Inflation and the Birth of the Universe (Part 2)

The Universe Before the Big Bang

Inflation and the Birth of the Universe

Finally, the reader may ask why the phase of pre-Big-Bang inflation has been represented in Fig. 5.3 by plotting two different types of behavior of the spatial radius, corresponding to the solid curves labeled by R (the inner one) and RE (the outer one), describing expanding and contracting phases, respectively. (Note that the specular symmetry characterizing the contraction with respect to the standard phase is not an essential feature, but only a special choice suggested for reasons of graphic convenience.) It is thus important to explain that those two different types of behavior do not correspond to different models of pre-Big-Bang evolution; rather, they are two different (but physically equivalent) kinematic representations of the same model in terms of two different space-time metrics. Both representations are useful to provide an effective il- lustration of different aspects of the same scenario.

The expanding geometry (perhaps providing us with the most intuitive representation) uses as space-time metric the same metric “felt”2 by a string present in the Universe during the pre-Big-Bang phase (the metric we also adopted at the end of the last chapter to discuss possible modifications of the Einstein equations introduced by string theory). This metric is also called the string metric, or string-frame metric. The contracting geometry (probably associated with the most conventional representation) uses instead the space-time metric felt by gravitons and dilatons (the fundamental particles of the theory), i.e., the same metric as would be used in the context of general relativity (it is in fact called the Einstein metric, or Einstein-frame metric).

It is always possible to switch from one representation to the other through a simple transformation which redefines the metric and other fields (performing, in particular, a local rescaling of such fields), without altering the physical phenomena, but simply describing them in terms of different variables. If we transform the geometry of a pre-Big-Bang model in this way we find, in particular, that the curvature keeps growing and the horizon radius keeps shrinking, but the expanding spatial radius R of the string metric becomes a contracting spatial radius RE in the Einstein metric, and vice versa. Thus, in the Einstein geometry the initial horizon is larger (and the duration of inflation is longer) than in the associated string geometry, as is clearly illustrated in Fig. 5.3. However, the two geometric descriptions are physically equivalent, and both provide a consistent description of the pre-Big-Bang scenario.

To complete the discussion of the main (practical and conceptual) differences between pre-Big-Bang inflation and conventional de Sitter inflation we should recall that a phase of conventional inflation cannot be extended arbitrarily far back in time (as al- ready stressed in the first chapter). In such a case, working within the framework of general relativity (i.e., of classical gravitational physics), it is in practice impossible to answer questions about how inflation began, or what happened before inflation. Trying to answer such questions would necessarily require the methods of quantum cosmology (see Chap. 8); but even in that case the choice of the state of the Universe preceding the inflationary epoch is completely arbitrary, yielding the so-called boundary condition problem. Various proposals have been made for the initial state, with contradictory outcomes.

Within string cosmology, on the other hand, the possibility that inflation can effectively last for an infinite amount of time is not forbidden as in the case of de Sitter-type inflation. But in the case of a finite duration, the above-mentioned questions concerning the origin of inflation are well-posed, and can be answered entirely within the framework of string theory. The initial configuration of the Universe is then identified without ambiguity with a configuration approaching the so called string perturbative vacuum, free of interactions,3 already introduced in Chap. 3.

Detailed studies of the evolution of an initial, non-homogeneous perturbative state, carried out by Alessandra Buonanno, Thibault Damour, and Gabriele Veneziano, and followed by other studies by Alexander Feinstein, Kerstin Kunze, Miguel Vasquez-Mozo, and Valerio Bozza, have shed light on the possible mechanism that could ignite the inflationary phase. Indeed, as already discussed in Chap. 3, the initial state of pre- Big-Bang cosmology should be seen as a quite extended portion of space-time without any matter or forces, and hence extremely flat, empty, and cold. The further back in time we go, the weaker the interactions become, and the more the space-time geometry looks similar to the rigid geometry of special relativity. This does not mean, however, that the whole Universe is rigidly crystallized in a static configuration. Generally, small (classical and quantum) in- homogeneities may always be present, producing space-time fluctuations in both the metric and the dilaton field (as well as in all possible background fields in principle allowed by the theory).

To make an analogy we may think of the surface of a very quiet ocean, where nothing seems to happen. Only a few tiny waves prop- agate over the surface, occasionally colliding with other waves. If some of these collisions are strong enough, or if some wave be- comes big enough to break up, some “foam” can be produced here and there, in a chaotic and random fashion. Similarly, in the primordial Universe, random fluctuations of the geometry (and of other background fields) could focus in a small spatial region a high enough energy density to trigger a local gravitational collapse, with a corresponding local “implosion” of both space-time and all forms of energy. A process of collapse similar to the one that, even today, could convert some dead stars into black holes,, i.e., “bottomless pits” of gravitational attraction where everything is swallowed up forever.

According to this representation of the initial cosmological state, our Universe could emerge from precisely this type of collapse, and thus correspond to the portion of the whole space contained within one of those black holes. Working with a simple but quantitative model one can then estimate that, in order to produce a Universe similar to the present one from the collapse, the initial size of the black hole must be at least of the order of the radius of an atomic nucleus (i.e., about 10−13 cm).

If we adopt the standard Einstein metric of general relativity, describing the process of collapse as a geometrical contraction, we find that the above initial size of the collapsing region progressively shrinks, so that the resulting Universe becomes more and more compact. If instead we adopt the string metric, describing an expanding cosmological geometry, we find that the initial size, instead of shrinking to a point, grows in an accelerated inflationary fashion. The shift from contraction to expansion is due to the dilaton, which is the field responsible for rescaling the Einstein metric (i.e., for transforming from the Einstein to the string geometry), and which grows during the phase of pre-Big-Bang inflation, as already stressed in previous chapters. In both representations the curvature keeps growing, so that the curvature radius eventually reaches the minimum allowed value Ls marking the end of the phase of accelerated evolution. At that final stage, the initial size of the collapsing portion of space, described in terms of the string geometry, has increased from the initial 10−13 cm to something of the order of one tenth of a millimeter, corresponding exactly to the initial size required at the string curvature scale to reproduce our currently observed Universe after the period of standard evolution.

To help the reader to visualize this scenario, Fig. 5.4 shows a qualitative sketch of the model just described, where (excluding one of the three spatial dimensions) we provide three subsequent space-time diagrams of the collapse/inflation process. In that figure we can see how, at successive times (the various planes from bot- tom to top), the wavy sea representing the initial state has produced various types of gravitational collapses at various points, within spatial regions of different sizes. We can also see that, subsequently, one of those regions has inflated until it has reached a spatial size of 0.1 millimeters in correspondence of the string curvature scale, where quantum string effects are expected to stop inflationary expansion, converting it into the dual process (not represented in the figure), characterized by standard decelerated evolution.

FIGURE 5.4 Inflation as collapse in pre-Big-Bang cosmology (the time co- ordinate increases up the axis). The various planes, from bottom to top, correspond to various sections of the space-time manifold filled with gravitational anddilaton fluctuations distorting the flat geometry of the perturbative vacuum. The space-time regions undergoing gravitational collapse are simultaneously represented using the string geometry (the outer expanding cones) and the Einstein geometry (the inner contracting cones)

According to this scenario, our Universe is included in one of those primordial “bubbles”, and separated from other, possibly different Universes born from gravitational collapses characterized by different sets of initial parameters. This model for the birth of the Universe is reminiscent of some similar scenarios suggested from time to time by a number of theoretical physicists such as Valery Frolov, Moisei Markov, and Viatcheslav Mukhanov in the Soviet Union, and Robert Brandenberger, John Wheeler, and Lee Smolin in the United States. However, it differs substantially from the previous scenarios through the key role played by string theory and its dual symmetries.

It is interesting to note that, in this context, some of the properties of the current Universe can be directly traced back to the properties of the initial state giving rise to the subsequent inflationary evolution. In other words, encoded into the current observational data we can find the imprint of the Universe before the Planck era (i.e., of the cosmological state preceding the Big Bang and the quantum gravity epochs) – just as the final particles produced in a decay process contain (encoded in their quantum numbers) the imprint of the state of the system before the decay.

This possibility can be illustrated by focusing our attention on a key property of the current Universe: the entropy stored in the cosmic microwave background (CMB), which is measured – in appropriate units called natural units, where the speed of light c and the Planck constant ¯h are both set to unity – by an extremely large dimensionless number, of the order of 1090. In fact, the cosmic back- ground of electromagnetic radiation is in thermal equilibrium (the temperature is the same everywhere, apart from tiny fluctuations that we can safely neglect for the purpose of this argument), and its energy thus follows the Planck statistical distribution (also known as the black-body spectrum). By applying statistical arguments one then finds that the entropy associated with this distribution is automatically nonvanishing. More precisely, the associated entropy is directly proportional to the spatial volume occupied by the radiation particles (in this case photons), and to the third power of the radiation temperature.

During the phase of standard (post-Big-Bang, post-inflationary) evolution, the volume increases as the third power of the spatial radius R (as the Universe is expanding), while the temperature decreases as the reciprocal of R (being redshifted like the frequency and the energy of the radiation). It turns out that the entropy is exactly conserved (i.e., the evolution is said to be adiabatic). Hence the standard cosmological model cannot explain the origin of the entropy currently associated with the CMB radiation. The value that we observe today was exactly the same value present at the beginning of the standard evolution.

An explanation of this entropy could possibly be provided by the inflationary dynamics and, indeed, the inflationary models were originally formulated to solve this entropy problem, along with the other problems already mentioned. To this end, the final stage of inflation is characterized by the occurrence of non- adiabatic processes that “heat up” the Universe, massively generating thermal radiation, and thus entropy, so as to agree with current observations. It should be noted, however, that the total value of the CMB entropy generated in this way, which could at a first glance appear to be a huge number when compared with macroscopic standards, is an extremely small quantity if compared with the entropy that could be associated with a Universe as large as the current one.

To understand this argument, pointed out by the famous mathematical physicist Roger Penrose in one of his books, let us extrapolate the evolution of our Universe back in time according to the standard cosmological model, until we reach an epoch (that we shall call the Planck epoch) during which the radius of the Hubble horizon c/H, and hence the space-time curvature radius, were as small as the Planck length LP (i.e., of the order of 10^−33 cm). We already know, on the other hand, that going backward in time, the spatial radius R of the Universe decreases more slowly than the horizon radius (see Fig. 5.1). Using general relativity, we can estimate that, at the Planck epoch, the radius R was bigger than the horizon radius c/H = LP by a factor of about 10^30. Therefore, at that time the spatial volume of our currently observed Universe was filled with 10^30 to the third power (i.e., 10^90) small, causally connected spheres of Planckian radius.

The currently observed entropy – measured by a number whose order of magnitude is just 10^90 – can thus be reproduced by assigning one degree of freedom (i.e., one “bit” of information, to use the jargon of computer science) to any portion of the horizon area of Planckian size. This prescription is equivalent to providing every “small Hubble sphere” of radius c/H with the maximum entropy allowed by the so-called holographic principle, first conjectured by the theoretical physicist Gerard t’Hooft (Nobel Prize winner in 1999), and subsequently applied to cosmology by many others. It is also equivalent, as pointed out by Gabriele Veneziano, to assigning to each spatial volume enclosed within a Hubble horizon the same entropy as would be carried by a black hole of equal spatial extension. However, the above arguments yield an entropy of the order of 10^90 only if they are applied to the CMB radiation during the Planck epoch. If we apply them during subsequent epochs, when the horizon radius was larger than Planckian, the corresponding value of the CMB entropy increases until it reaches today the extremely high value of 10^122, which is the value associated with the area of the present Hubble horizon (whose radius is equivalent to about 10^61 Planck radii). Thus, we are led to the following question: Why is the currently observed CMB entropy – which seems to be a “maximal” entropy (according to the holographic principle, or to the black hole entropy) if its value is computed at the Planck epoch – so small when compared with the entropy that could be associated with the current horizon?

Now, it is well known from the basic principles of statistical mechanics that the entropy somehow measures the amount of dis- order associated with a system. If the present entropy within our Universe is small, it means that our Universe behaves as a highly ordered system, i.e., a system that has not lost track of its origins, and which may still encode a lot of readable information about its past history. String cosmology, and pre-Big-Bang models in particular, seem to be able to provide a key to interpret the data about the CMB entropy in terms of the cosmological evolution preceding the Planck epoch.

Indeed, if the initial pre-Big-Bang phase is described as a con- traction (in terms of the Einstein metric, where gravitons move along geodesics), the unavoidable outcome is a collapse and subsequent formation of a black hole. The horizon of such a black hole depends on the portion of the space that has collapsed, and the horizon area determines the maximum entropy associated with that portion of space (according to the recipe of Bekenstein and Hawking), which remains constant until the cosmological evolution is adiabatic.
The initial black hole horizon, on the other hand, coincides with the Hubble horizon which appears in the context of the string metric, where the pre-Big-Bang phase is described as an inflationary expansion. During the pre-Big-Bang phase, the radius of the Hubble horizon shrinks linearly (as shown in Fig. 5.3). Hence, if at the beginning of the process the whole entropy of the system is encoded into the surface of a single, large Hubble sphere (also corresponding to the black hole horizon), when the Planck scale is reached the total entropy – which always has the same value, since the process is adiabatic – is distributed among a large number of small Hubble spheres of Planckian radius, causally disconnected from one another. The pre-Big-Bang scenario thus seems to be able to explain why one arrives at the Planck scale with an entropy which has the maximum value predicted by the holographic principle, applied to the total number of Hubble spheres contained in our Universe at that epoch. However, this entropy is of geometric type, i.e., it is associated with the horizon area and hence with geometric properties of the space-time under consideration. But why should the cosmic radiation, which is produced later and becomes dominant in the subsequent standard phase, be characterized by the same amount of entropy? String cosmology also seems to provide an answer to this question.

In the context of pre-Big-Bang models, the radiation that dominates the standard cosmological evolution is produced by amplification of the quantum fluctuations of the vacuum, according to a mechanism that will be described in detail in the next chapter.

The amplification process already begins during the pre-Big-Bang phase, when the wavelength of the quantum oscillations becomes larger than the Hubble radius. As the horizon radius shrinks to- wards smaller and smaller values, oscillations with smaller and smaller wavelengths (hence higher and higher frequencies) get amplified, so that the energy and entropy of the quantum radiation included in a given portion of space grow progressively larger and larger. At a given time, the amount of entropy of this quantum radiation thus depends on the reciprocal of the horizon radius.

Detailed computation (performed mainly by two groups, one including Robert Brandenberger, Viatcheslav Mukhanov, and Tomasz Prokopec, and the other Massimo Giovannini and the present author) have shown that, within a given spatial volume, the entropy of this radiation is proportional to the number of Hubble spheres (of radius c/H) contained in that volume. On the other hand, the geometric entropy is proportional to the number of such spheres times their area in Planck units. The ratio between geometric entropy and radiation entropy, at any given time, is therefore approximately determined by the area of the Hubble horizon in Planck units.

This ratio, initially quite high, tends to decrease during the pre-Big-Bang phase, approaching unity when the space-time curvature gets close to the Planck scale. On the other hand – as we shall see again in Chap. 8 – we ought to expect the phase of standard cosmological evolution to begin when the contribution of the quantum corrections to the gravitational equations becomes significant, in particular when we must take into account the “back-reaction” of the radiation produced by amplifying the quantum fluctuations of the vacuum.

The relative weight of the quantum corrections is determined by the square of the space-time curvature in Planck units, i.e., by the ratio H^2L^2P/c^2, which also expresses the reciprocal of the horizon area in Planck units, hence the ratio between radiation entropy and geometric entropy. Since the Universe tends to exit the accelerated pre-Big-Bang phase and become radiation-dominated just when the above ratio is of order one, it follows that the transition to the standard regime occurs precisely when the entropy stored in the produced radiation is equal to the geometric entropy, i.e., equal to the maximum entropy allowed by the holographic principle applied to the Hubble spheres.

If we accept the idea that the radiation corresponding to the currently observed cosmic background finds its primordial origin in the process of amplification of the quantum fluctuations of the vacuum, we can then explain why its entropy exactly saturates the maximum allowed value when it is evaluated at the Planck epoch.

In such a context, the current value of the radiation entropy can be interpreted as the imprint left by the cosmological evolution during the epochs preceding the Big Bang (and the Planck era), and in particular by the size of the horizon of the initial geometric configuration from which our Universe has evolved.

To conclude this chapter we can say that the phase of accelerated evolution and increasing curvature, which could represent the primordial stage of our Universe according to pre-Big-Bang models, is a phase of inflationary type, able to overcome the shortcomings of the standard cosmological model, despite a type of kinematics which is profoundly different from the one characterizing other inflationary models of more conventional type. If we are convinced by the arguments presented in this chapter (and by others that will not be reproduced here, for the sake of simplicity) that such a phase could fully describe, in a complete and logically consistent fashion, the state of the Universe before the Big Bang – and could therefore constitute a physically acceptable model of primordial cosmological evolution – we are led to a question that may be regarded as crucial from a physicist’s perspective: Are there phenomenological consequences, i.e., effects that are – at least in principle – observable, that could discriminate between pre-Big-Bang cosmology (or, more generally, string cosmology) and the more conventional inflationary cosmology? And are such differences observable today, in practice, given the current status of our technology?
The answer to these questions will be discussed in the following chapters.

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Inflation and the Birth of the Universe (Part 1)