropriate number of isotropic dimensions.

Going back to the anisotropy of the CMB radiation we note that, if this anisotropy is indirectly generated by the decay of the cosmic axions produced by a phase of pre-Big-Bang evolution, then any anisotropy measurement carries direct information regarding the parameters and kinematics of that primordial phase.

We refer, in particular, to the already mentioned measurements of the COBE satellite, which ﬁx the amplitude of the temperature ﬂuctuations at large angular scales, and to the more recent measurements of the WMAP satellite, which determine the height of the ﬁrst oscillation peaks, and the spectral slope of the geometric ﬂuctuations generating the CMB anisotropy. These measurements may give us direct information about the amplitude and slope of the primordial axion background, which in turn depend, respectively, on the values of the string mass Ms and the number of dimensions (and their kinematics) of the accelerated epoch. We can then obtain important experimental constraints on models of pre-Big-Bang inﬂation, as discussed by Valerio Bozza, Massimo Gio- vannini, Gabriele Veneziano, and the present author.

For this purpose, it is important to stress that an axionic origin of the CMB anisotropy could be experimentally conﬁrmed by its non-Gaussian statistical properties, arising if the primordial axion background decays early enough, before becoming the dominant cosmological source (as pointed out by David Lyth, Carlo Ungarelli, and David Wands). If, on the contrary, the mass is too small to forbid the decay, and the energy of the axion background always stay sub-dominant with respect to other grav- itational sources, the axions can still contribute to the CMB anisotropy as seeds, i.e., as quadratic sources of the geometric ﬂuctuations, as discussed by Ruth Durrer, Alessandro Melchiorri, and Filippo Vernizzi. In that case, however, the angular distribution of the temperature anisotropy turns out to be signiﬁcantly different from that obtained from the standard inﬂationary scenario and from the curvaton mechanism. As a consequence (and according to present observations), any seed-like contribution is possibly allowed only as a subdominant component of the total observed anisotropy.

To conclude this chapter, it is interesting to observe that the phenomenological consequences of pre-Big-Bang models can be classiﬁed into three main classes, according to their observational chances. To paraphrase using well known science-ﬁction jargon (concerning encounters with extraterrestrials), we may divide the phenomenological consequences into ﬁrst kind, second kind, and third kind. Effects of the ﬁrst kind would be discovered by observations to be carried out in the next twenty to thirty years or so; effects of the second kind are relative to observations to be made in the near future (within a few years); ﬁnally, effects of the third kind are relative to observations already realized or currently being carried out.

An effect of the ﬁrst kind, discussed in the last chapter, is the production of an intense background of relic gravitational radiation. Currently, gravitational antenna do not have enough sensitivity to detect such a background. However, the required sensitivity is expected to be reached in the not so distant future by detectors that are either being built or being planned.

As an effect of the second kind we may mention the total absence of contributions arising from the relic gravitational waves to the large scale anisotropy of the CMB radiation (because of the very steep slope of the graviton spectrum); but also, the presence of a small non-Gaussian feature in the spectral distribution of the anisotropy (because of its indirect axionic origin). There are satellites like PLANCK (see Fig. 7.2) which will be launched in the very near future, and will soon be able to perform high precision measurements of the ﬁne-structure properties of the CMB anisotropy, on various angular scales. In this way we may expect to obtain precise information about the two effects mentioned above, thus conﬁrming or disproving the predictions of different inﬂationary scenarios.

An effect of the third kind could be represented by the pro- duction of magnetic seeds. The presence of cosmological magnetic ﬁelds requiring those seeds could be interpreted as an indirect conﬁrmation of the string models able to produce them, in view of the difﬁculties in generating the magnetic seeds in the context of other inﬂationary models. Another example of an effect of the third kind could be represented by the CMB anisotropy measured by COBE and WMAP, under the hypothesis that its axionic (pre-Big-Bang) origin may be conﬁrmed by future observations.

Finally, let us note that the phenomenology associated with the dilatons does not seem to ﬁt any of the three above-mentioned kinds of effects, given the experimental difﬁculty involved in direct detection of such particles. However, there is one possible exception. The physical effects of the dilaton ﬁeld may already have been observed (i.e., they could be of the third kind), if this ﬁeld turned out to be the elusive quintessence dominating the present Universe and producing the observed large-scale acceleration (this possibility will be illustrated in Chap. 9). Up to now, recent and current studies do not seem to invalidate this interesting scenario.